A320015 Number of proper divisors of n that are either of the form 6*k+1 or 6*k + 5.

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

Offset

1,10

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) = A320001(n) + A320005(n).

a(n) = A035218(n) - ch15(n), where ch15 is the characteristic function of numbers of the form +-1 mod 6, i.e., ch15(n) = A232991(n-1).

Sum_{k=1..n} a(k) = n*log(n)/3 + c*n + O(n^(1/3)*log(n)), where c = 2*(gamma + log(12)/4 - 1)/3 = 0.132294..., and gamma is Euler's constant (A001620) (Smith and Subbarao, 1981). - Amiram Eldar, Nov 25 2023

Mathematica

a[n_] := DivisorSum[n, 1 &, # < n && MemberQ[{1, 5}, Mod[#, 6]] &]; Array[a, 100] (* Amiram Eldar, Nov 25 2023 *)

Prog

(PARI) A320015(n) = if(!n, n, sumdiv(n, d, (d<n)&&(d%2)&&(d%3)));

Crossrefs

Cf. A001620, A035218, A232991, A293895, A293896, A320001, A320005.

Sequence in context: A316790 A316789 A319661 * A342956 A241918 A276317

Adjacent sequences: A320012 A320013 A320014 * A320016 A320017 A320018

Keyword

nonn,easy

Author

Antti Karttunen, Oct 03 2018

Status

approved