The quasi-periodic A094756

[Don Reble]

Seqfans:

I found this one interesting.

%I A094756
%S A094756 2,4,2,7,2,4,2,7,2,4,2,16,2,4,2,7,2,4,2,7,2,4,2,16,2,4,2,7,2,
%T A094756 4,2,7,2,4,2,16,2,4,2,7,2,4,2,7,2,4,2,22,2,4,2,7,2,4,2,7,2,4,
%U A094756 2,16,2,4,2,7,2,4,2,7,2,4,2,16,2,4,2,7,2,4,2,7,2,4,2,16,2,4,2
%N A094756 a(n) = least k>1 such that (1+2+3+...+k) divides (1^n+2^n+3^n +...+k^n).
%C A094756 Conjecture: No entry is zero.
%Y A094756 Cf. A094755.
%K A094756 nonn,new
%O A094756 1,1
%A A094756 Amarnath Murthy (amarnath_murthy(AT)yahoo.com), May 29 2004

You may have discerned a periodic tendency in the above. It turns out
that the first part of the sequence can be described this way:

           If N is not divisible by 2, a(N) = 2.
Otherwise, if N is not divisible by 4, a(N) = 4.
Otherwise, if N is not divisible by 12, a(N) = 7.
Otherwise, if N is not divisible by 48, a(N) = 16.
Otherwise, if N is not divisible by 240, a(N) = 22 or 31.
    (31 if N is divisible by 528=11*48; otherwise 22).
Otherwise, if N is not divisible by 720, a(N) = 37.
Otherwise, if N is not divisible by 11 nor 23, a(N) = 46.
Otherwise, if N is not divisible by 77, a(N) = 58.
Otherwise, if N is not divisible by 13 nor 53, a(N) = 106.
Otherwise, if N is not divisible by 13, a(N) = 157.
Otherwise, if N is not divisible by 41 nor 83, a(N) = 166.
    ...

That works for N < 120,000,000 or so. But I don't expect that any finite
description of that kind is complete.

--
Don Reble       djr at nk.ca