[math-fun] 9 * 99 * 999 * ...
One side-alley that I glanced down when I was preparing my current Mathematical Enchantments essay, but then resolutely marched myself past, involves the behavior of the decimal expansion of the product (10^1-1)(10^2-1)...(10^n-1); the initial and final digits are quite orderly, while the digits in the middle look random. Has anyone looked into this? It'd be especially interesting if one could establish some sort of phase transition. I realize I'm using vocabulary loosely; there's no randomness involved, so we're not really doing stat mech. Still, some of the concepts of stat mech might apply. Jim Propp
Note that product(n>=1, 1-10^{-n}) is essentially Dedekind's eta function and Euler's pentagonal number theorem (to name just one) may be helpful with spotting patterns in the expansion. Best regards, jj * James Propp <jamespropp@gmail.com> [Jul 13. 2016 08:00]:
One side-alley that I glanced down when I was preparing my current Mathematical Enchantments essay, but then resolutely marched myself past, involves the behavior of the decimal expansion of the product (10^1-1)(10^2-1)...(10^n-1); the initial and final digits are quite orderly, while the digits in the middle look random. Has anyone looked into this? It'd be especially interesting if one could establish some sort of phase transition.
I realize I'm using vocabulary loosely; there's no randomness involved, so we're not really doing stat mech. Still, some of the concepts of stat mech might apply.
Jim Propp _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
Yes, absolutely. That's why the initial and final digits behave so nicely. I should've mentioned that. Jim On Wednesday, July 13, 2016, Joerg Arndt <arndt@jjj.de> wrote:
Note that product(n>=1, 1-10^{-n}) is essentially Dedekind's eta function and Euler's pentagonal number theorem (to name just one) may be helpful with spotting patterns in the expansion.
Best regards, jj
* James Propp <jamespropp@gmail.com <javascript:;>> [Jul 13. 2016 08:00]:
One side-alley that I glanced down when I was preparing my current Mathematical Enchantments essay, but then resolutely marched myself past, involves the behavior of the decimal expansion of the product (10^1-1)(10^2-1)...(10^n-1); the initial and final digits are quite orderly, while the digits in the middle look random. Has anyone looked into this? It'd be especially interesting if one could establish some sort of phase transition.
I realize I'm using vocabulary loosely; there's no randomness involved, so we're not really doing stat mech. Still, some of the concepts of stat mech might apply.
Jim Propp _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com <javascript:;> https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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The sequence in question is https://oeis.org/A027878. Please also add any useful comments to that entry! Best regards Neil Neil J. A. Sloane, President, OEIS Foundation. 11 South Adelaide Avenue, Highland Park, NJ 08904, USA. Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ. Phone: 732 828 6098; home page: http://NeilSloane.com Email: njasloane@gmail.com On Wed, Jul 13, 2016 at 11:45 AM, James Propp <jamespropp@gmail.com> wrote:
Yes, absolutely. That's why the initial and final digits behave so nicely. I should've mentioned that.
Jim
On Wednesday, July 13, 2016, Joerg Arndt <arndt@jjj.de> wrote:
Note that product(n>=1, 1-10^{-n}) is essentially Dedekind's eta function and Euler's pentagonal number theorem (to name just one) may be helpful with spotting patterns in the expansion.
Best regards, jj
* James Propp <jamespropp@gmail.com <javascript:;>> [Jul 13. 2016 08:00]:
One side-alley that I glanced down when I was preparing my current Mathematical Enchantments essay, but then resolutely marched myself past, involves the behavior of the decimal expansion of the product (10^1-1)(10^2-1)...(10^n-1); the initial and final digits are quite orderly, while the digits in the middle look random. Has anyone looked into this? It'd be especially interesting if one could establish some sort of phase transition.
I realize I'm using vocabulary loosely; there's no randomness involved, so we're not really doing stat mech. Still, some of the concepts of stat mech might apply.
