A123391 - OEIS

%I #19 Sep 29 2023 04:01:50

%S 0,0,0,2,0,0,0,3,2,0,0,2,0,0,0,0,0,2,0,2,0,0,0,3,2,0,3,2,0,0,0,5,0,0,

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

%U 0,0,0,5,0,0,2,2,0,0,0,0,0,0,0,2,0,0,0,3,0,2,0,2,0,0,0,5,0,2,2,4,0,0,0,3,0

%N a(n) = sum of exponents that are primes in the prime factorization of n.

%H Antti Karttunen, <a href="/A123391/b123391.txt">Table of n, a(n) for n = 1..10000</a>

%H <a href="/index/Eu#epf">Index entries for sequences computed from exponents in factorization of n</a>.

%F Additive with a(p^e) = A010051(e)*e. - _Antti Karttunen_, Jul 19 2017

%F Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{p prime} p*(P(p)-P(p+1)) = 0.97487020987790163735..., where P(s) is the prime zeta function. - _Amiram Eldar_, Sep 29 2023

%e 36 = 2^2*3^2. Both exponents in this prime factorization are primes. So a(36) = 2+2 = 4.

%t f[n_] := Plus @@ Select[Last /@ FactorInteger[n], PrimeQ];Table[f[n], {n, 120}] (* _Ray Chandler_, Nov 11 2006*)

%o (PARI) {m=105; for(n=1, m, v=factor(n)[,2]; s=0; for(j=1, #v, if(isprime(v[j]), s=s+v[j])); print1(s,","))} \\ _Klaus Brockhaus_, Nov 14 2006

%o (PARI) A123391(n) = vecsum(apply(e -> isprime(e)*e, factorint(n)[, 2])); \\ _Antti Karttunen_, Jul 19 2017

%Y Cf. A010051, A101436, A125030, A125071, A125073.

%K nonn,easy

%O 1,4

%A _Leroy Quet_, Nov 10 2006

%E Extended by _Ray Chandler_ and _Klaus Brockhaus_, Nov 11 2006