A227848 Numbers n such that Sum_{i=1..n} (i')^i == 0 (mod n), where i' is the arithmetic derivative of i.

1, 9, 71, 120, 331, 393, 728, 1223, 3697, 4123, 6791, 7391, 23911, 25099, 35287, 86442, 86716, 118034, 292411, 352970, 527255, 606425

Offset: 1

Author

Paolo P. Lava, Aug 01 2013

Keywords

• nonn
• more

Comments

a(19) > 200000. - Giovanni Resta, Aug 01 2013

Examples

			1'^1 + 2'^2 + 3'^3 + 4'^4 + 5'^5 + 6'^6 + 7'^7 + 8'^8 + 9'^9 = 0^1 + 1^2 + 1^3 + 4^4 + 1^5 + 5^6 + 1^7 + 12^8 + 6^9 = 440075277 and 440075277 / 9 = 48897253.
		

Crossrefs

Cf. A003415, A227502.

Programs

Maple

with(numtheory); ListA227848:=proc(q) local a, n, p;  a:=0;
for n from 1 to q do a:=a+(n*add(op(2, p)/op(1, p), p=ifactors(n)[2]))^n;
if a mod n=0 then print(n); fi; od; end: ListA227848(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[d[k]^k, {k, 1, n}], n] == 0, Print[n]; Sow[n]]]][[2, 1]] (* Jean-François Alcover, Feb 21 2014 *)

Extensions

a(16)-a(18) from Giovanni Resta, Aug 01 2013

a(19)-a(22) from Bert Dobbelaere, Dec 23 2018