A125029 - OEIS

%I #22 Sep 30 2023 21:55:30

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

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

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

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

%H Antti Karttunen, <a href="/A125029/b125029.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 From _Amiram Eldar_, Sep 30 2023: (Start)

%F Additive with a(p^e) = A080339(e).

%F Sum_{k=1..n} a(k) ~ n * (log(log(n)) + B - C), where B is Mertens's constant (A077761) and C = P(3) - Sum_{p prime >= 3} (P(p) - P(p+1)) = 0.05377157198303445809..., where P(s) is the prime zeta function. (End)

%e a(720) = 2, since the prime factorization of 720 is 2^4 * 3^2 * 5^1 and two of the exponents in this factorization are noncomposites (the exponents 2 and 1).

%t f[n_] := Length @ Select[Last /@ FactorInteger[n], # == 1 || PrimeQ[ # ] &];Table[f[n], {n, 110}] (* _Ray Chandler_, Nov 19 2006 *)

%o (PARI) A125029(n) = vecsum(apply(e -> if((1==e)||isprime(e),1,0), factorint(n)[, 2])); \\ _Antti Karttunen_, Jul 07 2017

%Y Cf. A001221, A077761, A080339, A125030, A125070.

%K nonn,easy

%O 1,6

%A _Leroy Quet_, Nov 16 2006

%E Extended by _Ray Chandler_, Nov 19 2006