A333842 - OEIS

%I #17 Jul 25 2024 03:11:58

%S 0,0,0,1,0,0,0,1,2,0,0,1,0,0,0,1,0,2,0,1,0,0,0,1,3,0,2,1,0,0,0,1,0,0,

%T 0,3,0,0,0,1,0,0,0,1,2,0,0,1,4,3,0,1,0,2,0,1,0,0,0,1,0,0,2,1,0,0,0,1,

%U 0,0,0,3,0,0,3,1,0,0,0,1,2,0,0,1,0,0,0,1,0,2,0,1,0,0,0,1,0,4,2,4,0,0,0,1

%N G.f.: Sum_{k>=1} k * x^(prime(k)^2) / (1 - x^(prime(k)^2)).

%C Sum of indices of non-unitary prime factors of n (prime factors for which the exponent exceeds 1).

%H Antti Karttunen, <a href="/A333842/b333842.txt">Table of n, a(n) for n = 1..65537</a>

%H <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>.

%F a(n) = A056239(A071773(n)) = A066328(A003557(n)). - _Peter Munn_ and _Antti Karttunen_, Jun 13 2020

%F Additive with a(p^e) = primepi(p) = A000720(p) if e >= 2, and 0 otherwise. - _Amiram Eldar_, Jul 24 2024

%e a(450) = a(2 * 3^2 * 5^2) = a(prime(1) * prime(2)^2 * prime(3)^2) = 2 + 3 = 5.

%t nmax = 104; CoefficientList[Series[Sum[k x^(Prime[k]^2)/(1 - x^(Prime[k]^2)), {k, 1, nmax}], {x,0, nmax}], x] // Rest

%t f[p_, e_] := If[e == 1, 0, PrimePi[p]]; a[1] = 0; a[n_] := Plus @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Jul 24 2024 *)

%o (PARI) A333842(n) = { my(f=factor(n)); sum(k=1, #f~, if(1==f[k, 2],0,1)*primepi(f[k, 1])); }; \\ _Antti Karttunen_, Jun 12 2020

%Y Cf. A000720, A003557, A005117 (positions of 0's), A056170, A056239, A063958, A066328, A071773.

%K nonn

%O 1,9

%A _Ilya Gutkovskiy_, Apr 07 2020