A320001 - OEIS

A320001

Number of proper divisors of n of the form 6*k + 1.

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

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OFFSET

1,14

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) = A279060(n) - [+1 = n (mod 6)], where [ ] is the Iverson bracket, giving 1 only when n = 1 mod 6, and 0 otherwise.

a(n) = A320015(n) - A320005(n).

a(n) = A007814(A319991(n)).

G.f.: Sum_{k>=1} x^(12*k-10) / (1 - x^(6*k-5)). - 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(1,6) - (2 - gamma)/6 = 0.519597..., gamma(1,6) = -(psi(1/6) + log(6))/6 is a generalized Euler constant, and gamma is Euler's constant (A001620) (Smith and Subbarao, 1981). - Amiram Eldar, Nov 25 2023

MATHEMATICA

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

PROG

(PARI) A320001(n) = if(!n, n, sumdiv(n, d, (d<n)*(1==(d%6))));

CROSSREFS

Cf. A279060, A293895, A319991, A320003, A320005, A320015.

Cf. A001620, A016629, A222457 (psi(1/6)).

Sequence in context: A337473 A039738 A191515 * A168201 A263000 A075774

Adjacent sequences: A319998 A319999 A320000 * A320002 A320003 A320004

KEYWORD

nonn,easy

AUTHOR

Antti Karttunen, Oct 03 2018

STATUS

approved