A091304 a(n) = Omega(2n-1) (number of prime factors of the n-th odd number, counted with multiplicity).

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

Offset

1,5

Comments

Omega(n) of the odd integers follows a pattern similar to A001222, with 4 maxima instead of 2 - i.e. between 2^n and (2^(n+1) - 1) there are two numbers with exactly n factors (2^n and 2^(n-1) * 3) while the odd integers have 4 maxima (3^n, 3^(n-1) * 5, 3^(n-1) * 7, 5^2*3^(n-2)) between 3^n and 3^(n+1) - 1.

Links

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

Formula

a(n) = Omega(2n-1). [Odd bisection of A001222.]

From Antti Karttunen, May 31 2017: (Start)

For n >= 1, a(n) = A000120(A244153(n)).

For n >= 2, a(n) = 1+A285716(n).

(End)

Example

Omega(1) = 0, Omega(9) = 2 (3 * 3 = 9), Omega (243) = 5 (3 * 3 * 3 * 3 * 3 = 243), Omega(51) = 2 (3 * 17 = 51).
For n = 92, A001222(2*92 - 1) = A001222(183) = 2 as 183 = 3*61, thus a(92) = 2. - Antti Karttunen, May 31 2017

Mathematica

a[n_] := PrimeOmega[2*n - 1]; Array[a, 100] (* Amiram Eldar, Jul 23 2023 *)

Prog

(PARI) a(n) = bigomega(2*n-1) \\ Michel Marcus, Jul 26 2013, edited to reflect the changed starting offset by Antti Karttunen, May 31 2017

Crossrefs

One more than A285716 (after the initial term).

Cf. A001222, A000120, A092523, A099774, A099990, A244153, A278223, A286582.

Cf. A006254 (positions of ones).

Sequence in context: A091853 A193773 A369377 * A340101 A049847 A255274

Adjacent sequences: A091301 A091302 A091303 * A091305 A091306 A091307

Keyword

easy,nonn

Author

Andrew S. Plewe, Feb 20 2004

Extensions

Starting offset changed to 1 and the definition modified respectively. Also values of the initial term and of term a(92) (= 2, previously a(91) = 1) corrected by Antti Karttunen, May 31 2017

Status

approved