Jim Propp _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com <javascript:;> https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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Quicky: The product of palindromes is a palindrome, if there are no carries. In the 9*99*... case, if we allow m = -1 as a digit, the palindrome-product theorem needs a bit of fixup for nega-dromes like 1m*10m, but still seems to be relevant. Re JJs remark: There's a finite version of Euler's pentagonal number product. It's a special case of Thm 348 in Hardy & Wright's "Introduction to the Theory of Numbers", with a=-1: (1-x)(1-x^2)(1-x^3)...(1-x^j) = 1 - x (1-x^j)/(1-x) + x^3 (1-x^j)(1-x^(j-1))/(1-x)(1-x^2) - ... + (-1)^j x^(j(j+1)/2). The terms in the sum have x^triangle and the ratio of initial and final sections of the product. This ratio has a nice symmetry between the beginning and ending terms of the sum, a kind of power-series palindrome. We can plug in x = .1 or x = 10 for useful results. This should lead to regularities even in the middle digits of the product, until carries spoil the patterns. Q: If the quotient of two palindomes is an integer, must the quotient be a shoehorn-palindrome? SP is a palindrome with digits outside the normal range, with the carries propagated. 301 is an SP, with decimal representation 1*100 + 20*10 + 1, middle digit 20. It's tempting to do cancellations on the product-ratios from the sum, but there are some hazards: (1-x)(1-x^2)(1-x^3) is a divisor of (1-x^j)(1-x^(j-1))(1-x^(j-2)), both as integers and as polynomials, but if j = 7, the the ratio can't be split into individual polynomial terms (1-x^a)/(1-x^b) with b dividing a. Rich ------- Quoting Neil Sloane <njasloane@gmail.com>:
The sequence in question is https://oeis.org/A027878. Please also add any useful comments to that entry!
Best regards Neil
Neil J. A. Sloane, President, OEIS Foundation. 11 South Adelaide Avenue, Highland Park, NJ 08904, USA. Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ. Phone: 732 828 6098; home page: http://NeilSloane.com Email: njasloane@gmail.com
On Wed, Jul 13, 2016 at 11:45 AM, James Propp <jamespropp@gmail.com> wrote:
Yes, absolutely. That's why the initial and final digits behave so nicely. I should've mentioned that.
Jim
On Wednesday, July 13, 2016, Joerg Arndt <arndt@jjj.de> wrote:
Note that product(n>=1, 1-10^{-n}) is essentially Dedekind's eta function and Euler's pentagonal number theorem (to name just one) may be helpful with spotting patterns in the expansion.
Best regards, jj
* James Propp <jamespropp@gmail.com <javascript:;>> [Jul 13. 2016 08:00]:
One side-alley that I glanced down when I was preparing my current Mathematical Enchantments essay, but then resolutely marched myself past, involves the behavior of the decimal expansion of the product (10^1-1)(10^2-1)...(10^n-1); the initial and final digits are quite orderly, while the digits in the middle look random. Has anyone looked into this? It'd be especially interesting if one could establish some sort of phase transition.
I realize I'm using vocabulary loosely; there's no randomness involved, so we're not really doing stat mech. Still, some of the concepts of stat mech might apply.
Jim Propp _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com <javascript:;> https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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Speaking of equidistribution/normality, I wonder what's known about https://oeis.org/A018247 (the digits of a 10-adic fixed point of the squaring map). Has anyone read any of the references? REFERENCES W. W. R. Ball, Mathematical Recreations & Essays, N.Y. Macmillan Co, 1947. V. deGuerre and R. A. Fairbairn, Automorphic numbers, J. Rec. Math., 1 (No. 3, 1968), 173-179. M. Kraitchik, Sphinx, 1935, p. 1. I'm guessing that nobody has done a statistical analysis of the digits. Jim Propp On Wednesday, July 13, 2016, Neil Sloane <njasloane@gmail.com <javascript:_e(%7B%7D,'cvml','njasloane@gmail.com');>> wrote:
The sequence in question is https://oeis.org/A027878. Please also add any useful comments to that entry!
Best regards Neil
Neil J. A. Sloane, President, OEIS Foundation. 11 South Adelaide Avenue, Highland Park, NJ 08904, USA. Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ. Phone: 732 828 6098; home page: http://NeilSloane.com Email: njasloane@gmail.com
On Wed, Jul 13, 2016 at 11:45 AM, James Propp <jamespropp@gmail.com> wrote:
Yes, absolutely. That's why the initial and final digits behave so nicely. I should've mentioned that.
