A226565 Numbers k such that Sum_{d|k} sigma(d)^3 is a multiple of k.

1, 2, 14, 32, 39, 42, 78, 96, 105, 117, 126, 133, 189, 195, 210, 224, 234, 266, 288, 378, 390, 399, 465, 480, 546, 585, 672, 793, 798, 930, 975, 1170, 1197, 1248, 1365, 1470, 1586, 1638, 1862, 1950, 1995, 2016, 2379, 2394, 2646, 2730, 3255, 3360, 3393, 3591

Offset: 1

Author

Paolo P. Lava, Jun 11 2013

Keywords

• nonn

Examples

			Divisors of 189 are 1, 3, 7, 9, 21, 27, 63, 189, sigma(1) = 1, sigma(3) = 4, sigma(7) = 8, sigma(9) = 13, sigma(21) = 32, sigma(27) = 40, sigma(63) = 104, sigma(189) = 320. (1^3 + 4^3 + 8^3 + 13^3 + 32^3 + 40^3 + 104^3 + 320^3) / 189 = 179854.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..5000 (terms 1..100 from Paolo P. Lava)

Crossrefs

Cf. A068978, A068986, A226563, A226564, A226566, A344044.

Programs

Maple

with(numtheory); ListA226565:=proc(q) local a,b,k,n;
for n from 1 to q do a:=[op(divisors(n))]; b:=add(sigma(a[k])^3/n,k=1..nops(a));
if type(b,integer) then print(n); fi; od; end: ListA226565 (10^6);

Mathematica

Select[Range[4000],Divisible[Total[DivisorSigma[1,#]^3&/@Divisors[#]],#]&] (* Harvey P. Dale, Sep 17 2019 *)
s[n_] := DivisorSum[n, DivisorSigma[1, #]^3 &]; Select[Range[3600], Divisible[s[#], #] &] (* Amiram Eldar, Jul 01 2022 *)