cp's OEIS Frontend

A353062 Non-prime-powers k such that sigma(k^2) is divisible by d(k^2), where d = A000005, sigma = A000203; non-prime-powers k such that k^2 is in A003601.

%I A353062 #18 Jul 19 2024 08:47:19
%S A353062 1,91,133,217,247,259,296,301,403,427,469,481,511,536,553,559,589,632,
%T A353062 679,703,721,763,793,817,847,871,872,889,949,973,999,1027,1057,1099,
%U A353062 1141,1147,1159,1208,1261,1267,1273,1304,1333,1339,1351,1387,1393,1417,1477
%N A353062 Non-prime-powers k such that sigma(k^2) is divisible by d(k^2), where d = A000005, sigma = A000203; non-prime-powers k such that k^2 is in A003601.
%C A353062 Here prime powers means the numbers in A246655.
%C A353062 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 this sequence is (A107924 U A107925) \ {p^((k-1)/2): p prime, k odd, k | (p^k-1)/(p-1)}.
%C A353062 It is standard that k does not divide 2^k-1 for k > 1, so no term > 1 in A003601 can be a power of 2, hence A107924 is a subsequence.
%C A353062 Since a,b in A003601 (resp. A107924 U A107925) and gcd(a,b) = 1 implies that a*b is in A003601 (resp. A107924 U A107925), this sequence is infinite. For example, all numbers of the form (p_1)*(p_2)*...*(p_k) are here, where p_i's are distinct primes congruent to 1 modulo 3, k >= 2.
%H A353062 Amiram Eldar, <a href="/A353062/b353062.txt">Table of n, a(n) for n = 1..10000</a>
%e A353062 91 is a term since sigma(91^2) = 10431 is divisible by d(91^2) = 9.
%e A353062 296 is a term since sigma(296^2) = 178689 is divisible by d(296^2) = 21. 296 is the smallest term that is not a product of coprime numbers > 1 in A107924 U A107925.
%e A353062 999 is a term since sigma(999^2) = 1537851 is divisible by d(999^2) = 21. 999 is the smallest odd term that is not a product of coprime numbers > 1 in A107924 U A107925.
%t A353062 Select[Range[1500], !PrimePowerQ[#] && Divisible @@ DivisorSigma[{1, 0}, #^2] &] (* _Amiram Eldar_, Jul 19 2024 *)
%o A353062 (PARI) isA353062(n) = sigma(n^2)%numdiv(n^2)==0 && !isprimepower(n)
%Y A353062 Equals (A107924 U A107925) \ A246655. The even terms are listed in A107924.
%Y A353062 Cf. A003601, A000005, A000203.
%K A353062 nonn
%O A353062 1,2
%A A353062 _Jianing Song_, Apr 20 2022