A293895 Number of proper divisors of n of the form 3k+1.

0, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 3, 2, 1, 1, 2, 1, 2, 1, 3, 1, 2, 1, 3, 1, 1, 2, 2, 1, 2, 2, 3, 1, 2, 1, 3, 1, 1, 1, 3, 2, 3, 1, 3, 1, 1, 1, 4, 2, 1, 1, 3, 1, 2, 2, 3, 2, 2, 1, 3, 1, 3, 1, 2, 1, 2, 2, 3, 2, 2, 1, 5, 1, 1, 1, 4, 1, 2, 1, 3, 1, 2, 3, 3, 2, 1, 2, 3, 1, 3, 1, 4, 1, 2, 1, 4, 2

Offset: 1

Author

Antti Karttunen, Nov 06 2017

Keywords

• nonn
• easy

Links

• Antti Karttunen, Table of n, a(n) for n = 1..20000
• 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.

Crossrefs

Cf. A001620, A001817, A256425, A293451, A293896, A293897, A293899.

Programs

Mathematica

Table[DivisorSum[n, 1 &, And[Mod[#, 3] == 1, # != n] &], {n, 105}] (* Michael De Vlieger, Nov 08 2017 *)

PARI

A293895(n) = sumdiv(n,d,(d
				

Formula

a(n) = A001817(n) - [n == 1 (mod 3)].

G.f.: Sum_{k>=1} x^(6*k-4) / (1 - x^(3*k-2)). - Ilya Gutkovskiy, Apr 14 2021

Sum_{k=1..n} a(k) = n*log(n)/3 + c*n + O(n^(1/3)*log(n)), where c = gamma(1,3) - (2 - gamma)/3 = A256425 - (2 - A001620)/3 = 0.203545... (Smith and Subbarao, 1981). - Amiram Eldar, Nov 25 2023