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