[math-fun] A puzzle inspired by cartalk
On the program cartalk last week, there was a puzzle of the following form: A list of numbers was given, and one was asked what they had in common (I give the actual puzzler at the end). The answer was that each of these numbers was divisible by the number of letters (excluding spaces) in the standard spelling out of the number in words. This got me to thinking of the following modification: Suppose that we map each positive integer into an integer by the following means: write the number in decimal (no leading zeros). The value of the function will be to total number of letters that one gets by mapping each digit to its English equivalent (e.g. 13 -> length("OneThree") = 8. Or more generally give a fixed map g : {0,1,...,9} -> positive integers, and define f(n) = sum_{d digits in n} g(d). So are there an infinite number of positive integers n such that n is divisible by g(n)? One can generalize this to other bases. If there are an infinite number of them how dense is the set? Victor Here's the original puzzler: When my kids were in school, they, like all the other kids I guess, had to learn their numbers. So each day for homework, they would bring home a list of numbers on a piece of paper, and they were asked to write out the letters that spelled that number, right next to each of them. So the number seven would be there, there'd be a blank space, the kids would have to write S - E - V - E - N. And of course they were also asked which numbers were spelled out by the various combinations of letters, so they'd see S - I - X - T - Y and write Sixty, etc. One day, son number two presented me with a list of numbers and he said, "These numbers are different. There's something special about them." Here are the numbers: Four, Six, Twelve, Thirty, Thirty Three, Thirty Six, Forty, Forty Five, Fifty, Fifty Four, Fifty Six, Sixty, Seventy, Eighty One, Eighty Eight, Ninety, and a Hundred. Now there are no other numbers between one and a hundred inclusive that share this same characteristic. There's something unusual about these numbers that son number two figured out. And I'll give you an additional hint that order does not matter. The best hint is that he determined that these numbers should be on the list perhaps from his homework assignment. What is special about this list of numbers?
FYI, my column "Four, Twenty Four,...", in Mathematical Intelligencer 24 #2 (Sprint 2002), is on a similar topic: instead I wrote number names in English and then *multiplied* the word lengths together, and checked whether they were equal to the original number. So after "four" comes "twenty four", with word lengths 6 and 4, and 6*4=24. I came up with a heuristic argument that there should only be finitely many numbers for which it worked, though. http://oeis.org/A058230 --Michael On Sun, Sep 25, 2011 at 1:33 PM, Victor Miller <victorsmiller@gmail.com>wrote:
On the program cartalk last week, there was a puzzle of the following form:
A list of numbers was given, and one was asked what they had in common (I give the actual puzzler at the end). The answer was that each of these numbers was divisible by the number of letters (excluding spaces) in the standard spelling out of the number in words. This got me to thinking of the following modification:
Suppose that we map each positive integer into an integer by the following means: write the number in decimal (no leading zeros). The value of the function will be to total number of letters that one gets by mapping each digit to its English equivalent (e.g. 13 -> length("OneThree") = 8. Or more generally give a fixed map g : {0,1,...,9} -> positive integers, and define f(n) = sum_{d digits in n} g(d). So are there an infinite number of positive integers n such that n is divisible by g(n)? One can generalize this to other bases. If there are an infinite number of them how dense is the set?
Victor
Here's the original puzzler:
When my kids were in school, they, like all the other kids I guess, had to learn their numbers. So each day for homework, they would bring home a list of numbers on a piece of paper, and they were asked to write out the letters that spelled that number, right next to each of them. So the number seven would be there, there'd be a blank space, the kids would have to write S - E - V - E - N. And of course they were also asked which numbers were spelled out by the various combinations of letters, so they'd see S - I - X - T - Y and write Sixty, etc.
One day, son number two presented me with a list of numbers and he said, "These numbers are different. There's something special about them." Here are the numbers:
Four, Six, Twelve, Thirty, Thirty Three, Thirty Six, Forty, Forty Five, Fifty, Fifty Four, Fifty Six, Sixty, Seventy, Eighty One, Eighty Eight, Ninety, and a Hundred.
