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%I #37 Sep 08 2022 08:46:05
%S 1,5,55,529,12501,94835,4941259,67240193,2324562301,40039063525,
%T 2853116706111,35668789979107,3634501279107037,66676110291801575,
%U 3503151245145885315,147575078498173255681,13235844190181388226833,236079349222711695887225,35611553801885644604231623
%N a(n) = Sum_{d|n} d^n * phi(d), where phi(n) is the Euler totient function A000010(n).
%C Compare formula to the identity: Sum_{d|n} phi(d) = n.
%H Robert Israel, <a href="/A226561/b226561.txt">Table of n, a(n) for n = 1..385</a>
%F Logarithmic derivative of A226560.
%F a(n) = Sum_{d|n} d * phi(d^n).
%F a(n) = Sum_{d|n} phi(d^(n+1)).
%F a(n) = Sum_{d|n} phi(d^(n+2))/d.
%F a(n) = Sum_{d|n} d^(n-k+1) * phi(d^k) for k >= 1.
%F G.f.: Sum_{k>=1} phi(k)*(k*x)^k/(1 - (k*x)^k). - _Ilya Gutkovskiy_, Nov 06 2018
%F a(n) = Sum_{k=1..n} (n/gcd(k,n))^n. - _Seiichi Manyama_, Mar 11 2021
%F 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
%e 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 + ...
%e where
%e 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 + ...
%p f:= n -> add(d^n * numtheory:-phi(d), d = numtheory:-divisors(n)):
%p map(f, [$1..40]); # _Robert Israel_, Jun 16 2017
%t 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 *)
%o (PARI) {a(n)=sumdiv(n, d, d^n*eulerphi(d))}
%o for(n=1,30,print1(a(n),", "))
%o (PARI) a(n) = sum(k=1, n, (n/gcd(k, n))^n); \\ _Seiichi Manyama_, Mar 11 2021
%o (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
%o (Python)
%o from sympy import totient, divisors
%o def A226561(n):
%o return sum(totient(d)*d**n for d in divisors(n,generator=True)) # _Chai Wah Wu_, Feb 15 2020
%Y Cf. A226560, A226459, A000010, A321349, A332517.
%K nonn
%O 1,2
%A _Paul D. Hanna_, Jun 10 2013
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