A083917 Number of divisors of n that are congruent to 7 modulo 10.

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

Offset

1,77

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) - A083918(n) - A083919(n).

G.f.: Sum_{k>=1} x^(7*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(7,10) - (1 - gamma)/10 = -0.150534..., gamma(7,10) = -(psi(7/10) + 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

Table[Count[Divisors[n], _?(Mod[#, 10]==7&)], {n, 110}] (* Harvey P. Dale, Sep 15 2023 *)
a[n_] := DivisorSum[n, 1 &, Mod[#, 10] == 7 &]; Array[a, 100] (* Amiram Eldar, Dec 30 2023 *)

Prog

(PARI) a(n) = sumdiv(n, d, d % 10 == 7); \\ Amiram Eldar, Dec 30 2023

Crossrefs

Cf. A000005, A001227, A010879.

Cf. A001620, A002392, A354643 (psi(7/10)).

Cf. A083910, A083911, A083912, A083913, A083914, A083915, A083916, A083918, A083919.

Sequence in context: A085981 A243224 A127324 * A117974 A369927 A328037

Adjacent sequences: A083914 A083915 A083916 * A083918 A083919 A083920

Keyword

nonn,easy

Author

Reinhard Zumkeller, May 08 2003

Status

approved