cp's OEIS Frontend

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

%I A275123 #42 Mar 07 2025 04:15:17
%S A275123 4,16,64,100,196,484,676,1024,1156,1296,1444,1936,2116,3364,3844,4096,
%T A275123 4900,5476,5776,6400,6724,7396,8836,10816,11236,12100,13456,13924,
%U A275123 14884,15376,16900,17956,20164,21316,23716,24964,26896,27556,28900,31684,33124,36100
%N A275123 Even numbers n such that sigma(n) divides sigma(n^n).
%C A275123 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
%C A275123 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.
%H A275123 Robert Israel, <a href="/A275123/b275123.txt">Table of n, a(n) for n = 1..500</a>
%e A275123 4 is a term because sigma(4^4) = 511 is divisible by sigma(4) = 7.
%p A275123 filter:= proc(n) local F,t,b,r;
%p A275123   F:= ifactors(n)[2];
%p A275123   b:= mul(t[1]^(t[2]+1)-1, t=F);
%p A275123   r:= 1;
%p A275123   for t in F do r:= r * (t[1] &^ (t[2]*n+1)-1) mod b od;
%p A275123   r = 0;
%p A275123 end proc:
%p A275123 select(filter, [seq(i,i=2..10^5,2)]); # _Robert Israel_, Jul 19 2016
%t A275123 Select[Range[2, 10^4, 2], Divisible[DivisorSigma[1, #^#], DivisorSigma[1, #]] &] (* _Michael De Vlieger_, Jul 19 2016 *)
%o A275123 (PARI) /* Requires a large PARI stack to return even the first few terms */
%o A275123 is(n) = Mod(n, 2)==0 && Mod(sigma(n^n), sigma(n))==0 \\ _Felix Fröhlich_, Jul 19 2016
%Y A275123 Cf. A000203, A062727.
%K A275123 nonn
%O A275123 1,1
%A A275123 _Altug Alkan_, Jul 18 2016
%E A275123 a(8)-a(22) from _Michel Marcus_, Jul 19 2016
%E A275123 More terms from _Robert Israel_, Jul 19 2016