%I #27 Mar 09 2017 19:00:32
%S 1,2,4,6,9,15,16,21,24,25,27,28,30,33,35,39,42,48,51,54,55,57,64,65,
%T 66,69,70,77,78,85,87,90,91,93,95,100,102,105,110,111,112,114,115,119,
%U 120,123,125,129,130,133,135,138,141,143,145,154,155,159,161,165
%N Natural numbers n such that the number of primes of the form d(n)/x is equal to the number of primes of the form sigma(n)/y where x, y are divisors of n.
%C In this sequence there are no odd prime numbers, but there is even prime number 2.
%e 2 is in this sequence because d(2)/1 = 2 is prime and sigma(2)/1 = 3 is prime, d(2)/2 = 1 is no prime and sigma(2)/2 = 3/2 is no prime, where 1, 2 are divisors of 2.
%t okQ[n_] := Block[{d = Divisors[n], t0, t1=DivisorSigma[1, n]}, t0 = Length@ d; Length @Select[d, PrimeQ[ t0/#] &] == Length@ Select[d, PrimeQ[ t1/#] &]]; Select[Range@ 1000, okQ] (* _Giovanni Resta_, Mar 05 2017 *)
%o (PARI) is(n)=my(f=factor(n),d=numdiv(f),fd=factor(d)[,1],s=sigma(f),fs=factor(s)[,1]); sum(i=1,#fd,n%(d/fd[i])==0)==sum(i=1,#fs,n%(s/fs[i])==0) \\ _Charles R Greathouse IV_, Feb 27 2017
%Y Cf. A000005, A000203.
%K nonn
%O 1,2
%A _Juri-Stepan Gerasimov_, Feb 27 2017
%E a(7)-a(44) from _Charles R Greathouse IV_, Feb 27 2017
%E a(45)-a(60) from _Giovanni Resta_, Mar 05 2017