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A359100 a(n) = (1/4) * Sum_{d|n} phi(5 * d).

%I A359100 #28 Feb 16 2025 08:34:04
%S A359100 1,2,3,4,6,6,7,8,9,12,11,12,13,14,18,16,17,18,19,24,21,22,23,24,31,26,
%T A359100 27,28,29,36,31,32,33,34,42,36,37,38,39,48,41,42,43,44,54,46,47,48,49,
%U A359100 62,51,52,53,54,66,56,57,58,59,72,61,62,63,64,78,66,67,68,69,84,71,72,73,74,93,76
%N A359100 a(n) = (1/4) * Sum_{d|n} phi(5 * d).
%H A359100 Michael De Vlieger, <a href="/A359100/b359100.txt">Table of n, a(n) for n = 1..10000</a>
%H A359100 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/TotientFunction.html">Totient Function</a>.
%F A359100 G.f.: Sum_{k>=1} phi(5 * k) * x^k / (4 * (1 - x^k)).
%F A359100 G.f.: Sum_{k>=0} x^(5^k) / (1 - x^(5^k))^2.
%F A359100 From _Amiram Eldar_, Dec 17 2022: (Start)
%F A359100 Multiplicative with a(5^e) = (5^(e+1)-1)/4, and a(p^e) = p if p != 5.
%F A359100 Dirichlet g.f.: zeta(s-1)*(1+1/(5^s-1)).
%F A359100 Sum_{k=1..n} a(k) ~ (25/48) * n^2. (End)
%F A359100 From _Seiichi Manyama_, Jun 04 2024: (Start)
%F A359100 G.f. A(x) satisfies A(x) = x/(1 - x)^2 + A(x^5).
%F A359100 If n == 0 (mod 5), a(n) = n + a(n/5) otherwise a(n) = n. (End)
%t A359100 Array[DivisorSum[#, EulerPhi[5 #] &]/4 &, 76] (* _Michael De Vlieger_, Dec 16 2022 *)
%t A359100 f[p_, e_] := If[p == 5, (5^(e + 1) - 1)/4, p^e]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Dec 17 2022 *)
%o A359100 (PARI) a(n) = sumdiv(n, d, eulerphi(5*d))/4;
%o A359100 (PARI) my(N=80, x='x+O('x^N)); Vec(sum(k=1, N, eulerphi(5*k)*x^k/(1-x^k))/4)
%Y A359100 Cf. A129527, A327625, A359099, A373188.
%Y A359100 Cf. A000010, A055457.
%K A359100 nonn,mult
%O A359100 1,2
%A A359100 _Seiichi Manyama_, Dec 16 2022