Now there are no other numbers between one and a hundred inclusive that share this same characteristic. There's something unusual about these numbers that son number two figured out. And I'll give you an additional hint that order does not matter. The best hint is that he determined that these numbers should be on the list perhaps from his homework assignment.
What is special about this list of numbers?
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-- Forewarned is worth an octopus in the bush.
I see that two versions of that sequence are already in OEIS: A092320 A126259. I avoided the standard spelling, because it's not clear how one should do it for arbitrarily large integers. Here's the list of terms <= 1000 (it's not in the OEIS): 4, 6, 12, 14, 32, 40, 42, 56, 66, 72, 91, 108, 110, 120, 126, 132, 154, 160, 162, 176, 180, 209, 210, 216, 220, 231, 260, 261, 273, 286, 308, 324, 360, 390, 396, 403, 429, 432, 444, 451, 468, 504, 506, 507, 528, 540, 576, 585, 588, 598, 605, 610, 612, 620, 621, 627, 637, 648, 649, 660, 666, 671, 682, 684, 720, 726, 728, 754, 756, 767, 770, 784, 792, 798, 832, 845, 852, 864, 871, 900, 902, 924, 936, 938, 946, 972, 975 Victor On Sun, Sep 25, 2011 at 3:18 PM, Michael Kleber <michael.kleber@gmail.com> wrote:
FYI, my column "Four, Twenty Four,...", in Mathematical Intelligencer 24 #2 (Sprint 2002), is on a similar topic: instead I wrote number names in English and then *multiplied* the word lengths together, and checked whether they were equal to the original number. So after "four" comes "twenty four", with word lengths 6 and 4, and 6*4=24. I came up with a heuristic argument that there should only be finitely many numbers for which it worked, though. http://oeis.org/A058230
--Michael
On Sun, Sep 25, 2011 at 1:33 PM, Victor Miller <victorsmiller@gmail.com>wrote:
On the program cartalk last week, there was a puzzle of the following form:
A list of numbers was given, and one was asked what they had in common (I give the actual puzzler at the end). The answer was that each of these numbers was divisible by the number of letters (excluding spaces) in the standard spelling out of the number in words. This got me to thinking of the following modification:
Suppose that we map each positive integer into an integer by the following means: write the number in decimal (no leading zeros). The value of the function will be to total number of letters that one gets by mapping each digit to its English equivalent (e.g. 13 -> length("OneThree") = 8. Or more generally give a fixed map g : {0,1,...,9} -> positive integers, and define f(n) = sum_{d digits in n} g(d). So are there an infinite number of positive integers n such that n is divisible by g(n)? One can generalize this to other bases. If there are an infinite number of them how dense is the set?
Victor
Here's the original puzzler:
When my kids were in school, they, like all the other kids I guess, had to learn their numbers. So each day for homework, they would bring home a list of numbers on a piece of paper, and they were asked to write out the letters that spelled that number, right next to each of them. So the number seven would be there, there'd be a blank space, the kids would have to write S - E - V - E - N. And of course they were also asked which numbers were spelled out by the various combinations of letters, so they'd see S - I - X - T - Y and write Sixty, etc.
One day, son number two presented me with a list of numbers and he said, "These numbers are different. There's something special about them." Here are the numbers:
Four, Six, Twelve, Thirty, Thirty Three, Thirty Six, Forty, Forty Five, Fifty, Fifty Four, Fifty Six, Sixty, Seventy, Eighty One, Eighty Eight, Ninety, and a Hundred.
Now there are no other numbers between one and a hundred inclusive that share this same characteristic. There's something unusual about these numbers that son number two figured out. And I'll give you an additional hint that order does not matter. The best hint is that he determined that these numbers should be on the list perhaps from his homework assignment.
What is special about this list of numbers?
