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.

1, 91, 133, 217, 247, 259, 296, 301, 403, 427, 469, 481, 511, 536, 553, 559, 589, 632, 679, 703, 721, 763, 793, 817, 847, 871, 872, 889, 949, 973, 999, 1027, 1057, 1099, 1141, 1147, 1159, 1208, 1261, 1267, 1273, 1304, 1333, 1339, 1351, 1387, 1393, 1417, 1477

Offset: 1

Author

Jianing Song, Apr 20 2022

Keywords

• nonn

Comments

Here prime powers means the numbers in A246655.

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)}.

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.

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.

Examples

			91 is a term since sigma(91^2) = 10431 is divisible by d(91^2) = 9.
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.
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.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Equals (A107924 U A107925) \ A246655. The even terms are listed in A107924.

Cf. A003601, A000005, A000203.

Programs

Mathematica

Select[Range[1500], !PrimePowerQ[#] && Divisible @@ DivisorSigma[{1, 0}, #^2] &] (* Amiram Eldar, Jul 19 2024 *)

PARI

isA353062(n) = sigma(n^2)%numdiv(n^2)==0 && !isprimepower(n)