A046870 Numbers k such that sigma_1(k) divides sigma_4(k).

1, 4, 9, 16, 20, 25, 36, 49, 50, 64, 81, 100, 117, 121, 144, 169, 180, 196, 225, 242, 256, 289, 324, 325, 361, 400, 441, 450, 468, 484, 500, 529, 576, 578, 605, 625, 650, 676, 729, 784, 800, 841, 900, 961, 980, 1024, 1025, 1058, 1089, 1156, 1225, 1280, 1296

Offset: 1

Author

Labos Elemer

Keywords

• nonn

Comments

sigma_1(n) is the sum of the divisors of n [same as sigma(n)] (A000203).

sigma_2(n) is the sum of the squares of the divisors of n (A001157).

sigma_4(n) is the sum of the 4th powers of the divisors of n (A001159).

Examples

			k = a(18) = 196 of which divisor power sums for k=0,1,2,3,4 are 9,399,51471, 8613489, 1574446419. sigma_1(k) = 399 and sigma_4(k) = 51471*30589=399*129*30589. Thus both sigma_2(k) and sigma_1(k) divide sigma_4(k).
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Paolo P. Lava)

Crossrefs

Cf. A000203, A001157, A001159.

Has large overlap with A020487.

Programs

Mathematica

Select[Range[1300], Divisible @@ DivisorSigma[{4, 1}, #] &] (* Amiram Eldar, Jun 15 2024 *)

PARI

is(k) = {my(f = factor(k)); !(sigma(f, 4) % sigma(f)); } \\ Amiram Eldar, Jun 15 2024