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-- Forewarned is worth an octopus in the bush. _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
I see that I had a small typo in the original formulation: I said "So are there an infinite number of positive integers n such that n is divisible by g(n)?". I meant "So are there an infinite number of positive integers n such that n is divisible by f(n)?" On Sun, Sep 25, 2011 at 3:27 PM, Victor Miller <victorsmiller@gmail.com> wrote:
I see that two versions of that sequence are already in OEIS: A092320 A126259. I avoided the standard spelling, because it's not clear how one should do it for arbitrarily large integers. Here's the list of terms <= 1000 (it's not in the OEIS):
4, 6, 12, 14, 32, 40, 42, 56, 66, 72, 91, 108, 110, 120, 126, 132, 154, 160, 162, 176, 180, 209, 210, 216, 220, 231, 260, 261, 273, 286, 308, 324, 360, 390, 396, 403, 429, 432, 444, 451, 468, 504, 506, 507, 528, 540, 576, 585, 588, 598, 605, 610, 612, 620, 621, 627, 637, 648, 649, 660, 666, 671, 682, 684, 720, 726, 728, 754, 756, 767, 770, 784, 792, 798, 832, 845, 852, 864, 871, 900, 902, 924, 936, 938, 946, 972, 975
Victor
On Sun, Sep 25, 2011 at 3:18 PM, Michael Kleber <michael.kleber@gmail.com> wrote:
FYI, my column "Four, Twenty Four,...", in Mathematical Intelligencer 24 #2 (Sprint 2002), is on a similar topic: instead I wrote number names in English and then *multiplied* the word lengths together, and checked whether they were equal to the original number. So after "four" comes "twenty four", with word lengths 6 and 4, and 6*4=24. I came up with a heuristic argument that there should only be finitely many numbers for which it worked, though. http://oeis.org/A058230
--Michael
On Sun, Sep 25, 2011 at 1:33 PM, Victor Miller <victorsmiller@gmail.com>wrote:
On the program cartalk last week, there was a puzzle of the following form:
A list of numbers was given, and one was asked what they had in common (I give the actual puzzler at the end). The answer was that each of these numbers was divisible by the number of letters (excluding spaces) in the standard spelling out of the number in words. This got me to thinking of the following modification:
Suppose that we map each positive integer into an integer by the following means: write the number in decimal (no leading zeros). The value of the function will be to total number of letters that one gets by mapping each digit to its English equivalent (e.g. 13 -> length("OneThree") = 8. Or more generally give a fixed map g : {0,1,...,9} -> positive integers, and define f(n) = sum_{d digits in n} g(d). So are there an infinite number of positive integers n such that n is divisible by g(n)? One can generalize this to other bases. If there are an infinite number of them how dense is the set?
Victor
Here's the original puzzler:
When my kids were in school, they, like all the other kids I guess, had to learn their numbers. So each day for homework, they would bring home a list of numbers on a piece of paper, and they were asked to write out the letters that spelled that number, right next to each of them. So the number seven would be there, there'd be a blank space, the kids would have to write S - E - V - E - N. And of course they were also asked which numbers were spelled out by the various combinations of letters, so they'd see S - I - X - T - Y and write Sixty, etc.
One day, son number two presented me with a list of numbers and he said, "These numbers are different. There's something special about them." Here are the numbers:
Four, Six, Twelve, Thirty, Thirty Three, Thirty Six, Forty, Forty Five, Fifty, Fifty Four, Fifty Six, Sixty, Seventy, Eighty One, Eighty Eight, Ninety, and a Hundred.
Now there are no other numbers between one and a hundred inclusive that share this same characteristic. There's something unusual about these numbers that son number two figured out. And I'll give you an additional hint that order does not matter. The best hint is that he determined that these numbers should be on the list perhaps from his homework assignment.
What is special about this list of numbers?
