login
A125029
a(n) = number of exponents in the prime factorization of n that are noncomposite.
4
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, 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, 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
OFFSET
1,6
FORMULA
From Amiram Eldar, Sep 30 2023: (Start)
Additive with a(p^e) = A080339(e).
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)
EXAMPLE
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).
MATHEMATICA
f[n_] := Length @ Select[Last /@ FactorInteger[n], # == 1 || PrimeQ[ # ] &]; Table[f[n], {n, 110}] (* Ray Chandler, Nov 19 2006 *)
PROG
(PARI) A125029(n) = vecsum(apply(e -> if((1==e)||isprime(e), 1, 0), factorint(n)[, 2])); \\ Antti Karttunen, Jul 07 2017
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Leroy Quet, Nov 16 2006
EXTENSIONS
Extended by Ray Chandler, Nov 19 2006
STATUS
approved