A083910 Number of divisors of n that are congruent to 0 modulo 10.

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

Offset: 1

Author

Reinhard Zumkeller, May 08 2003

Keywords

• nonn
• easy

Links

• Reinhard Zumkeller, 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.

Crossrefs

Cf. A000005, A000007, A001227, A010879, A027750, A183063.

Cf. A001620, A002392.

Cf. A083911, A083912, A083913, A083914, A083915, A083916, A083917, A083918, A083919.

Programs

Haskell

a083910 = sum . map (a000007 . a010879) . a027750_row
-- Reinhard Zumkeller, Jan 15 2013

Mathematica

ndc10[n_]:=Count[Divisors[n],?(Divisible[#,10]&)]; Array[ndc10,110] (* _Harvey P. Dale, Jan 05 2013 *)
a[n_] := If[Divisible[n, 10], DivisorSigma[0, n/10], 0]; Array[a, 100] (* Amiram Eldar, Dec 30 2023 *)

PARI

a(n)=if(n%10,0,numdiv(n/10)) \\ Charles R Greathouse IV, Sep 27 2015

Formula

a(n) = A000005(n) - A083911(n) - A083912(n) - A083913(n) - A083914(n) - A083915(n) - A083916(n) - A083917(n) - A083918(n) - A083919(n).

a(10k) = tau(k) = A000005(k); a(n) = 0 if 10 does not divide n. - Franklin T. Adams-Watters, Apr 15 2007

G.f.: Sum_{k>=1} x^(10*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 = (2*gamma - 1 - log(10))/10 = -0.214815..., and gamma is Euler's constant (A001620) (Smith and Subbarao, 1981). - Amiram Eldar, Dec 30 2023