A275123 Even numbers n such that sigma(n) divides sigma(n^n).

4, 16, 64, 100, 196, 484, 676, 1024, 1156, 1296, 1444, 1936, 2116, 3364, 3844, 4096, 4900, 5476, 5776, 6400, 6724, 7396, 8836, 10816, 11236, 12100, 13456, 13924, 14884, 15376, 16900, 17956, 20164, 21316, 23716, 24964, 26896, 27556, 28900, 31684, 33124, 36100

Offset: 1

Author

Altug Alkan, Jul 18 2016

Keywords

• nonn

Comments

A number n with prime factorization Product_i p_i^(e_i) is in the sequence iff Product_i ((p_i^(e_i*n+1)-1)/(p_i^(e_i+1)-1)) is an integer. - Robert Israel, Jul 19 2016

Does this sequence consist of the even numbers n such that A000005(n) divides A000005(n^n)? The answer is no according to the b-file since 50176 is missing (((2^(10*50176+1)-1)*(7^(2*50176+1)-1)) mod ((2^11-1)*(7^3-1)) = 372438 and (10*50176+1)*(2*50176+1) mod (11*3) = 0). Note that 50176 is the least number with this property.

Examples

			4 is a term because sigma(4^4) = 511 is divisible by sigma(4) = 7.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..500

Crossrefs

Cf. A000203, A062727.

Programs

Maple

filter:= proc(n) local F,t,b,r;
  F:= ifactors(n)[2];
  b:= mul(t[1]^(t[2]+1)-1, t=F);
  r:= 1;
  for t in F do r:= r * (t[1] &^ (t[2]*n+1)-1) mod b od;
  r = 0;
end proc:
select(filter, [seq(i,i=2..10^5,2)]); # Robert Israel, Jul 19 2016

Mathematica

Select[Range[2, 10^4, 2], Divisible[DivisorSigma[1, #^#], DivisorSigma[1, #]] &] (* Michael De Vlieger, Jul 19 2016 *)

PARI

/* Requires a large PARI stack to return even the first few terms */
is(n) = Mod(n, 2)==0 && Mod(sigma(n^n), sigma(n))==0 \\ Felix Fröhlich, Jul 19 2016

Extensions

a(8)-a(22) from Michel Marcus, Jul 19 2016

More terms from Robert Israel, Jul 19 2016