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A226561
a(n) = Sum_{d|n} d^n * phi(d), where phi(n) is the Euler totient function A000010(n).
11
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
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
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jun 10 2013
STATUS
approved