-----Original Message----- From: math-fun-bounces@mailman.xmission.com [mailto:math-fun- bounces@mailman.xmission.com] On Behalf Of Neil Sloane Sent: Friday, February 15, 2013 2:05 PM To: math-fun Subject: Re: [math-fun] Sums of two cubes
Allan, last month you said:
After staring at these numbers for a couple of days, I have come up with a hard question related to this spooky para-multiplicativity. For any k, let Ck be the set of integers n such that kn is the sum of two squares. C1, for instance, is A045980 itself. C2 is {1, 4, 8, 13, 14 ...}, which is not in OEIS. (In fact, none of the Ck's I looked at were in OEIS except C1.) Here is my question: do there exist *any* two integers j and k such that Cj = Ck?
[David Wilson] I presume you mean distinct integers j, k >= 1. Define f(j, k) to be the smallest integer in exactly one of Cj and Ck. If Cj = Ck, f(j, k) is undefined. Empirically (testing j, k <= 300), it looks as if f(j, k) is always defined and appears to satisfy f(j, k) <= (j+k)^2/4.