Jim
On Wednesday, July 13, 2016, Joerg Arndt <arndt@jjj.de> wrote:
Note that product(n>=1, 1-10^{-n}) is essentially Dedekind's eta function and Euler's pentagonal number theorem (to name just one) may be helpful with spotting patterns in the expansion.
Best regards, jj
* James Propp <jamespropp@gmail.com <javascript:;>> [Jul 13. 2016 08:00]:
One side-alley that I glanced down when I was preparing my current Mathematical Enchantments essay, but then resolutely marched myself past, involves the behavior of the decimal expansion of the product (10^1-1)(10^2-1)...(10^n-1); the initial and final digits are quite orderly, while the digits in the middle look random. Has anyone looked into this? It'd be especially interesting if one could establish some sort of phase transition.
I realize I'm using vocabulary loosely; there's no randomness involved, so we're not really doing stat mech. Still, some of the concepts of stat mech might apply.
Jim Propp _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com <javascript:;> https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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The left digits of A027878 <https://oeis.org/A027878> seem to form a limiting sequence, growing about one digit per term. At the 30th term, the sequence is 890010099998999000000100010000. Clearly, this sequence is related to the base being 10 (9 * 99 * 999 * ...), so I looked at other bases. For example with base 5, the product would be 4 * 44 * 444 * ..., and for base 16, f * ff * fff * ... Here are the beginnings of other limiting sequences: - base 5: 340010044443444000000100004444... - base 6: 450010055554555000000100005555... - base 7: 560010066665666000000100006666... - base 8: 670010077776777000000100007777... - base 9: 780010088887888000000100008888... - base 10: 890010099998999000000100009999... - base 16: ef00100ffffefff00000010000ffff... For any base b > 2, the string seems to start with b-2, b-1, 0, 0, 1, 0, 0, etc. Up through base 72, the first 100 terms are: b-2, b-1, 0, 0, 1, 0, 0, b-1, b-1, b-1, b-1, b-2, b-1, b-1, b-1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, b-1, b-1, b-1, b-1, b-1, b-1, b-1, b-1, b-2, b-1, b-1, b-1, b-1, b-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, b-1, b-1, b-1, b-1, b-1, b-1, b-1, b-1, b-1, b-1, b-1, b-1, b-2, b-1, b-1, b-1, b-1, b-1, b-1, b-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, … The digits on the right of the terms of A027878 oscillate between two limiting sequences: ...000100009999998999000010099891 and ...999899990000001000999989900109 The terms for other bases (b > 2) behave similarly. The general patterns for the last terms appear to be: …, 0, 0, 0, 0, 0, 0, 0, 0, b-1, b-1, b-1, b-1, b-1, b-1, b-1, b-1, b-1, b-1, b-1, b-1, b-1, b-1, b-2, b-1, b-1, b-1, b-1, b-1, b-1, b-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, b-1, b-1, b-1, b-1, b-1, b-1, b-1, b-1, b-1, b-1, b-2, b-1, b-1, b-1, b-1, b-1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, b-1, b-1, b-1, b-1, b-1, b-1, b-2, b-1, b-1, b-1, 0, 0, 0, 0, 1, 0, 0, b-1, b-1, b-2, b-1, 1 and …, b-1, b-1, b-1, b-1, b-1, b-1, b-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, b-1, b-1, b-1, b-1, b-1, b-1, b-1, b-1, b-1, b-1, b-1, b-1, b-2, b-1, b-1, b-1, b-1, b-1, b-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, b-1, b-1, b-1, b-1, b-1, b-1, b-1, b-1, b-2, b-1, b-1, b-1, b-1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, b-1, b-1, b-1, b-1, b-2, b-1, b-1, 0, 0, 1, 0, b-1 Kerry On Wed, Jul 13, 2016 at 8:51 AM, Neil Sloane <njasloane@gmail.com> wrote:
The sequence in question is https://oeis.org/A027878. Please also add any useful comments to that entry!
Best regards Neil
Neil J. A. Sloane, President, OEIS Foundation. 11 South Adelaide Avenue, Highland Park, NJ 08904, USA. Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ. Phone: 732 828 6098; home page: http://NeilSloane.com Email: njasloane@gmail.com
On Wed, Jul 13, 2016 at 11:45 AM, James Propp <jamespropp@gmail.com> wrote:
Yes, absolutely. That's why the initial and final digits behave so nicely. I should've mentioned that.
