A125029 - OEIS

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A125029

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

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

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OFFSET

1,6

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..10000

Index entries for sequences computed from exponents in factorization of n.

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

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

Sequence in context: A103765 A344772 A307610 * A339839 A062893 A328512

Adjacent sequences: A125026 A125027 A125028 * A125030 A125031 A125032

KEYWORD

nonn,easy

AUTHOR

Leroy Quet, Nov 16 2006

EXTENSIONS

Extended by Ray Chandler, Nov 19 2006

STATUS

approved