A272981 Least prime k>1 such that the sum of divisors of powers k^e, 1 <= e <= n, are divisible by the number their divisors, d(k^e).

3, 7, 7, 31, 31, 211, 211, 211, 211, 2311, 2311, 120121, 120121, 120121, 120121, 4084081, 4084081, 106696591, 106696591, 106696591, 106696591, 892371481, 892371481, 892371481, 892371481, 892371481, 892371481, 71166625531, 71166625531, 200560490131, 200560490131

Offset: 1

Author

Paolo P. Lava, May 12 2016

Keywords

• nonn

Comments

For 1A272981(n) = A092967(n+1).

The different numbers are listed in A073917.

Examples

			sigma(3) / d(3) = 4 / 2 = 2 but sigma(3^2) / d(3^2) = 13 / 3;
sigma(7) / d(7) = 8 / 2 = 4, sigma(7^2) / d(7^2) = 57 / 3 = 19, sigma(7^3) / d(7^3) = 400 / 4 = 100 but sigma(7^4) / d(7^4) = 2801 / 5.
		

Links

• Paolo P. Lava, Table of n, a(n) for n = 1..100

Crossrefs

Cf. A000005, A000203, A073917, A092967, A272980.

Programs

Maple

with(numtheory): P:= proc(q) local a,j,k,ok,p; global n; a:=2;
for k from 1 to q do for n from a to q do ok:=1;
for j from 1 to k do if not type(sigma(n^j)/tau(n^j),integer) then ok:=0; break; fi; od;
if ok=1 then a:=n; print(n); break; fi; od; od; end: P(10^9);

Mathematica

Table[SelectFirst[Range[2, 10^6], AllTrue[#^Range@ n, Divisible[DivisorSigma[1, #], DivisorSigma[0, #]] &] &], {n, 15}] (* Michael De Vlieger, May 12 2016, Version 10 *)