%I #18 Jul 01 2022 05:26:02
%S 1,2,14,32,39,42,78,96,105,117,126,133,189,195,210,224,234,266,288,
%T 378,390,399,465,480,546,585,672,793,798,930,975,1170,1197,1248,1365,
%U 1470,1586,1638,1862,1950,1995,2016,2379,2394,2646,2730,3255,3360,3393,3591
%N Numbers k such that Sum_{d|k} sigma(d)^3 is a multiple of k.
%H Amiram Eldar, <a href="/A226565/b226565.txt">Table of n, a(n) for n = 1..5000</a> (terms 1..100 from Paolo P. Lava)
%e 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.
%p with(numtheory); ListA226565:=proc(q) local a,b,k,n;
%p for n from 1 to q do a:=[op(divisors(n))]; b:=add(sigma(a[k])^3/n,k=1..nops(a));
%p if type(b,integer) then print(n); fi; od; end: ListA226565 (10^6);
%t Select[Range[4000],Divisible[Total[DivisorSigma[1,#]^3&/@Divisors[#]],#]&] (* _Harvey P. Dale_, Sep 17 2019 *)
%t s[n_] := DivisorSum[n, DivisorSigma[1, #]^3 &]; Select[Range[3600], Divisible[s[#], #] &] (* _Amiram Eldar_, Jul 01 2022 *)
%Y Cf. A068978, A068986, A226563, A226564, A226566, A344044.
%K nonn
%O 1,2
%A _Paolo P. Lava_, Jun 11 2013