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

%I A083910 #26 Dec 30 2023 09:33:46
%S A083910 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,
%T A083910 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,
%U A083910 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
%N A083910 Number of divisors of n that are congruent to 0 modulo 10.
%H A083910 Reinhard Zumkeller, <a href="/A083910/b083910.txt">Table of n, a(n) for n = 1..10000</a>
%H A083910 R. A. Smith and M. V. Subbarao, <a href="https://doi.org/10.4153/CMB-1981-005-3">The average number of divisors in an arithmetic progression</a>, Canadian Mathematical Bulletin, Vol. 24, No. 1 (1981), pp. 37-41.
%F A083910 a(n) = A000005(n) - A083911(n) - A083912(n) - A083913(n) - A083914(n) - A083915(n) - A083916(n) - A083917(n) - A083918(n) - A083919(n).
%F A083910 a(10k) = tau(k) = A000005(k); a(n) = 0 if 10 does not divide n. - _Franklin T. Adams-Watters_, Apr 15 2007
%F A083910 G.f.: Sum_{k>=1} x^(10*k)/(1 - x^(10*k)). - _Ilya Gutkovskiy_, Sep 11 2019
%F A083910 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
%t A083910 ndc10[n_]:=Count[Divisors[n],_?(Divisible[#,10]&)]; Array[ndc10,110] (* _Harvey P. Dale_, Jan 05 2013 *)
%t A083910 a[n_] := If[Divisible[n, 10], DivisorSigma[0, n/10], 0]; Array[a, 100] (* _Amiram Eldar_, Dec 30 2023 *)
%o A083910 (Haskell)
%o A083910 a083910 = sum . map (a000007 . a010879) . a027750_row
%o A083910 -- _Reinhard Zumkeller_, Jan 15 2013
%o A083910 (PARI) a(n)=if(n%10,0,numdiv(n/10)) \\ _Charles R Greathouse IV_, Sep 27 2015
%Y A083910 Cf. A000005, A000007, A001227, A010879, A027750, A183063.
%Y A083910 Cf. A001620, A002392.
%Y A083910 Cf. A083911, A083912, A083913, A083914, A083915, A083916, A083917, A083918, A083919.
%K A083910 nonn,easy
%O A083910 1,20
%A A083910 _Reinhard Zumkeller_, May 08 2003