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.

2, 5, 5, 7, 7, 29, 37, 37, 37, 37, 37, 43, 43, 43, 53, 79, 101, 101, 101, 101, 101, 101, 101, 101, 131, 131, 131, 131, 131, 131, 131, 173, 173, 173, 173, 173, 173, 173, 173, 173, 173, 173, 173, 173, 173, 173, 173, 173, 173, 283, 317, 389, 389, 389, 389, 389, 389, 389, 389, 389

Offset: 1

Author

Zhi-Wei Sun, Oct 14 2015

Keywords

• nonn

Comments

Conjecture: a(n) exists for any n > 0, and a(n) < n^2 for all n > 2.

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.

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.

Examples

			a(1) = 2 since 2 is the least prime greater than sigma(1^2)/1^2 = 1.
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.
		

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 1..2000

Crossrefs

Cf. A000203, A000290, A001157, A065764.

Programs

Mathematica

rMod[m_,n_]:=rMod[m,n]=Mod[Numerator[m]*PowerMod[Denominator[m],-1,n],n,-n/2]
f[n_]:=f[n]=DivisorSigma[1,n^2]/n^2
Le[n_,m_]:=Le[m,n]=Length[Union[Table[rMod[f[k],Prime[m]],{k,1,n}]]]
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]]]

Extensions

Definition corrected by Omar E. Pol, Oct 24 2015