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

1,2

Links

Charles R Greathouse IV, Table of n, a(n) for n = 1..200

Paolo P. Lava, First 30 terms with k value

Formula

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

Example

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.

Maple

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

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 *)

Prog

(PARI) is(n)=my(e=factor(n)[, 2], k, t); while(t<n, k++; t += prod(i=1, #e, k*e[i]+1)); t==n \\ Charles R Greathouse IV, Mar 31 2016

Crossrefs

Cf. A000005, A270713.

Sequence in context: A012949 A027667 A076630 * A350956 A086560 A305292

Adjacent sequences: A270386 A270387 A270388 * A270390 A270391 A270392

Keyword

nonn

Author

Paolo P. Lava, Mar 16 2016

Status

approved