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

1,1

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, ...}.

Links

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

Example

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.

Mathematica

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

Prog

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

Crossrefs

Cf. A001358, A025475, A076290, A164722.

Sequence in context: A088949 A134600 A134601 * A319163 A134611 A134612

Adjacent sequences: A227473 A227474 A227475 * A227477 A227478 A227479

Keyword

nonn

Author

Michel Lagneau, Jul 13 2013

Extensions

Definition corrected by Michel Marcus, Sep 16 2017

Status

approved