A227427 Numbers k such that Sum_{i=1..k} i^sigma(i) == 0 (mod k).

1, 3, 17, 64, 223, 816, 3536, 22704, 27549, 401311

Offset: 1

Author

Paolo P. Lava, Jul 11 2013

Keywords

• nonn
• more

Examples

			1^sigma(1) + 2^sigma(2) + ... + 16^sigma(16) + 17^sigma(17) =
1^1 + 2^3 + 3^4 + 4^7 + 5^6 + 6^12 + 7^8 + 8^15 + 9^13 + 10^18 + 11^12 + 12^28 + 13^14 + 14^24 + 15^24 + 16^31 + 17^18 = 21267649601053603536507860737702745369 and 21267649601053603536507860737702745369 / 17 = 1251038211826682560971050631629573257.
		

Crossrefs

Cf. A000203, A227429.

Programs

Maple

with(numtheory); ListA227427:=proc(q) local i,n;
for n from 1 to q do if add(i^sigma(i),i=1..n) mod n=0 then print(n);
fi; od; end: ListA227427(10^6);

Mathematica

With[{nn=40000},Transpose[Select[Thread[{Accumulate[Table[n^DivisorSigma[ 1,n],{n,nn}]],Range[nn]}],Divisible[#[[1]],#[[2]]]&]][[2]]] (* Harvey P. Dale, Jun 08 2016 *)

PARI

isok(k) = sum(i=1, k, Mod(i,k)^sigma(i)) == 0; \\ Michel Marcus, Feb 18 2021

Extensions

a(7)-a(9) from Giovanni Resta, Jul 11 2013

a(10) from Jinyuan Wang, Feb 18 2021