Reminds me of the Pillai sequence http://oeis.org/A066352 which is the special case K = r = 1. The first cases I get for +_ are 1 1 2 176 3 352 4 2109 5 5832 6 28037 7 290789 8 290790 9 1255508 10 4325179 11 11135847 12 18567920 13 28794696 14 28794696 15 28794696
What is the first case with more than two terms altogether?
Pretty big, I'd wager. Charles Greathouse Analyst/Programmer Case Western Reserve University On Fri, Mar 29, 2013 at 10:04 PM, Dan Asimov <dasimov@earthlink.net> wrote:
Let N be a positive integer.
As long as N > 1, we can always find the largest integer of the form
K * p^r
with p prime, r >= 1, 1 <= K <= p-1
that is <= N.
Letting K*p^r be that largest such integer, now find the corresponding number for N - K*p^r, iteratively, until what is left is 0 or 1.
If it's 0, we're done, and if it's 1, just add 1 at the end.
In this way we can represent all positive integers as sums of such K*p^r with possibly a 1 tacked on at the end:
N = K_1*p_1^r_1 + . . . + K_d*p_d^r_d (+ 1).
For instance 46 = 4*11 + 2, 126 = 5^3 + 1, 144 = 11*13 + 1.
There seem to be interesting statistics lurking here, such as how often the representation ends with a 1 at the end, or how many terms are required.
Having checked up to only 150 (by hand), things I don't know yet include
* What is the 2nd case ending with + 2 (other than 2 itself) ? (46 is the first.)
* What is the first case ending with + 3 (other than 3) ?
* What is the first case having a term K*2^r with r >= 1 (other than powers of 2 themselves) ? (This is not counting terms of the form K*p^r where p > 2 and K is a power of 2, such as 108 = 4*3^3.)
* What is the first case with more than two terms altogether?
--Dan
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun