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A285896 Sum of divisors d of n such that n/d is not congruent to 0 mod 5. 5

%I #31 Dec 30 2022 03:55:38

%S 1,3,4,7,5,12,8,15,13,15,12,28,14,24,20,31,18,39,20,35,32,36,24,60,25,

%T 42,40,56,30,60,32,63,48,54,40,91,38,60,56,75,42,96,44,84,65,72,48,

%U 124,57,75,72,98,54,120,60,120,80,90,60,140,62,96,104,127,70,144

%N Sum of divisors d of n such that n/d is not congruent to 0 mod 5.

%H Seiichi Manyama, <a href="/A285896/b285896.txt">Table of n, a(n) for n = 1..10000</a>

%F a(n) = (A000203(5*n)-A000203(n))/5.

%F G.f.: Sum_{k>=1} k*x^k*(1 + x^k + x^(2*k) + x^(3*k))/(1 - x^(5*k)). - _Ilya Gutkovskiy_, Sep 12 2019

%F From _Amiram Eldar_, Oct 30 2022: (Start)

%F Multiplicative with a(5^e) = 5^e and a(p^e) = (p^(e+1)-1)/(p-1) if p != 5.

%F Sum_{k=1..n} a(k) ~ c * n^2, where c = 2*Pi^2/25 = 0.789568... . (End)

%F Dirichlet g.f.: zeta(s)*zeta(s-1)*(1-1/5^s). - _Amiram Eldar_, Dec 30 2022

%e The divisors of 10 are 1, 2, 5, and 10. 10/1 == 0 (mod 5) and 10/2 == 0 (mod 5). Hence, a(10) = 5 + 10 = 15.

%t f[p_, e_] := If[p == 5, 5^e, (p^(e+1)-1)/(p-1)]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Oct 30 2022 *)

%o (PARI) a(n)=sumdiv(n, d, if(n/d%5, d, 0)); \\ _Andrew Howroyd_, Jul 20 2018

%Y Cf. A002131 (k=2), A078708 (k=3), A285895 (k=4), this sequence (k=5).

%Y Cf. A000203.

%K nonn,mult

%O 1,2

%A _Seiichi Manyama_, Apr 28 2017

%E Keyword:mult added by _Andrew Howroyd_, Jul 20 2018

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Last modified March 29 09:44 EDT 2024. Contains 371268 sequences. (Running on oeis4.)