A083915 Number of divisors of n that are congruent to 5 modulo 10.

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

Offset

1,15

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

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) = A000005(n) - A083910(n) - A083911(n) - A083912(n) - A083913(n) - A083914(n) - A083916(n) - A083917(n) - A083918(n) - A083919(n).

G.f.: Sum_{k>=1} x^(5*k)/(1 - x^(10*k)). - Ilya Gutkovskiy, Sep 11 2019

Sum_{k=1..n} a(k) = n*log(n)/10 + c*n + O(n^(1/3)*log(n)), where c = gamma(5,10) - (1 - gamma)/10 = 0.0761859..., gamma(5,10) = -(psi(1/2) + log(10))/10 is a generalized Euler constant, and gamma is Euler's constant (A001620) (Smith and Subbarao, 1981). - Amiram Eldar, Dec 30 2023

Mathematica

Table[Count[Divisors[n], _?(Mod[#, 10]==5&)], {n, 120}] (* Harvey P. Dale, Jan 26 2018 *)
a[n_] := DivisorSum[n, 1 &, Mod[#, 10] == 5 &]; Array[a, 100] (* Amiram Eldar, Dec 30 2023 *)

Prog

(PARI) a(n) = sumdiv(n, d, d % 10 == 5); \\ Amiram Eldar, Dec 30 2023

Crossrefs

Cf. A000005, A001227, A010879.

Cf. A001620, A002392, A020759 (psi(1/2)).

Cf. A083910, A083911, A083912, A083913, A083914, A083916, A083917, A083918, A083919.

Sequence in context: A105966 A318950 A319000 * A083892 A354451 A089814

Adjacent sequences: A083912 A083913 A083914 * A083916 A083917 A083918

Keyword

nonn,easy

Author

Reinhard Zumkeller, May 08 2003

Status

approved