A270389 Numbers that are equal to the sum of the number of divisors of their k first powers, for some k.

1, 2, 5, 64, 203, 505, 524, 649, 818, 1295, 2469, 2869, 4355, 5048, 6083, 10415, 14909, 15021, 22329, 27433, 29189, 29369, 35719, 38023, 44099, 48229, 56372, 85329, 85343, 89270

Offset: 1

Author

Paolo P. Lava, Mar 16 2016

Keywords

• nonn

Examples

			d(1^1) = 1;
d(2^1) = 2;
d(5^1) + d(5^2) = 2 + 3 = 5;
d(64^1) + d(64^2) + d(64^3) + d(64^4) = 7 + 13 + 19 + 25 = 64;
d(203^1) + d(203^2) + d(203^3)+ d(203^4)+ d(203^5)+ d(203^6)+ d(203^7) = 4 + 9 + 16 + 25 + 36 + 49 + 64 = 203.
		

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 1..200
• Paolo P. Lava, First 30 terms with k value

Crossrefs

Cf. A000005, A270713.

Programs

Maple

with(numtheory): P:=proc(q) local a,k,n;
for n from 1 to q do a:=0; k:=0;
while a
				

Mathematica

Select[Range[10^4], Function[n, IntegerQ@ SelectFirst[Range@ 25, Total@ Map[DivisorSigma[0, #] &, n^Range[#]] == n &]]] (* Michael De Vlieger, Mar 17 2016, Version 10 *)

PARI

is(n)=my(e=factor(n)[,2],k,t); while(tCharles R Greathouse IV, Mar 31 2016

Formula

Solutions of the equation n = Sum_{i=1..k}{d(n^k)}.