A359099 a(n) = (1/6) * Sum_{d|n} phi(7 * d).

1, 2, 3, 4, 5, 6, 8, 8, 9, 10, 11, 12, 13, 16, 15, 16, 17, 18, 19, 20, 24, 22, 23, 24, 25, 26, 27, 32, 29, 30, 31, 32, 33, 34, 40, 36, 37, 38, 39, 40, 41, 48, 43, 44, 45, 46, 47, 48, 57, 50, 51, 52, 53, 54, 55, 64, 57, 58, 59, 60, 61, 62, 72, 64, 65, 66, 67, 68, 69, 80, 71, 72, 73, 74, 75, 76, 88, 78, 79

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

Cf. A129527, A327625, A359100, A373188.

Cf. A000010, A373217.

Sequence in context: A330401 A145518 A256221 * A210253 A130916 A003965

Adjacent sequences: A359096 A359097 A359098 * A359100 A359101 A359102

Keyword

nonn,mult

Author

Seiichi Manyama, Dec 16 2022

Status

approved