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
-- Forewarned is worth an octopus in the bush. _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
Car Talk, huh? Who knew. Landon Curt Noll [1] is a fairly well respected computer scientist and mathematician (sort of) who worked with John Horton Conway on the system for names of arbitrarily high powers of 1000 which was published in (Conway and (Richard K) Guy)'s book "The Book of Numbers" [2]. So he ought to know. Noll published a Perl script that formats integers into English names, and an online version is here: http://www.isthe.com/cgi-bin/number.cgi He spells "1001" as "one thousand, one" and similarly for everything up to 999999. For example 27101 is "twenty seven thousand, one hundred one" which has 11+8+13=32 letters and one comma. You can try more examples on the website. For the purposes of this question I think you can just break the digit into groups of 3 digits, add the letter-length of both, plus 8 for "thousand". - Robert Munafo [1] see http://en.wikipedia.org/wiki/Landon_Curt_Noll [2] Conway, Johh Horton. The Book of Numbers. ISBN 0-387-97993-X. pages 13-15. On Sun, Sep 25, 2011 at 15:27, Victor Miller <victorsmiller@gmail.com>wrote:
I see that two versions of that sequence are already in OEIS: A092320 A126259. I avoided the standard spelling, because it's not clear how one should do it for arbitrarily large integers. [...]
On Sun, Sep 25, 2011 at 1:33 PM, Victor Miller <victorsmiller@gmail.com wrote:
On the program cartalk last week, there was a puzzle of the following form:
A list of numbers was given, and one was asked what they had in common (I give the actual puzzler at the end). The answer was that each of these numbers was divisible by the number of letters (excluding spaces) in the standard spelling out of the number in words. [...]
-- Robert Munafo -- mrob.com Follow me at: gplus.to/mrob - fb.com/mrob27 - twitter.com/mrob_27 - mrob27.wordpress.com - youtube.com/user/mrob143 - rilybot.blogspot.com
I have seen "one thousand one" and "one thousand and one", but all due respect to Dr. Noll, I am not sure if I have ever seen "one thousand, one" or any other number spelled out with a comma in modern times, although I can brook correction on that count. In other words, I wouldn't take this web page as the final word on number naming. English naming conventions for numbers is an interesting subject. There is a definite split along British/American lines. I believe that in Britain (correct me), the use of "and" is more or less universal in both spoken and written numbers (two thousand three hundred and seventeen), whereas in America, there is a formalism introduced some time in the last century in which the "and" was dropped (two thousand three hundred seventeen). The latter predominates in written material and in formal speech. In casual speech, the "and" form is more common, or else a shortened form (twenty-three seventeen). I believe the "andless" formalism was introduced in America circa 1900 (don't quite me). As I have heard it spoken, "and" is used as follows: (1) Within a block of three digits, to separate positive hundreds from a positive remainder, as in "three hundred and forty-seven" or "five hundred and one thousand six hundred and thirty-eight"; and (2) in a number of two or more blocks, before the last block if its value is between 1and 99, as in "one thousand and one" or "seventy-five million and thirty-eight". There is also the British use of the word "naught" (nought) for zero. In a numerical context, I understand this to be more of a name for the digit itself (similar to "cipher"), than for the number it represents, I am not sure if it is ever used to name a number. In America, the digit and the number are "zero." The meaning "nothing" for "naught" survives in America, but is obsolescent. The Americans corrupted "naught" to "aught" (ought) in times past: the year 1906 would have been abbreviated to '06 and spoken "aught eight". The .30-06 Springfield rifle, invented in 1906, is still called a "thirty aught six". In modern speech, "oh" is commonly used for the digit zero. There is also the famous divergence between British and Amercan zillions. The British zillions are based on blocks of six digits (base 1000000), the American on blocks of three digits (base 1000), the zillion names (thousand, million, billion, etc.) coincide. Formal zillion names extend to the vigintillion (British 10^120, American 10^63), Conway's system, based on a formal extension of the Greek zillion names, ostensibly names any positive integer (I am not fully convinced that Conway does not eventually run out of Greek prefixes), at any rate, Conway's system is at present a curiosity and not a standard; we have rare need and little motivation to name numbers so large that speaking them would consume entire lifetimes. On 9/25/2011 9:24 PM, Robert Munafo wrote:
Car Talk, huh? Who knew.
