A226561 a(n) = Sum_{d|n} d^n * phi(d), where phi(n) is the Euler totient function A000010(n).

1, 5, 55, 529, 12501, 94835, 4941259, 67240193, 2324562301, 40039063525, 2853116706111, 35668789979107, 3634501279107037, 66676110291801575, 3503151245145885315, 147575078498173255681, 13235844190181388226833, 236079349222711695887225, 35611553801885644604231623

Offset

1,2

Comments

Compare formula to the identity: Sum_{d|n} phi(d) = n.

Links

Robert Israel, Table of n, a(n) for n = 1..385

Formula

Logarithmic derivative of A226560.

a(n) = Sum_{d|n} d * phi(d^n).

a(n) = Sum_{d|n} phi(d^(n+1)).

a(n) = Sum_{d|n} phi(d^(n+2))/d.

a(n) = Sum_{d|n} d^(n-k+1) * phi(d^k) for k >= 1.

G.f.: Sum_{k>=1} phi(k)*(k*x)^k/(1 - (k*x)^k). - Ilya Gutkovskiy, Nov 06 2018

a(n) = Sum_{k=1..n} (n/gcd(k,n))^n. - Seiichi Manyama, Mar 11 2021

a(n) = Sum_{k=1..n} gcd(n,k)^n*phi(gcd(n,k))/phi(n/gcd(n,k)). - Richard L. Ollerton, May 10 2021

Example

L.g.f.: L(x) = x + 5*x^2/2 + 55*x^3/3 + 529*x^4/4 + 12501*x^5/5 + 94835*x^6/6 + ...
where
exp(L(x)) = 1 + x + 3*x^2 + 21*x^3 + 155*x^4 + 2691*x^5 + 18924*x^6 + 732230*x^7 + 9223166*x^8 + ... + A226560(n)*x^n + ...

Maple

f:= n -> add(d^n * numtheory:-phi(d), d = numtheory:-divisors(n)):
map(f, [$1..40]); # Robert Israel, Jun 16 2017

Mathematica

Table[DivisorSum[n, #*EulerPhi[#^n]  &], {n, 1, 30}]  (* or *) With[{nmax = 30}, Rest[CoefficientList[Series[Sum[EulerPhi[k]*(k*x)^k/(1 - (k*x)^k), {k, 1, 2*nmax}], {x, 0, nmax}], x]]]  (* G. C. Greubel, Nov 07 2018 *)

Prog

(PARI) {a(n)=sumdiv(n, d, d^n*eulerphi(d))}
for(n=1, 30, print1(a(n), ", "))
(PARI) a(n) = sum(k=1, n, (n/gcd(k, n))^n); \\ Seiichi Manyama, Mar 11 2021
(Magma) m:=30; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!(  (&+[EulerPhi(k)*(k*x)^k/(1-(k*x)^k): k in [1..2*m]]) )); // G. C. Greubel, Nov 07 2018
(Python)
from sympy import totient, divisors
def A226561(n):
    return sum(totient(d)*d**n for d in divisors(n, generator=True)) # Chai Wah Wu, Feb 15 2020

Crossrefs

Cf. A226560, A226459, A000010, A321349, A332517.

Sequence in context: A045640 A043040 A342420 * A062183 A002279 A119292

Adjacent sequences: A226558 A226559 A226560 * A226562 A226563 A226564

Keyword

nonn

Author

Paul D. Hanna, Jun 10 2013

Status

approved