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

1,6

Links

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

Formula

From Antti Karttunen, Oct 04 2017: (Start)

a(n) = Sum_{d|n, d<n} A000035(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

Example

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.

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 *)

Prog

(PARI) A091954(n) = sumdiv(n, d, (d<n)*(d%2)); \\ Antti 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

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).

Sequence in context: A007061 A001817 A214973 * A325167 A213408 A183953

Adjacent sequences: A091951 A091952 A091953 * A091955 A091956 A091957

Keyword

easy,nonn

Author

Mohammad K. Azarian, Mar 12 2004

Extensions

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

Status

approved