Landon Curt Noll [1] is a fairly well respected computer scientist and mathematician (sort of) who worked with John Horton Conway on the system for names of arbitrarily high powers of 1000 which was published in (Conway and (Richard K) Guy)'s book "The Book of Numbers" [2]. So he ought to know.
Noll published a Perl script that formats integers into English names, and an online version is here:
http://www.isthe.com/cgi-bin/number.cgi
He spells "1001" as "one thousand, one" and similarly for everything up to 999999. For example 27101 is "twenty seven thousand, one hundred one" which has 11+8+13=32 letters and one comma. You can try more examples on the website.
Writing out large numbers in words: http://www.grammarbook.com/numbers/numbers.asp See examples in Rule 7, and note that the comma is used in words whenever it is used in digits (but not in 4-digit numbers). But in any case there is no "and", and the comma when present replaces the missing "and". Numbers written digitally have a comma "27,101", so the use of the comma in words is closer to the way it is written in digits. "One thousand and one nights" is a contraction of the original title (in English) "The Book of the Thousand Nights and One Night" which implies that the 1000 nights are to be considered separate from the one night. In the story, 1000 nights are spend telling tales with cliffhanger endings and Scheherazade anticipating her execution, and then the final night, in which she has been pardoned (and will become the queen). It is as if "1000 vases and one jug" were poorly translated into "1000 and one bottles" which was then interpreted by the reader as "1001". - Robert On Mon, Sep 26, 2011 at 08:40, David Wilson <davidwwilson@comcast.net>wrote:
I have seen "one thousand one" and "one thousand and one", but all due respect to Dr. Noll, I am not sure if I have ever seen "one thousand, one" or any other number spelled out with a comma in modern times, although I can brook correction on that count. In other words, I wouldn't take this web page as the final word on number naming.
English naming conventions for numbers is an interesting subject. There is a definite split along British/American lines. I believe that in Britain (correct me), the use of "and" is more or less universal in both spoken and written numbers (two thousand three hundred and seventeen), whereas in America, there is a formalism introduced some time in the last century in which the "and" was dropped (two thousand three hundred seventeen). The latter predominates in written material and in formal speech. In casual speech, the "and" form is more common, or else a shortened form (twenty-three seventeen). I believe the "andless" formalism was introduced in America circa 1900 (don't quite me).
As I have heard it spoken, "and" is used as follows: (1) Within a block of three digits, to separate positive hundreds from a positive remainder, as in "three hundred and forty-seven" or "five hundred and one thousand six hundred and thirty-eight"; and (2) in a number of two or more blocks, before the last block if its value is between 1and 99, as in "one thousand and one" or "seventy-five million and thirty-eight".
There is also the British use of the word "naught" (nought) for zero. In a numerical context, I understand this to be more of a name for the digit itself (similar to "cipher"), than for the number it represents, I am not sure if it is ever used to name a number. In America, the digit and the number are "zero." The meaning "nothing" for "naught" survives in America, but is obsolescent. The Americans corrupted "naught" to "aught" (ought) in times past: the year 1906 would have been abbreviated to '06 and spoken "aught eight". The .30-06 Springfield rifle, invented in 1906, is still called a "thirty aught six". In modern speech, "oh" is commonly used for the digit zero. [...]
http://www.isthe.com/cgi-bin/**number.cgi<http://www.isthe.com/cgi-bin/number.cgi>
[Landon Curt Noll] spells "1001" as "one thousand, one" and similarly for everything up to 999999. For example 27101 is "twenty seven thousand, one hundred one" which has 11+8+13=32 letters and one comma. You can try more examples on the website.
