cp's OEIS Frontend

A263317 Least prime p > n such that the numbers sigma(k^2)/k^2 (k = 1,...,n) are pairwise incongruent modulo p, where sigma(m) is the sum of the divisors of m.

%I A263317 #21 Nov 25 2015 02:33:52
%S A263317 2,5,5,7,7,29,37,37,37,37,37,43,43,43,53,79,101,101,101,101,101,101,
%T A263317 101,101,131,131,131,131,131,131,131,173,173,173,173,173,173,173,173,
%U A263317 173,173,173,173,173,173,173,173,173,173,283,317,389,389,389,389,389,389,389,389,389
%N A263317 Least prime p > n such that the numbers sigma(k^2)/k^2 (k = 1,...,n) are pairwise incongruent modulo p, where sigma(m) is the sum of the divisors of m.
%C A263317 Conjecture: a(n) exists for any n > 0, and a(n) < n^2 for all n > 2.
%C A263317 This implies that all the rational numbers sigma(n^2)/n^2 = Sum_{d|n^2} 1/d (n = 1,2,3,...) are pairwise distinct. We have verified that the numbers sigma(n^2)/n^2 (n = 1..10^5) are indeed pairwise distinct, and noted that sigma(26334^2)/26334^2 - sigma(6^2)/6^2 = 127/36 - 91/36 = 1.
%C A263317 We guess that for each k = 2,3,... all the numbers sigma(n^k)/n^k = Sum_{d|n^k} 1/d (n = 1,2,3,...) are pairwise distinct. See also A001157 for a similar conjecture.
%H A263317 Zhi-Wei Sun, <a href="/A263317/b263317.txt">Table of n, a(n) for n = 1..2000</a>
%e A263317 a(1) = 2 since 2 is the least prime greater than sigma(1^2)/1^2 = 1.
%e A263317 a(2) = 5 since sigma(1^2)/1^2 = 1 and sigma(2^2)/2^2 = 7/4 are incongruent modulo the prime 5 > 2, but 1 is congruent to 7/4 modulo the prime 3.
%t A263317 rMod[m_,n_]:=rMod[m,n]=Mod[Numerator[m]*PowerMod[Denominator[m],-1,n],n,-n/2]
%t A263317 f[n_]:=f[n]=DivisorSigma[1,n^2]/n^2
%t A263317 Le[n_,m_]:=Le[m,n]=Length[Union[Table[rMod[f[k],Prime[m]],{k,1,n}]]]
%t A263317 Do[n=1;m=1;Label[aa];If[m>PrimePi[n]&&Le[n,m]==n,Goto[bb],m=m+1;Goto[aa]];Label[bb];Print[n," ",Prime[m]];If[n<60,n=n+1;Goto[aa]]]
%Y A263317 Cf. A000203, A000290, A001157, A065764.
%K A263317 nonn
%O A263317 1,1
%A A263317 _Zhi-Wei Sun_, Oct 14 2015
%E A263317 Definition corrected by _Omar E. Pol_, Oct 24 2015