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

Author

Paul D. Hanna, Jun 10 2013

Keywords

• nonn

Comments

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

Examples

			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 + ...
		

Links

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

Crossrefs

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

Programs

Magma

m:=30; R:=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

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 *)

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

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

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