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A226459 a(n) = Sum_{d|n} phi(d^d), where phi(n) is the Euler totient function A000010(n). 7
1, 3, 19, 131, 2501, 15573, 705895, 8388739, 258280345, 4000002503, 259374246011, 2972033498453, 279577021469773, 4762288640230761, 233543408203127519, 9223372036863164547, 778579070010669895697, 13115469358432437487707, 1874292305362402347591139 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

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

LINKS

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

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)

EXAMPLE

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 + ...

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 *)

PROG

(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

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

(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

CROSSREFS

Cf. A226458, A226561, A000010.

Sequence in context: A074713 A199484 A063395 * A256092 A074567 A332621

Adjacent sequences:  A226456 A226457 A226458 * A226460 A226461 A226462

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Jun 08 2013

STATUS

approved

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Last modified November 27 03:56 EST 2021. Contains 349345 sequences. (Running on oeis4.)