A162641 Number of even exponents in canonical prime factorization of n.

0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 1, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 1, 2, 0, 0, 0, 0, 0

Offset: 1

Author

Reinhard Zumkeller, Jul 08 2009

Keywords

• nonn

Links

• Antti Karttunen, Table of n, a(n) for n = 1..10000
• Index entries for sequences computed from exponents in factorization of n.

Crossrefs

Cf. A001221, A003557, A002035, A007913, A025487, A059841, A061742 (where records occur), A072587, A162642, A179119.

Cf. A268335 (positions of zeros), A295316.

Programs

Mathematica

Table[Count[FactorInteger[n][[All, -1]], ?EvenQ], {n, 105}] (* _Michael De Vlieger, Jul 23 2017 *)

PARI

A162641(n) = omega(n) - omega(core(n)); \\ Antti Karttunen, Jul 23 2017

Scheme

(define (A162641 n) (if (= 1 n) 0 (+ (A059841 (A067029 n)) (A162641 (A028234 n))))) ;; Antti Karttunen, Jul 23 2017

Formula

a(n) = A001221(n) - A162642(n).

a(A002035(n)) = 0.

a(A072587(n)) > 0.

Additive with a(p^e) = A059841(e). - Antti Karttunen, Jul 23 2017

From Antti Karttunen, Nov 28 2017: (Start)

a(n) = A162642(A003557(n)).

a(n) <= A056170(n).

(End)

Asymptotic mean: lim_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{p prime} 1/(p*(p+1)) = 0.3302299262... (A179119). - Amiram Eldar, Dec 25 2021