-- Robert Munafo -- mrob.com Follow me at: gplus.to/mrob - fb.com/mrob27 - twitter.com/mrob_27 - mrob27.wordpress.com - youtube.com/user/mrob143 - rilybot.blogspot.com
Here are some experimental results, especially on density. I used names with no "and"; length counts letters only; 0 is "zero"; N < 0 is "minus" N. I investigated only up to the neighborhood of abs(N) = 10^18. I agree with Bill Gosper that such problems are less interesting when they depend on decimal; and Victor Miller's original post noted that (his modification) can be generalized to other bases. That said, it's a cute problem. -- Mike Beeler // cases in range 1..10^0: 0 // cases in range 1..10^1: 2 // cases in range 1..10^2: 17 (the Puzzler) // cases in range 1..10^3: 56 // cases in range 1..10^4: 359 // cases in range 1..10^5: 2981 // cases in range 1..10^6: 22826 // cases in range 1..10^7: 194682 // cases in range 1..10^8: 1739872 // cases in range -(1..10^0): 0 // cases in range -(1..10^1): 1 // cases in range -(1..10^2): 7 (-9, -11, -13, -40, -50, -52, -60) // cases in range -(1..10^3): 42 // cases in range -(1..10^4): 314 // cases in range -(1..10^5): 2586 // cases in range -(1..10^6): 20759 // cases in range -(1..10^7): 177174 // cases in range -(1..10^8): 1575817 // count of both N and -N in list, N = 1..10^0: 0 // count of both N and -N in list, N = 1..10^1: 0 // count of both N and -N in list, N = 1..10^2: 3 (last = 60) // count of both N and -N in list, N = 1..10^3: 7 (last = 660) // count of both N and -N in list, N = 1..10^4: 27 (last = 9860) // count of both N and -N in list, N = 1..10^5: 156 (last = 99456) // count of both N and -N in list, N = 1..10^6: 1006 (last = 999450) // count of both N and -N in list, N = 1..10^7: 7934 (last = 9999660) // count of both N and -N in list, N = 1..10^8: 56435 (last = 99996600) // the 7 through 1000 are: 40, 50, 60, 240, 456, 600, 660 // count of pairs of consecutive numbers in list, N = 0..10^x: // for 0..10^0: 0 positive, 0 negative // for 0..10^1: 0 positive, 0 negative // for 0..10^2: 0 positive, 0 negative // for 0..10^3: 2 positive, 2 negative // for 0..10^4: 7 positive, 8 negative // for 0..10^5: 44 positive, 40 negative // for 0..10^6: 271 positive, 238 negative // for 0..10^7: 2020 positive, 1509 negative // for 0..10^8: 15063 positive, 12709 negative // First 10 positive (N, of N and N+1): // 405, 665, 2262, 3135, 4508, 5082, 8903, 10503, 11865, 13283 // First 10 negative (N, of N and N-1): // -160, -728, -1155, -1368, -2379, -3626, -6303, -6623, -16769, -17080 // count of triplets of consecutive numbers in list, N = 0..10^x: // for 0..10^0: 0 positive, 0 negative // for 0..10^1: 0 positive, 0 negative // for 0..10^2: 0 positive, 0 negative // for 0..10^3: 0 positive, 0 negative // for 0..10^4: 0 positive, 0 negative // for 0..10^5: 2 positive, 0 negative // for 0..10^6: 4 positive, 3 negative // for 0..10^7: 17 positive, 8 negative // for 0..10^8: 89 positive, 66 negative // First 15 positive (N, of N and N+1 and N+2): // 44608, 64638, 390382, 774639, 1062528, // 1882399, 2048542, 3408428, 3819879, 6731898, // 6731899, 6912788, 7514652, 8225972, 8622898 // First 15 negative (N, of N and N-1 and N-2): // -706102, -932958, -943102, -1304728, -5556042, // -7378239, -7394522, -8420048, -11374208, -13295679, // -14779798, -17226168, -19023398, -20187568, -21017202 // There is 1 positive quadruplet of consecutive numbers in the // list, in the range 1..10^8. It begins at 6731898. // There are no negative quadruplets of consecutive numbers in the // list, in the range -1..