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

1,20

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.

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

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 *)

Prog

(Haskell)
a083910 = sum . map (a000007 . a010879) . a027750_row
-- Reinhard Zumkeller, Jan 15 2013
(PARI) a(n)=if(n%10, 0, numdiv(n/10)) \\ Charles R Greathouse IV, Sep 27 2015

Crossrefs

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

Cf. A001620, A002392.

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

Sequence in context: A173667 A241541 A086008 * A069843 A069849 A138045

Adjacent sequences: A083907 A083908 A083909 * A083911 A083912 A083913

Keyword

nonn,easy

Author

Reinhard Zumkeller, May 08 2003

Status

approved