Jim
On Wednesday, July 13, 2016, Joerg Arndt <arndt@jjj.de> wrote:
Note that product(n>=1, 1-10^{-n}) is essentially Dedekind's eta function and Euler's pentagonal number theorem (to name just one) may be helpful with spotting patterns in the expansion.
Best regards, jj
* James Propp <jamespropp@gmail.com <javascript:;>> [Jul 13. 2016 08:00]:
One side-alley that I glanced down when I was preparing my current Mathematical Enchantments essay, but then resolutely marched myself past, involves the behavior of the decimal expansion of the product (10^1-1)(10^2-1)...(10^n-1); the initial and final digits are quite orderly, while the digits in the middle look random. Has anyone looked into this? It'd be especially interesting if one could establish some sort of phase transition.
I realize I'm using vocabulary loosely; there's no randomness involved, so we're not really doing stat mech. Still, some of the concepts of stat mech might apply.
Jim Propp _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com <javascript:;> https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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Kerry, I think the "starting limit" for bases 2 through 10 ought to be in the OEIS, as well as the "ending limit" written backwards. That's 27 sequences (two each for the terminating versions) - can you enter them? Or I can do it if you send them to me. Best regards Neil Neil J. A. Sloane, President, OEIS Foundation. 11 South Adelaide Avenue, Highland Park, NJ 08904, USA. Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ. Phone: 732 828 6098; home page: http://NeilSloane.com Email: njasloane@gmail.com On Wed, Jul 13, 2016 at 11:33 PM, Kerry Mitchell <lkmitch@gmail.com> wrote:
The left digits of A027878 <https://oeis.org/A027878> seem to form a limiting sequence, growing about one digit per term. At the 30th term, the sequence is 890010099998999000000100010000. Clearly, this sequence is related to the base being 10 (9 * 99 * 999 * ...), so I looked at other bases. For example with base 5, the product would be 4 * 44 * 444 * ..., and for base 16, f * ff * fff * ... Here are the beginnings of other limiting sequences:
- base 5: 340010044443444000000100004444... - base 6: 450010055554555000000100005555... - base 7: 560010066665666000000100006666... - base 8: 670010077776777000000100007777... - base 9: 780010088887888000000100008888... - base 10: 890010099998999000000100009999... - base 16: ef00100ffffefff00000010000ffff...
For any base b > 2, the string seems to start with b-2, b-1, 0, 0, 1, 0, 0, etc. Up through base 72, the first 100 terms are:
b-2, b-1, 0, 0, 1, 0, 0, b-1, b-1, b-1, b-1, b-2, b-1, b-1, b-1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, b-1, b-1, b-1, b-1, b-1, b-1, b-1, b-1, b-2, b-1, b-1, b-1, b-1, b-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, b-1, b-1, b-1, b-1, b-1, b-1, b-1, b-1, b-1, b-1, b-1, b-1, b-2, b-1, b-1, b-1, b-1, b-1, b-1, b-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, …
The digits on the right of the terms of A027878 oscillate between two limiting sequences: ...000100009999998999000010099891 and ...999899990000001000999989900109
The terms for other bases (b > 2) behave similarly. The general patterns for the last terms appear to be:
…, 0, 0, 0, 0, 0, 0, 0, 0, b-1, b-1, b-1, b-1, b-1, b-1, b-1, b-1, b-1, b-1, b-1, b-1, b-1, b-1, b-2, b-1, b-1, b-1, b-1, b-1, b-1, b-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, b-1, b-1, b-1, b-1, b-1, b-1, b-1, b-1, b-1, b-1, b-2, b-1, b-1, b-1, b-1, b-1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, b-1, b-1, b-1, b-1, b-1, b-1, b-2, b-1, b-1, b-1, 0, 0, 0, 0, 1, 0, 0, b-1, b-1, b-2, b-1, 1
and
…, b-1, b-1, b-1, b-1, b-1, b-1, b-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, b-1, b-1, b-1, b-1, b-1, b-1, b-1, b-1, b-1, b-1, b-1, b-1, b-2, b-1, b-1, b-1, b-1, b-1, b-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, b-1, b-1, b-1, b-1, b-1, b-1, b-1, b-1, b-2, b-1, b-1, b-1, b-1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, b-1, b-1, b-1, b-1, b-2, b-1, b-1, 0, 0, 1, 0, b-1
Kerry
On Wed, Jul 13, 2016 at 8:51 AM, Neil Sloane <njasloane@gmail.com> wrote:
The sequence in question is https://oeis.org/A027878. Please also add any useful comments to that entry!
