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

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, 27, 28, 29, 36, 31, 32, 33, 34, 42, 36, 37, 38, 39, 48, 41, 42, 43, 44, 54, 46, 47, 48, 49, 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

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

Cf. A129527, A327625, A359099, A373188.

Cf. A000010, A055457.

Sequence in context: A336406 A297351 A060019 * A093451 A106006 A364104

Adjacent sequences: A359097 A359098 A359099 * A359101 A359102 A359103

Keyword

nonn,mult

Author

Seiichi Manyama, Dec 16 2022

Status

approved