cp's OEIS Frontend

A107925 Odd numbers k such that k^2 is an arithmetic number.

%I A107925 #19 Apr 26 2022 05:30:03
%S A107925 1,7,13,19,31,37,43,61,67,73,79,91,97,103,109,121,127,133,139,151,157,
%T A107925 163,181,193,199,211,217,223,229,241,247,259,271,277,283,301,307,313,
%U A107925 331,337,349,367,373,379,397,403,409,421,427,433,439,457,463,469,481
%N A107925 Odd numbers k such that k^2 is an arithmetic number.
%C A107925 Also, odd numbers k such that the arithmetic and geometric means of the divisors of k^2 are both integer.
%C A107925 Even numbers with this property are more rare and given by A107924.
%C A107925 Contains A002476 as a subsequence.
%C A107925 From _Jianing Song_, Apr 25 2022: (Start)
%C A107925 For p prime, p^(k-1) is a term in A003601 if and only if (p^k-1)/(p-1) is divisible by k. So p^e is a term here if and only if k | (p^k-1)/(p-1) for k = 2*e+1. (Note that p cannot be equal to 2 if k | (p^k-1)/(p-1).)
%C A107925 If a,b are both here, gcd(a,b) = 1, then a*b is also here. If a is A107924 and b is here, gcd(a,b) = 1, then a*b is also in A107924.
%C A107925 Let r >= 1, p_1, p_2, ..., p_r be distinct primes, k_1, k_2, ..., k_r be odd numbers such that Product_{i=1..r} (p_i)^(k_i) is an arithmetic number. Then there exists a number i in 1..r such that (p_i)^(k_i) is an arithmetic number. See my link for a proof. (End)
%H A107925 Amiram Eldar, <a href="/A107925/b107925.txt">Table of n, a(n) for n = 1..10000</a>
%H A107925 Jianing Song, <a href="/A107925/a107925.pdf">Notes on A107925</a>
%t A107925 Select[Range[1, 500, 2], Mod[DivisorSigma[1, #^2], DivisorSigma[0, #^2]]==0&]
%Y A107925 Cf. A002476, A003601, A107924.
%K A107925 easy,nonn
%O A107925 1,2
%A A107925 _Zak Seidov_, Jun 10 2005