A087436 Number of odd prime factors of n, counted with repetitions.

0, 0, 1, 0, 1, 1, 1, 0, 2, 1, 1, 1, 1, 1, 2, 0, 1, 2, 1, 1, 2, 1, 1, 1, 2, 1, 3, 1, 1, 2, 1, 0, 2, 1, 2, 2, 1, 1, 2, 1, 1, 2, 1, 1, 3, 1, 1, 1, 2, 2, 2, 1, 1, 3, 2, 1, 2, 1, 1, 2, 1, 1, 3, 0, 2, 2, 1, 1, 2, 2, 1, 2, 1, 1, 3, 1, 2, 2, 1, 1, 4, 1, 1, 2, 2, 1, 2, 1, 1, 3, 2, 1, 2, 1, 2, 1, 1, 2, 3, 2, 1, 2

Offset

1,9

Comments

Number of parts larger than 1 in the partition with Heinz number n. The Heinz number of an integer partition p = [p_1, p_2, ..., p_r] is defined as Product(p_j-th prime, j=1...r) (concept used by Alois P. Heinz in A215366 as an "encoding" of a partition). Example: a(9) = 2 because the partition with Heinz number 9 (=3*3) is [2,2]. - Emeric Deutsch Oct 02 2015

Totally additive because both A001222 and A007814 are. a(2) = 0, and a(p) = 1 for odd primes p, a(m*n) = a(m)+a(n) for m, n > 1. - Antti Karttunen, Jul 10 2020

Links

Antti Karttunen, Table of n, a(n) for n = 1..65537 (first 1000 terms from G. C. Greubel)

Formula

a(n) = A001222(n) - A007814(n).

a(n) = A001222(A000265(n)). - Antti Karttunen, Jul 10 2020

Example

a(9) = 2 because 9 = 3*3 has 2 odd prime factors. - Emeric Deutsch Oct 02 2015

Maple

seq(bigomega(n) - padic[ordp](n, 2), n=1..102); # Peter Luschny, Dec 06 2017

Mathematica

Join[{0}, Table[Length[Select[Flatten[Table[#[[1]], {#[[2]]}]&/@ FactorInteger[ n]], OddQ]], {n, 2, 110}]] (* Harvey P. Dale, Feb 01 2013 *)

Prog

(PARI) a(n) = bigomega(n) - valuation(n, 2); \\ Michel Marcus, Sep 10 2019
(PARI) A087436(n) = (bigomega(n>>valuation(n, 2))); \\ Antti Karttunen, Jul 10 2020

Crossrefs

Cf. A000244 (the first occurrence of each n, and also the positions of records).

Cf. A000265, A001222, A005087, A007814, A215366, A336118, A336161.

Sequence in context: A176564 A237717 A154338 * A340831 A334862 A329801

Adjacent sequences: A087433 A087434 A087435 * A087437 A087438 A087439

Keyword

nonn

Author

Reinhard Zumkeller, Sep 03 2003

Status

approved