A083919 Number of divisors of n that are congruent to 9 modulo 10.

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

Offset

1,99

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) - A083915(n) - A083916(n) - A083917(n) - A083918(n).

G.f.: Sum_{k>=1} x^(9*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(9,10) - (1 - gamma)/10 = -0.197044..., gamma(9,10) = -(psi(9/10) + 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

a[n_] := DivisorSum[n, 1 &, Mod[#, 10] == 9 &]; Array[a, 100] (* Amiram Eldar, Dec 30 2023 *)

Prog

(PARI) a(n) = sumdiv(n, d, d % 10 == 9); \\ Amiram Eldar, Dec 30 2023

Crossrefs

Cf. A000005, A001227, A010879.

Cf. A001620, A002392, A354644 (psi(9/10)).

Cf. A083910, A083911, A083912, A083913, A083914, A083915, A083916, A083917, A083918.

Sequence in context: A102448 A102683 A122840 * A063665 A276306 A340851

Adjacent sequences: A083916 A083917 A083918 * A083920 A083921 A083922

Keyword

nonn,easy

Author

Reinhard Zumkeller, May 08 2003

Status

approved