A227502 Numbers n such that Sum_{i=1..n} i^(i') == 0 (mod n), where i' is the arithmetic derivative of i.

1, 3, 7, 19, 32, 57, 99, 103, 439, 540, 2656, 18156, 179171, 235056

Offset: 1

Author

Paolo P. Lava, Jul 13 2013

Keywords

• nonn
• more

Comments

a(14) > 200000. - Giovanni Resta, Jul 15 2013

a(15) > 1000000. - Bert Dobbelaere, Dec 24 2018

Examples

			1^1' + 2^2' + 3^3' = 1^0 + 2^1 + 3^1 = 6 and 6 == 0 (mod 3).
		

Crossrefs

Cf. A003415, A227848.

Programs

Maple

with(numtheory); ListA227502:=proc(q)  local a,n,p;  a:=0;
for n from 1 to q do a:=a+n^(n*add(op(2,p)/op(1,p),p=ifactors(n)[2]));
if a mod n=0 then print(n);  fi; od; end: ListA227502(10^6);

Mathematica

d[n_] := d[n] = n*Total[Apply[#2/#1 &, FactorInteger[n], {1}]]; Reap[For[n = 1, n <= 2*10^5, n++, If[Mod[Sum[k^d[k], {k, 1, n}], n] == 0, Print[n]; Sow[n]]]][[2, 1]] (* Jean-François Alcover, Feb 21 2014 *)

Extensions

a(13) from Giovanni Resta, Jul 15 2013

a(14) from Bert Dobbelaere, Dec 24 2018