A309369 a(n) = Sum_{d|n} phi(n/d)^d, where phi = Euler totient function (A000010).

1, 2, 3, 4, 5, 8, 7, 10, 15, 22, 11, 34, 13, 44, 105, 42, 17, 116, 19, 314, 357, 112, 23, 426, 1045, 158, 747, 1474, 29, 5290, 31, 594, 3069, 274, 24185, 6082, 37, 344, 9945, 67922, 41, 63542, 43, 12170, 303225, 508, 47, 74834, 279979, 1050022, 135201, 29098, 53, 309872, 4294345

Offset

1,2

Links

Seiichi Manyama, Table of n, a(n) for n = 1..5000

Formula

G.f.: Sum_{k>=1} phi(k)*x^k/(1 - phi(k)*x^k).

L.g.f.: -log(Product_{k>=1} (1 - phi(k)*x^k)^(1/k)).

a(p) = p for p prime.

a(n) = Sum_{k=1..n} phi(n/gcd(k, n))^(gcd(k, n) - 1). - Seiichi Manyama, Mar 13 2021

Mathematica

Table[Sum[EulerPhi[n/d]^d, {d, Divisors[n]}], {n, 1, 55}]
nmax = 55; CoefficientList[Series[Sum[EulerPhi[k] x^k/(1 - EulerPhi[k] x^k), {k, 1, nmax}], {x, 0, nmax}], x] // Rest
nmax = 55; CoefficientList[Series[-Log[Product[(1 - EulerPhi[k] x^k)^(1/k), {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax] // Rest

Prog

(PARI) a(n) = sum(k=1, n, eulerphi(n/gcd(k, n))^(gcd(k, n)-1)); \\ Seiichi Manyama, Mar 13 2021

Crossrefs

Cf. A000010, A055225, A164941, A264782, A279789.

Sequence in context: A158979 A233249 A330573 * A091893 A074756 A240221

Adjacent sequences: A309366 A309367 A309368 * A309370 A309371 A309372

Keyword

nonn

Author

Ilya Gutkovskiy, Jul 25 2019

Status

approved