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(7 * k) * x^k / (6 * (1 - x^k)).
G.f.: Sum_{k>=0} x^(7^k) / (1 - x^(7^k))^2.
From Amiram Eldar, Dec 17 2022: (Start)
Multiplicative with a(7^e) = (7^(e+1)-1)/6, and a(p^e) = p if p != 7.
Dirichlet g.f.: zeta(s-1)*(1+1/(7^s-1)).
Sum_{k=1..n} a(k) ~ (49/96) * n^2. (End)
From Seiichi Manyama, Jun 04 2024: (Start)
G.f. A(x) satisfies A(x) = x/(1 - x)^2 + A(x^7).
If n == 0 (mod 7), a(n) = n + a(n/7) otherwise a(n) = n. (End)
MATHEMATICA
Array[DivisorSum[#, EulerPhi[7 #] &]/6 &, 79] (* Michael De Vlieger, Dec 16 2022 *)
f[p_, e_] := If[p == 7, (7^(e + 1) - 1)/6, 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(7*d))/6;
(PARI) my(N=80, x='x+O('x^N)); Vec(sum(k=1, N, eulerphi(7*k)*x^k/(1-x^k))/6)
CROSSREFS
KEYWORD
nonn,mult
AUTHOR
Seiichi Manyama, Dec 16 2022
STATUS
approved