A091304 - OEIS

%I #25 Jul 23 2023 06:07:40

%S 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,

%T 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,

%U 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

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

%C 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.

%H Antti Karttunen, <a href="/A091304/b091304.txt">Table of n, a(n) for n = 1..10000</a>

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

%F From _Antti Karttunen_, May 31 2017: (Start)

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

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

%F (End)

%e Omega(1) = 0, Omega(9) = 2 (3 * 3 = 9), Omega (243) = 5 (3 * 3 * 3 * 3 * 3 = 243), Omega(51) = 2 (3 * 17 = 51).

%e For n = 92, A001222(2*92 - 1) = A001222(183) = 2 as 183 = 3*61, thus a(92) = 2. - _Antti Karttunen_, May 31 2017

%t a[n_] := PrimeOmega[2*n - 1]; Array[a, 100] (* _Amiram Eldar_, Jul 23 2023 *)

%o (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

%Y One more than A285716 (after the initial term).

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

%Y Cf. A006254 (positions of ones).

%K easy,nonn

%O 1,5

%A _Andrew S. Plewe_, Feb 20 2004

%E 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