A270389 - OEIS

%I #27 Jun 09 2017 09:19:56

%S 1,2,5,64,203,505,524,649,818,1295,2469,2869,4355,5048,6083,10415,

%T 14909,15021,22329,27433,29189,29369,35719,38023,44099,48229,56372,

%U 85329,85343,89270

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

%H Charles R Greathouse IV, <a href="/A270389/b270389.txt">Table of n, a(n) for n = 1..200</a>

%H Paolo P. Lava, <a href="/A270389/a270389.txt">First 30 terms with k value</a>

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

%e d(1^1) = 1;

%e d(2^1) = 2;

%e d(5^1) + d(5^2) = 2 + 3 = 5;

%e d(64^1) + d(64^2) + d(64^3) + d(64^4) = 7 + 13 + 19 + 25 = 64;

%e 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.

%p with(numtheory): P:=proc(q) local a,k,n;

%p for n from 1 to q do a:=0; k:=0;

%p while a<n do k:=k+1; a:=a+tau(n^k); od; if n=a then print(n); fi;

%p od; end: P(10^6);

%t 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 *)

%o (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

%Y Cf. A000005, A270713.

%K nonn

%O 1,2

%A _Paolo P. Lava_, Mar 16 2016