A091954 Number of odd proper divisors of n. That is, the number of odd divisors of n that are less than n.

0, 1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 2, 1, 2, 3, 1, 1, 3, 1, 2, 3, 2, 1, 2, 2, 2, 3, 2, 1, 4, 1, 1, 3, 2, 3, 3, 1, 2, 3, 2, 1, 4, 1, 2, 5, 2, 1, 2, 2, 3, 3, 2, 1, 4, 3, 2, 3, 2, 1, 4, 1, 2, 5, 1, 3, 4, 1, 2, 3, 4, 1, 3, 1, 2, 5, 2, 3, 4, 1, 2, 4, 2, 1, 4, 3, 2, 3, 2, 1, 6, 3, 2, 3, 2, 3, 2, 1, 3, 5, 3, 1, 4, 1, 2, 7, 2, 1, 4, 1, 4

Offset: 1

Author

Mohammad K. Azarian, Mar 12 2004

Keywords

• easy
• nonn

Examples

			The odd divisors of 15 that are less than 15 are 1, 3 and 5. Therefore there are three odd divisors of 15 that are less than 15.
		

Links

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

Crossrefs

Cf. A000005, A000035, A001620, A032741, A001227, A293214, A293216.

Sum of the k-th powers of the odd proper divisors of n for k=0..10: this sequence (k=0), A091570 (k=1), A351647 (k=2), A352031 (k=3), A352032 (k=4), A352033 (k=5), A352034 (k=6), A352035 (k=7), A352036 (k=8), A352037 (k=9), A352038 (k=10).

Sum of the k-th powers of the divisor complements of the odd proper divisors of n for k=0..10: this sequence (k=0), A352047 (k=1), A352048 (k=2), A352049 (k=3), A352050 (k=4), A352051 (k=5), A352052 (k=6), A352053 (k=7), A352054 (k=8), A352055 (k=9), A352056 (k=10).

Programs

Mathematica

Count[Most[Divisors[#]],?OddQ]&/@Range[100] (* _Harvey P. Dale, Sep 28 2012 *)
a[n_] := DivisorSigma[0, n/2^IntegerExponent[n, 2]] - Boole[OddQ[n]]; Array[a, 100] (* Amiram Eldar, Jun 11 2022 *)

PARI

A091954(n) = sumdiv(n,d,(dAntti Karttunen, Oct 04 2017

PARI

my(N=66, x='x+O('x^N)); concat(0, Vec(sum(k=2, N, x^k/(1-x^(2*k))))) \\ Seiichi Manyama, Jan 23 2021

Formula

From Antti Karttunen, Oct 04 2017: (Start)

a(n) = Sum_{d|n, dA000035(n).

a(n) = A001227(n) - A000035(n).

a(n) = A007814(A293214(n)) = A007814(A293216(n)).

(End)

G.f.: Sum_{k>=2} x^k/(1 - x^(2*k)). - Seiichi Manyama, Jan 23 2021

Sum_{k=1..n} a(k) ~ n*log(n)/2 + (gamma + log(2)/2 - 1)*n, where gamma is Euler's constant (A001620). - Amiram Eldar, Nov 26 2023

Extensions

Corrected and extended by Harvey P. Dale, Sep 28 2012