A320003 Number of proper divisors of n of the form 6*k + 3.

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

Offset

1,18

Comments

Number of divisors of n that are odd multiples of 3 and less than n.

Links

Antti Karttunen, Table of n, a(n) for n = 1..65537

R. A. Smith and M. V. Subbarao, The average number of divisors in an arithmetic progression, Canadian Mathematical Bulletin, Vol. 24, No. 1 (1981), pp. 37-41.

Formula

a(n) = Sum_{d|n, d<n} (1-A000035(d))*A079978(d).

a(n) = A007814(A319990(n)).

a(4*n) = a(2*n). - David A. Corneth, Oct 03 2018

G.f.: Sum_{k>=1} x^(12*k-6) / (1 - x^(6*k-3)). - Ilya Gutkovskiy, Apr 14 2021

Sum_{k=1..n} a(k) = n*log(n)/6 + c*n + O(n^(1/3)*log(n)), where c = gamma(3,6) - (2 - gamma)/6 = -0.208505..., gamma(3,6) = -(psi(1/2) + log(6))/6 is a generalized Euler constant, and gamma is Euler's constant (A001620) (Smith and Subbarao, 1981). - Amiram Eldar, Nov 25 2023

Example

For n = 18, of its five proper divisors [1, 2, 3, 6, 9] only 3 and 9 are odd multiples of three, thus a(18) = 2.
For n = 108, the odd part is 27 for which 27/3 has 3 divisors. As 108 is even, we don't subtract 1 from that 3 to get a(108) = 3. - David A. Corneth, Oct 03 2018

Mathematica

a[n_] := DivisorSum[n, 1 &, # < n && Mod[#, 6] == 3 &]; Array[a, 100] (* Amiram Eldar, Nov 25 2023 *)

Prog

(PARI) A320003(n) = if(!n, n, sumdiv(n, d, (d<n)*(3==(d%6))));
(PARI) a(n) = if(n%3==0, my(v=valuation(n, 2)); n>>=v; numdiv(n/3)-(!v), 0) \\ David A. Corneth, Oct 03 2018

Crossrefs

Cf. A319990, A320001, A320005.

Cf. A001620, A016629, A020759 (psi(1/2)).

Sequence in context: A016232 A007949 A191265 * A291749 A370599 A253786

Adjacent sequences: A320000 A320001 A320002 * A320004 A320005 A320006

Keyword

nonn,easy

Author

Antti Karttunen, Oct 03 2018

Status

approved