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

1,36

Links

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

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

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

Mathematica

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

Prog

(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

Crossrefs

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

Cf. A268335 (positions of zeros), A295316.

Sequence in context: A369427 A304819 A304820 * A333487 A348380 A185374

Adjacent sequences: A162638 A162639 A162640 * A162642 A162643 A162644

Keyword

nonn

Author

Reinhard Zumkeller, Jul 08 2009

Status

approved