Best regards Neil
Neil J. A. Sloane, President, OEIS Foundation. 11 South Adelaide Avenue, Highland Park, NJ 08904, USA. Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ. Phone: 732 828 6098; home page: http://NeilSloane.com Email: njasloane@gmail.com
On Wed, Jul 13, 2016 at 11:45 AM, James Propp <jamespropp@gmail.com> wrote:
Yes, absolutely. That's why the initial and final digits behave so nicely. I should've mentioned that.
Jim
On Wednesday, July 13, 2016, Joerg Arndt <arndt@jjj.de> wrote:
Note that product(n>=1, 1-10^{-n}) is essentially Dedekind's eta function and Euler's pentagonal number theorem (to name just one) may be helpful with spotting patterns in the expansion.
Best regards, jj
* James Propp <jamespropp@gmail.com <javascript:;>> [Jul 13. 2016 08:00]:
One side-alley that I glanced down when I was preparing my current Mathematical Enchantments essay, but then resolutely marched myself past, involves the behavior of the decimal expansion of the product (10^1-1)(10^2-1)...(10^n-1); the initial and final digits are quite orderly, while the digits in the middle look random. Has anyone looked into this? It'd be especially interesting if one could establish some sort of phase transition.
I realize I'm using vocabulary loosely; there's no randomness involved, so we're not really doing stat mech. Still, some of the concepts of stat mech might apply.
Jim Propp _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com <javascript:;> https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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Hi Neil, I'll get them submitted over the next few days. Thanks, Kerry On Wed, Jul 13, 2016 at 9:10 PM, Neil Sloane <njasloane@gmail.com> wrote:
Kerry, I think the "starting limit" for bases 2 through 10 ought to be in the OEIS, as well as the "ending limit" written backwards.
That's 27 sequences (two each for the terminating versions) - can you enter them? Or I can do it if you send them to me.
Best regards Neil
Neil J. A. Sloane, President, OEIS Foundation. 11 South Adelaide Avenue, Highland Park, NJ 08904, USA. Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ. Phone: 732 828 6098; home page: http://NeilSloane.com Email: njasloane@gmail.com
On Wed, Jul 13, 2016 at 11:33 PM, Kerry Mitchell <lkmitch@gmail.com> wrote:
The left digits of A027878 <https://oeis.org/A027878> seem to form a limiting sequence, growing about one digit per term. At the 30th term, the sequence is 890010099998999000000100010000. Clearly, this sequence is related to the base being 10 (9 * 99 * 999 * ...), so I looked at other bases. For example with base 5, the product would be 4 * 44 * 444 * ..., and for base 16, f * ff * fff * ... Here are the beginnings of other limiting sequences:
- base 5: 340010044443444000000100004444... - base 6: 450010055554555000000100005555... - base 7: 560010066665666000000100006666... - base 8: 670010077776777000000100007777... - base 9: 780010088887888000000100008888... - base 10: 890010099998999000000100009999... - base 16: ef00100ffffefff00000010000ffff...
