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A083910 Number of divisors of n that are congruent to 0 modulo 10. 12

%I #26 Dec 30 2023 09:33:46

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

%H Reinhard Zumkeller, <a href="/A083910/b083910.txt">Table of n, a(n) for n = 1..10000</a>

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

%F a(10k) = tau(k) = A000005(k); a(n) = 0 if 10 does not divide n. - _Franklin T. Adams-Watters_, Apr 15 2007

%F G.f.: Sum_{k>=1} x^(10*k)/(1 - x^(10*k)). - _Ilya Gutkovskiy_, Sep 11 2019

%F 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 ndc10[n_]:=Count[Divisors[n],_?(Divisible[#,10]&)]; Array[ndc10,110] (* _Harvey P. Dale_, Jan 05 2013 *)

%t a[n_] := If[Divisible[n, 10], DivisorSigma[0, n/10], 0]; Array[a, 100] (* _Amiram Eldar_, Dec 30 2023 *)

%o (Haskell)

%o a083910 = sum . map (a000007 . a010879) . a027750_row

%o -- _Reinhard Zumkeller_, Jan 15 2013

%o (PARI) a(n)=if(n%10,0,numdiv(n/10)) \\ _Charles R Greathouse IV_, Sep 27 2015

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

%Y Cf. A001620, A002392.

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

%K nonn,easy

%O 1,20

%A _Reinhard Zumkeller_, May 08 2003

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Last modified March 28 09:04 EDT 2024. Contains 371240 sequences. (Running on oeis4.)