A226561
a(n) = Sum_{d|n} d^n * phi(d), where phi(n) is the Euler totient function A000010(n).
Original entry on oeis.org
1, 5, 55, 529, 12501, 94835, 4941259, 67240193, 2324562301, 40039063525, 2853116706111, 35668789979107, 3634501279107037, 66676110291801575, 3503151245145885315, 147575078498173255681, 13235844190181388226833, 236079349222711695887225, 35611553801885644604231623
Offset: 1
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 + ...
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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
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f:= n -> add(d^n * numtheory:-phi(d), d = numtheory:-divisors(n)):
map(f, [$1..40]); # Robert Israel, Jun 16 2017
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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 *)
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{a(n)=sumdiv(n, d, d^n*eulerphi(d))}
for(n=1,30,print1(a(n),", "))
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a(n) = sum(k=1, n, (n/gcd(k, n))^n); \\ Seiichi Manyama, Mar 11 2021
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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
A342433
a(n) = Sum_{k=1..n} gcd(k,n)^(n-1).
Original entry on oeis.org
1, 3, 11, 74, 629, 8085, 117655, 2113796, 43059849, 1001955177, 25937424611, 743379914746, 23298085122493, 793811662313709, 29192938251553759, 1152956691126550536, 48661191875666868497, 2185928270773974154773
Offset: 1
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a[n_] := Sum[GCD[k, n]^(n - 1), {k, 1, n}]; Array[a, 20] (* Amiram Eldar, Mar 12 2021 *)
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a(n) = sum(k=1, n, gcd(k, n)^(n-1));
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a(n) = sumdiv(n, d, eulerphi(n/d)*d^(n-1));
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a(n) = sumdiv(n, d, moebius(n/d)*d*sigma(d, n-2));
A332621
a(n) = (1/n) * Sum_{k=1..n} n^(n/gcd(n, k)).
Original entry on oeis.org
1, 3, 19, 133, 2501, 15631, 705895, 8389641, 258280489, 4000040011, 259374246011, 2972033984173, 279577021469773, 4762288684702095, 233543408203327951, 9223372037928525841, 778579070010669895697, 13115469358498302735067, 1874292305362402347591139
Offset: 1
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[(1/n)*&+[n^(n div Gcd(n,k)):k in [1..n]]:n in [1..20]]; // Marius A. Burtea, Feb 17 2020
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Table[(1/n) Sum[n^(n/GCD[n, k]), {k, 1, n}], {n, 1, 19}]
Table[(1/n) Sum[EulerPhi[d] n^d, {d, Divisors[n]}], {n, 1, 19}]
Table[SeriesCoefficient[Sum[Sum[EulerPhi[j] n^(j - 1) x^(k j), {j, 1, n}], {k, 1, n}], {x, 0, n}], {n, 1, 19}]
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a(n) = sum(k=1, n, n^(n/gcd(n, k)))/n; \\ Michel Marcus, Mar 10 2021
A332620
a(n) = Sum_{k=1..n} n^(n/gcd(n, k)).
Original entry on oeis.org
1, 6, 57, 532, 12505, 93786, 4941265, 67117128, 2324524401, 40000400110, 2853116706121, 35664407810076, 3634501279107049, 66672041585829330, 3503151123049919265, 147573952606856413456, 13235844190181388226849, 236078448452969449231206, 35611553801885644604231641
Offset: 1
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[&+[n^(n div Gcd(n,k)):k in [1..n]]:n in [1..20]]; // Marius A. Burtea, Feb 17 2020
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Table[Sum[n^(n/GCD[n, k]), {k, 1, n}], {n, 1, 19}]
Table[Sum[EulerPhi[d] n^d, {d, Divisors[n]}], {n, 1, 19}]
Table[SeriesCoefficient[Sum[Sum[EulerPhi[j] n^j x^(k j), {j, 1, n}], {k, 1, n}], {x, 0, n}], {n, 1, 19}]
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a(n) = sum(k=1, n, n^(n/gcd(n, k))); \\ Michel Marcus, Mar 10 2021
A342412
a(n) = Sum_{k=1..n} (n/gcd(k,n))^(n-2).
Original entry on oeis.org
1, 2, 7, 37, 501, 2771, 100843, 1056833, 28702189, 401562757, 23579476911, 247792605523, 21505924728445, 340246521979079, 15569565432876147, 576478345026355201, 45798768824157052689, 728648310343004595593, 98646963440126439346903
Offset: 1
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a[n_] := Sum[(n/GCD[k, n])^(n - 2), {k, 1, n}]; Array[a, 20] (* Amiram Eldar, Mar 11 2021 *)
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a(n) = sum(k=1, n, (n/gcd(k, n))^(n-2));
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a(n) = sumdiv(n, d, eulerphi(d^(n-1)));
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a(n) = sumdiv(n, d, eulerphi(d)*d^(n-2));
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my(N=20, x='x+O('x^N)); Vec(sum(k=1, N, eulerphi(k^(k-1))*x^k/(1-(k*x)^k)))
Showing 1-5 of 5 results.
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