For any base b > 2, the string seems to start with b-2, b-1, 0, 0, 1, 0, 0, etc. Up through base 72, the first 100 terms are:
b-2, b-1, 0, 0, 1, 0, 0, b-1, b-1, b-1, b-1, b-2, b-1, b-1, b-1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, b-1, b-1, b-1, b-1, b-1, b-1, b-1, b-1, b-2, b-1, b-1, b-1, b-1, b-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, b-1, b-1, b-1, b-1, b-1, b-1, b-1, b-1, b-1, b-1, b-1, b-1, b-2, b-1, b-1, b-1, b-1, b-1, b-1, b-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, …
The digits on the right of the terms of A027878 oscillate between two limiting sequences: ...000100009999998999000010099891 and ...999899990000001000999989900109
The terms for other bases (b > 2) behave similarly. The general patterns for the last terms appear to be:
…, 0, 0, 0, 0, 0, 0, 0, 0, b-1, b-1, b-1, b-1, b-1, b-1, b-1, b-1, b-1, b-1, b-1, b-1, b-1, b-1, b-2, b-1, b-1, b-1, b-1, b-1, b-1, b-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, b-1, b-1, b-1, b-1, b-1, b-1, b-1, b-1, b-1, b-1, b-2, b-1, b-1, b-1, b-1, b-1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, b-1, b-1, b-1, b-1, b-1, b-1, b-2, b-1, b-1, b-1, 0, 0, 0, 0, 1, 0, 0, b-1, b-1, b-2, b-1, 1
and
…, b-1, b-1, b-1, b-1, b-1, b-1, b-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, b-1, b-1, b-1, b-1, b-1, b-1, b-1, b-1, b-1, b-1, b-1, b-1, b-2, b-1, b-1, b-1, b-1, b-1, b-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, b-1, b-1, b-1, b-1, b-1, b-1, b-1, b-1, b-2, b-1, b-1, b-1, b-1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, b-1, b-1, b-1, b-1, b-2, b-1, b-1, 0, 0, 1, 0, b-1
Kerry
On Wed, Jul 13, 2016 at 8:51 AM, Neil Sloane <njasloane@gmail.com> wrote:
The sequence in question is https://oeis.org/A027878. Please also add any useful comments to that entry!
Best regards Neil
Neil J. A. Sloane, President, OEIS Foundation. 11 South Adelaide Avenue, Highland Park, NJ 08904, USA. Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ. Phone: 732 828 6098; home page: http://NeilSloane.com Email: njasloane@gmail.com
On Wed, Jul 13, 2016 at 11:45 AM, James Propp <jamespropp@gmail.com> wrote:
Yes, absolutely. That's why the initial and final digits behave so nicely. I should've mentioned that.
Jim
On Wednesday, July 13, 2016, Joerg Arndt <arndt@jjj.de> wrote:
Note that product(n>=1, 1-10^{-n}) is essentially Dedekind's eta function and Euler's pentagonal number theorem (to name just one) may be helpful with spotting patterns in the expansion.
Best regards, jj
* James Propp <jamespropp@gmail.com <javascript:;>> [Jul 13. 2016 08:00]:
One side-alley that I glanced down when I was preparing my current Mathematical Enchantments essay, but then resolutely marched myself past, involves the behavior of the decimal expansion of the product (10^1-1)(10^2-1)...(10^n-1); the initial and final digits are quite orderly, while the digits in the middle look random. Has anyone looked into this? It'd be especially interesting if one could establish some sort of phase transition.
I realize I'm using vocabulary loosely; there's no randomness involved, so we're not really doing stat mech. Still, some of the concepts of stat mech might apply.
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Kerry, that's great (especially as i'm going on vac. starting tomorrow) if you let me know the A-numbers when you've submitted them, I can keep an eye on them Best regards Neil Neil J. A. Sloane, President, OEIS Foundation. 11 South Adelaide Avenue, Highland Park, NJ 08904, USA. Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ. Phone: 732 828 6098; home page: http://NeilSloane.com Email: njasloane@gmail.com On Thu, Jul 14, 2016 at 12:19 AM, Kerry Mitchell <lkmitch@gmail.com> wrote:
Hi Neil,
I'll get them submitted over the next few days.
Thanks, Kerry
On Wed, Jul 13, 2016 at 9:10 PM, Neil Sloane <njasloane@gmail.com> wrote:
Kerry, I think the "starting limit" for bases 2 through 10 ought to be in the OEIS, as well as the "ending limit" written backwards.
That's 27 sequences (two each for the terminating versions) - can you enter them? Or I can do it if you send them to me.
Best regards Neil
Neil J. A. Sloane, President, OEIS Foundation. 11 South Adelaide Avenue, Highland Park, NJ 08904, USA. Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ. Phone: 732 828 6098; home page: http://NeilSloane.com Email: njasloane@gmail.com
On Wed, Jul 13, 2016 at 11:33 PM, Kerry Mitchell <lkmitch@gmail.com> wrote:
The left digits of A027878 <https://oeis.org/A027878> seem to form a limiting sequence, growing about one digit per term. At the 30th term, the sequence is 890010099998999000000100010000. Clearly, this sequence is related to the base being 10 (9 * 99 * 999 * ...), so I looked at other bases. For example with base 5, the product would be 4 * 44 * 444 * ..., and for base 16, f * ff * fff * ... Here are the beginnings of other limiting sequences:
- base 5: 340010044443444000000100004444... - base 6: 450010055554555000000100005555... - base 7: 560010066665666000000100006666... - base 8: 670010077776777000000100007777... - base 9: 780010088887888000000100008888... - base 10: 890010099998999000000100009999... - base 16: ef00100ffffefff00000010000ffff...
