A227476 Numbers whose sum of semiprime divisors (A076290) is a positive square.

4, 8, 9, 16, 25, 27, 32, 49, 64, 81, 121, 125, 128, 138, 169, 225, 243, 256, 289, 306, 343, 361, 426, 512, 516, 529, 625, 644, 675, 729, 841, 918, 961, 975, 1002, 1024, 1032, 1125, 1140, 1146, 1150, 1220, 1230, 1288, 1305, 1316, 1331, 1369, 1681, 1849, 2025

Offset: 1

Author

Michel Lagneau, Jul 13 2013

Keywords

• nonn

Comments

Except for the number 1, the sequence A195942 (Zeroless prime powers (excluding primes)) is a subsequence of this sequence because the set of divisors of the numbers of the form p^m with p prime and m >= 2 contains only one semiprime divisor, p^2.

The subset of the nonprime powers is {138, 225, 306, 426, 516, 644, 675, 918, ...}.

Examples

			138 is in the sequence because the divisors of 138 are {1, 2, 3, 6, 23, 46, 69, 138} and the sum of the semiprime divisors is 2*3 + 2*23 + 3*23 = 11^2.
		

Links

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

Crossrefs

Cf. A001358, A025475, A076290, A164722.

Programs

Mathematica

semipSigma[n_] := DivisorSum[n, # &, PrimeOmega[#] == 2 &]; Select[Range[2000], (s = semipSigma[#]) > 0 && IntegerQ @ Sqrt[s] &] (* Amiram Eldar, May 10 2020 *)

PARI

isok(n) = issquare(s = sumdiv(n, d, d*(bigomega(d)==2))) && (s>0); \\ Michel Marcus, Sep 16 2017

Extensions

Definition corrected by Michel Marcus, Sep 16 2017