A260654 Numbers k such that Sum_{i=1..k} sigma(i)^d(i) == 0 (mod k), where sigma = A000203 and d = A000005.

1, 2, 5, 56, 59, 60, 75, 122, 743, 2835, 3951, 5712, 6866, 7884, 14754, 18751, 292123, 465289, 1921892, 3902477, 7609760, 21855984, 22013406, 358753359, 570535294, 582046711, 1846338478, 13691385818

Offset: 1

Author

Paolo P. Lava, Nov 13 2015

Keywords

• nonn
• more

Examples

			sigma(1)^tau(1) + sigma(2)^tau(2) + sigma(3)^tau(3) + sigma(4)^tau(4) + sigma(5)^tau(5) = 1^1 + 3^2 + 4^2 + 7^3 + 6^2 = 1 + 9 + 16 + 343 + 36 = 405 and 405 / 5 = 81.
		

Crossrefs

Cf. A000005, A000203, A227427, A227429, A227502, A227848, A229095, A229207, A229208, A229209, A229210, A229211.

Programs

Maple

with(numtheory): P:=proc(q) local a,n; a:=0;
for n from 1 to q do a:=a+sigma(n)^tau(n);
if a mod n=0 then print(n); fi; od; end: P(10^6);

PARI

for(n=1, 1e4, if(sum(k=1, n, sigma(k)^numdiv(k))%n==0, print1(n", "))) \\ Altug Alkan, Nov 13 2015

PARI

list(lim) = {my(s = 0, f); for(k = 1, lim, f = factor(k); s += sigma(f)^numdiv(f); if(!(s % k), print1(k, ", ")));} \\ Amiram Eldar, Dec 29 2024

Extensions

Incorrect terms removed by and more terms from Jinyuan Wang, Feb 18 2021

a(24)-a(28) from Amiram Eldar, Dec 29 2024