For any base b > 2, the string seems to start with b-2, b-1, 0, 0, 1, 0, 0, etc. Up through base 72, the first 100 terms are:
b-2, b-1, 0, 0, 1, 0, 0, b-1, b-1, b-1, b-1, b-2, b-1, b-1, b-1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, b-1, b-1, b-1, b-1, b-1, b-1, b-1, b-1, b-2, b-1, b-1, b-1, b-1, b-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, b-1, b-1, b-1, b-1, b-1, b-1, b-1, b-1, b-1, b-1, b-1, b-1, b-2, b-1, b-1, b-1, b-1, b-1, b-1, b-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, …
The digits on the right of the terms of A027878 oscillate between two limiting sequences: ...000100009999998999000010099891 and ...999899990000001000999989900109
The terms for other bases (b > 2) behave similarly. The general patterns for the last terms appear to be:
…, 0, 0, 0, 0, 0, 0, 0, 0, b-1, b-1, b-1, b-1, b-1, b-1, b-1, b-1, b-1, b-1, b-1, b-1, b-1, b-1, b-2, b-1, b-1, b-1, b-1, b-1, b-1, b-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, b-1, b-1, b-1, b-1, b-1, b-1, b-1, b-1, b-1, b-1, b-2, b-1, b-1, b-1, b-1, b-1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, b-1, b-1, b-1, b-1, b-1, b-1, b-2, b-1, b-1, b-1, 0, 0, 0, 0, 1, 0, 0, b-1, b-1, b-2, b-1, 1
and
…, b-1, b-1, b-1, b-1, b-1, b-1, b-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, b-1, b-1, b-1, b-1, b-1, b-1, b-1, b-1, b-1, b-1, b-1, b-1, b-2, b-1, b-1, b-1, b-1, b-1, b-1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, b-1, b-1, b-1, b-1, b-1, b-1, b-1, b-1, b-2, b-1, b-1, b-1, b-1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, b-1, b-1, b-1, b-1, b-2, b-1, b-1, 0, 0, 1, 0, b-1
Kerry
On Wed, Jul 13, 2016 at 8:51 AM, Neil Sloane <njasloane@gmail.com> wrote:
The sequence in question is https://oeis.org/A027878. Please also add any useful comments to that entry!
Best regards Neil
Neil J. A. Sloane, President, OEIS Foundation. 11 South Adelaide Avenue, Highland Park, NJ 08904, USA. Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ. Phone: 732 828 6098; home page: http://NeilSloane.com Email: njasloane@gmail.com
On Wed, Jul 13, 2016 at 11:45 AM, James Propp <jamespropp@gmail.com> wrote:
Yes, absolutely. That's why the initial and final digits behave so nicely. I should've mentioned that.
Jim
On Wednesday, July 13, 2016, Joerg Arndt <arndt@jjj.de> wrote:
Note that product(n>=1, 1-10^{-n}) is essentially Dedekind's eta function and Euler's pentagonal number theorem (to name just one) may be helpful with spotting patterns in the expansion.
Best regards, jj
* James Propp <jamespropp@gmail.com <javascript:;>> [Jul 13. 2016 08:00]: > One side-alley that I glanced down when I was preparing my > current Mathematical Enchantments essay, but then resolutely marched > myself past, involves the behavior of the decimal expansion of the > product (10^1-1)(10^2-1)...(10^n-1); the initial and final digits are quite > orderly, while the digits in the middle look random. Has anyone looked into > this? It'd be especially interesting if one could establish some sort of > phase transition. > > I realize I'm using vocabulary loosely; there's no randomness involved, so > we're not really doing stat mech. Still, some of the concepts of stat mech > might apply. > > Jim Propp > _______________________________________________ > math-fun mailing list > math-fun@mailman.xmission.com <javascript:;> > https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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