-10^8. // Above results suggest numbers in list become sparser as the numbers // get larger (either positive or negative). That requires the gap // between numbers in list grows. The results below show that the // max gap size does grow. The gap size is the count of consecutive // numbers not in the list, between two numbers that are in the list. // // gap search going positive // gap of 3 after 0 // gap of 5 after 6 // gap of 17 after 12 // gap of 37 after 112 // gap of 39 after 200 // gap of 47 after 252 // gap of 64 after 340 // gap of 75 after 1071 // gap of 83 after 2178 // gap of 105 after 4048 // gap of 141 after 5536 // gap of 169 after 17610 // gap of 194 after 29202 // gap of 233 after 38532 // gap of 245 after 50904 // gap of 289 after 84490 // gap of 310 after 185856 // gap of 327 after 198842 // gap of 332 after 261712 // gap of 390 after 378400 // gap of 401 after 535950 // gap of 433 after 1205946 // gap of 530 after 1552320 // gap of 549 after 3622960 // gap of 570 after 7523191 // gap of 583 after 17188600 // gap of 602 after 22357364 // gap of 602 after 24674883 // gap of 653 after 27346200 // gap of 709 after 28373233 // gap of 712 after 36238357 // gap of 716 after 39391176 // gap of 724 after 49277495 // gap of 830 after 65840289 // gap analysis done through 10^8 // // gap search going negative // gap of 8 after 0 // gap of 26 after -13 // gap of 64 after -60 // gap of 70 after -832 // gap of 86 after -903 // gap of 184 after -1443 // gap of 199 after -8050 // gap of 208 after -17081 // gap of 209 after -17880 // gap of 261 after -36320 // gap of 275 after -79684 // gap of 411 after -118512 // gap of 476 after -149354 // gap of 534 after -716375 // gap of 551 after -4824320 // gap of 625 after -5583370 // gap of 698 after -11658024 // gap of 710 after -35366383 // gap of 758 after -50834025 // gap of 860 after -61538100 // gap analysis done through -10^8 // For various powers of 10, what is the number in the list just // less than the 10^x, and just greatert than the 10^x? Also note // whether the power of 10 itself is in the list. // // for positive N: // bracketing 10^0 (N not in list): 10^0-1, 10^0+3 // bracketing 10^1 (N not in list): 10^1-4, 10^1+2 // bracketing 10^2 (N is in list): 10^2-10, 10^2+12 // bracketing 10^3 (N not in list): 10^3-40, 10^3+5 // bracketing 10^4 (N not in list): 10^4-1, 10^4+5 // bracketing 10^5 (N not in list): 10^5-40, 10^5+2 // bracketing 10^6 (N is in list): 10^6-8, 10^6+50 // bracketing 10^7 (N is in list): 10^7-206, 10^7+4 // bracketing 10^8 (N not in list): 10^8-152, 10^8+5 // bracketing 10^9 (N is in list): 10^9-67, 10^9+1 // bracketing 10^10 (N is in list): 10^10-56, 10^10+10 // bracketing 10^11 (N not in list): 10^11-102, 10^11+22 // bracketing 10^12 (N not in list): 10^12-16, 10^12+5 // bracketing 10^13 (N not in list): 10^13-70, 10^13+5 // bracketing 10^14 (N not in list): 10^14-28, 10^14+56 // bracketing 10^15 (N not in list): 10^15-158, 10^15+20 // bracketing 10^16 (N not in list): 10^16-137, 10^16+20 // bracketing 10^17 (N not in list): 10^17-54, 10^17+14 // bracketing 10^18 (N not in list): 10^18-180, 10^18+2 // // for negative N: // bracketing -10^0 (N not in list): -10^0-8, -10^0+1 // bracketing -10^1 (N not in list): -10^1-1, -10^1+1 // bracketing -10^2 (N not in list): -10^2-25, -10^2+40 // bracketing -10^3 (N not in list): -10^3-8, -10^3+10 // bracketing -10^4 (N is in list): -10^4-56, -10^4+6 // bracketing -10^5 (N not in list): -10^5-28, -10^5+32 // bracketing -10^6 (N not in list): -10^6-20, -10^6+49 // bracketing -10^7 (N not in list): -10^7-4, -10^7+20 // bracketing -10^8 (N not in list): -10^8-4, -10^8+136 // bracketing -10^9 (N not in list): -10^9-25, -10^9+78 // bracketing -10^10 (N not in list): -10^10-25, -10^10+10 // bracketing -10^11 (N not in list): -10^11-8, -10^11+100 // bracketing -10^12 (N is in list): -10^12-56, -10^12+1 // bracketing -10^13 (N is in list): -10^13-6, -10^13+16 // bracketing -10^14 (N not in list): -10^14-30, -10^14+125 // bracketing -10^15 (N not in list): -10^15-8, -10^15+80 // bracketing -10^16 (N not in list): -10^16-8, -10^16+144 // bracketing -10^17 (N not in list): -10^17-65, -10^17+362 // bracketing -10^18 (N not in list): -10^18-8, -10^18+343 // From various results above, we would expect the density of // numbers in list to drop as the size of the numbers increases. // Here, we start with 10^x, for x=6..18, and count how many // of the next million cases are in list. Positive and negative. // // sample block 10^ 6 + 1..10^6: 19562 in list // sample block 10^ 7 + 1..10^6: 19485 in list // sample block 10^ 8 + 1..10^6: 16628 in list // sample block 10^ 9 + 1..10^6: 19515 in list // sample block 10^10 + 1..10^6: 19737 in list // sample block 10^11 + 1..10^6: 16614 in list // sample block 10^12 + 1..10^6: 18859 in list // sample block 10^13 + 1..10^6: 19223 in list // sample block 10^14 + 1..10^6: 16387 in list // sample block 10^15 + 1..10^6: 18099 in list // sample block 10^16 + 1..10^6: 18037 in list // sample block 10^17 + 1..10^6: 15236 in list // sample block 10^18 + 1..10^6: 18038 in list // // sample block -10^ 6 - 1..10^6: 17681 in list // sample block -10^ 7 - 1..10^6: 17429 in list // sample block -10^ 8 - 1..10^6: 15580 in list // sample block -10^ 9 - 1..10^6: 17463 in list // sample block -10^10 - 1..10^6: 17442 in list // sample block -10^11 - 1..10^6: 15416 in list // sample block -10^12 - 1..10^6: 17342 in list // sample block -10^13 - 1..10^6: 17579 in list // sample block -10^14 - 1..10^6: 15305 in list // sample block -10^15 - 1..10^6: 16133 in list // sample block -10^16 - 1..10^6: 16145 in list // sample block -10^17 - 1..10^6: 14937 in list // sample block -10^18 - 1..10^6: 16269 in list // // If the density is decreasing, it is doing so very slowly. // Perhaps starting the sample block on a 10^x value skews // the results; that is, perhaps in-list cases are unevenly // distributed within each power of 10. To investigate, count // the in-list cases among each of the 100 samples of 1 million // numbers per sample, from 1 to 10^8. Those counts are: // // 22826, 19562, 19594, 18439, 19290, 19208, 19494, 18468, 18538, 19263, // 19485, 18004, 17864, 17954, 17870, 17868, 17635, 17304, 17783, 17732, // 17914, 17466, 17368, 16291, 16979, 16905, 17504, 16286, 16291, 16825, // 17866, 17303, 17451, 16543, 16895, 16943, 17355, 16439, 16650, 16811, // 18439, 17888, 17604, 16729, 17466, 17211, 17708, 16827, 16829, 17521, // 17953, 17682, 17638, 16824, 17449, 17273, 17585, 17059, 17057, 17399, // 17968, 17641, 17934, 16952, 17257, 17423, 17580, 16773, 16798, 17199, // 17583, 16743, 16832, 16257, 16213, 16142, 16680, 15908, 16219, 16248, // 17660, 17433, 17380, 16604, 16898, 16985, 17289, 16602, 16627, 17087, // 17569, 17203, 17360, 16494, 16685, 16855, 17013, 16283, 16403, 16685 // // There is a trend toward lower density, but it is very slow after the // initial million, and is quite noisy.
participants (5)
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David Wilson -
Michael Beeler -
Michael Kleber -
Robert Munafo -
Victor Miller