OFFSET
1,2
LINKS
Michael De Vlieger, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Totient Function.
FORMULA
G.f.: Sum_{k>=1} phi(5 * k) * x^k / (4 * (1 - x^k)).
G.f.: Sum_{k>=0} x^(5^k) / (1 - x^(5^k))^2.
From Amiram Eldar, Dec 17 2022: (Start)
Multiplicative with a(5^e) = (5^(e+1)-1)/4, and a(p^e) = p if p != 5.
Dirichlet g.f.: zeta(s-1)*(1+1/(5^s-1)).
Sum_{k=1..n} a(k) ~ (25/48) * n^2. (End)
From Seiichi Manyama, Jun 04 2024: (Start)
G.f. A(x) satisfies A(x) = x/(1 - x)^2 + A(x^5).
If n == 0 (mod 5), a(n) = n + a(n/5) otherwise a(n) = n. (End)
MATHEMATICA
Array[DivisorSum[#, EulerPhi[5 #] &]/4 &, 76] (* Michael De Vlieger, Dec 16 2022 *)
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 *)
PROG
(PARI) a(n) = sumdiv(n, d, eulerphi(5*d))/4;
(PARI) my(N=80, x='x+O('x^N)); Vec(sum(k=1, N, eulerphi(5*k)*x^k/(1-x^k))/4)
CROSSREFS
KEYWORD
nonn,mult
AUTHOR
Seiichi Manyama, Dec 16 2022
STATUS
approved