A226459 a(n) = Sum_{d|n} phi(d^d), where phi(n) is the Euler totient function A000010(n).

1, 3, 19, 131, 2501, 15573, 705895, 8388739, 258280345, 4000002503, 259374246011, 2972033498453, 279577021469773, 4762288640230761, 233543408203127519, 9223372036863164547, 778579070010669895697, 13115469358432437487707, 1874292305362402347591139

Offset: 1

Author

Paul D. Hanna, Jun 08 2013

Keywords

• nonn

Comments

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

Examples

			L.g.f.: L(x) = x + 3*x^2/2 + 19*x^3/3 + 131*x^4/4 + 2501*x^5/5 + ...
where
exp(L(x)) = 1 + x + 2*x^2 + 8*x^3 + 41*x^4 + 547*x^5 + 3193*x^6 + ... + A226458(n)*x^n + ...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 1..385

Crossrefs

Cf. A226458, A226561, A000010.

Programs

Magma

m:=30; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(  (&+[EulerPhi(k^k)*x^k/(1-x^k): k in [1..3*m]]) )); // G. C. Greubel, Nov 07 2018

Mathematica

ttf[n_]:=Module[{d=Divisors[n]},Total[EulerPhi[d^d]]]; Array[ttf,20] (* Harvey P. Dale, Aug 21 2013 *)
With[{nmax = 30}, Rest[CoefficientList[Series[Sum[EulerPhi[k^k]*x^k/(1 - x^k), {k, 1, 2*nmax}], {x, 0, nmax}], x]]] (* G. C. Greubel, Nov 07 2018 *)

PARI

{a(n)=sumdiv(n,d, eulerphi(d^d))}
for(n=1,30,print1(a(n),", "))

PARI

a(n) = sum(k=1, n, (n/gcd(k, n))^(n/gcd(k, n)-1)); \\ Seiichi Manyama, Mar 11 2021

Python

from sympy import totient, divisors
def A226459(n):
    return sum(totient(d)*d**(d-1) for d in divisors(n,generator=True)) # Chai Wah Wu, Feb 15 2020

Formula

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

Equals the logarithmic derivative of A226458.

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

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

From Richard L. Ollerton, May 08 2021: (Start)

a(n) = Sum_{k=1..n} phi(gcd(n,k)^gcd(n,k))/phi(n/gcd(n,k)).

a(n) = Sum_{k=1..n} phi((n/gcd(n,k))^(n/gcd(n,k)))/phi(n/gcd(n,k)).

a(n) = Sum_{k=1..n} gcd(n,k)^(gcd(n,k)-1)*phi(gcd(n,k))/phi(n/gcd(n,k)). (End)