A158947
Inverse Euler transform of A156217.
Original entry on oeis.org
1, 4, 21, 598, 1555, 497652, 299593, 320361028, 1178277701, 357046721884, 67546215517, 19351522090518670, 61054982558011, 1502551437369035044, 33657152197069739919, 45463945109198918808616, 128583032925805678351
Offset: 1
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A158947 := proc(n) add(numtheory[sigma](d)^d*numtheory[mobius](n/d),d=numtheory[divisors](n))/n ; end: seq( A158947(n),n=1..40) ; # R. J. Mathar, Apr 02 2009
# The function EulerInvTransform is defined in A358451.
a := EulerInvTransform(A156217):
seq(a(n), n = 1..17); # Peter Luschny, Nov 21 2022
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f[n_] := Block[{d = Divisors@n}, Plus @@ (DivisorSigma[1, d]^d*MoebiusMu[n/d])/n]; Array[f, 17] (* Robert G. Wilson v, May 04 2009 *)
A217872
a(n) = sigma(n)^n.
Original entry on oeis.org
1, 9, 64, 2401, 7776, 2985984, 2097152, 2562890625, 10604499373, 3570467226624, 743008370688, 232218265089212416, 793714773254144, 21035720123168587776, 504857282956046106624, 727423121747185263828481, 2185911559738696531968, 43567528752021332753202420081
Offset: 1
L.g.f.: L(x) = x + 3^2*x^2/2 + 4^3*x^3/3 + 7^4*x^4/4 + 6^5*x^5/5 + 12^6*x^6/6 +...
where exponentiation yields the g.f. of A156217:
exp(L(x)) = 1 + x + 5*x^2 + 26*x^3 + 634*x^4 + 2273*x^5 + 502568*x^6 +...
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Table[DivisorSigma[1, n]^n, {n, 1, 20}] (* Amiram Eldar, Nov 16 2020 *)
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{a(n)=sigma(n)^n}
for(n=1,20,print1(a(n),", "))
A224439
G.f.: A(x) = exp( Sum_{n>=1} sigma(n)^(n-1) * x^n/n ).
Original entry on oeis.org
1, 1, 2, 7, 93, 357, 41927, 80065, 21483964, 112388242, 19973468103, 25813956365, 691174602929572, 695655501206181, 63995738768530056, 1469847380380956056, 1468171845473348201557, 1477216529008886240457, 62064992121198579569054696, 62086294811417506896412871
Offset: 0
G.f.: A(x) = 1 + x + 2*x^2 + 7*x^3 + 93*x^4 + 357*x^5 + 41927*x^6 + ... where
log(A(x)) = x + 3^1*x^2/2 + 4^2*x^3/3 + 7^3*x^4/4 + 6^4*x^5/5 + 12^5*x^6/6 + ...
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{a(n)=polcoeff(exp(sum(m=1, n, sigma(m)^(m-1)*x^m/m)+x*O(x^n)), n)}
for(n=0, 20, print1(a(n), ", "))
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{a(n)=if(n==0, 1, (1/n)*sum(k=1, n, sigma(k)^(k-1)*a(n-k)))}
A159604
G.f.: A(x) = exp( Sum_{n>=1} [ Sum_{k>=1} sigma(n,k)*x^k ]^n/n ).
Original entry on oeis.org
1, 1, 6, 43, 856, 10744, 608375, 14284223, 551011548, 19119025101, 874788949035, 37896009869060, 20683158266928833, 1799893777863733707, 93147805938921355288, 3757831283217050847983, 180287028377782585130749
Offset: 0
G.f.: A(x) = 1 + x + 6*x^2 + 43*x^3 + 856*x^4 + 10744*x^5 +...
log(A(x)) = Sum_{n>=1} [sigma(n)*x + sigma(n,2)*x^2 + sigma(n,3)*x^3 +...]^n/n.
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{a(n)=local(A=1+x);for(i=1,n,A=exp(sum(m=1,n,sum(k=1,n,sigma(m,k)*x^k+x*O(x^n))^m/m)));polcoeff(A,n)}
for(n=0,20,print1(a(n),", "))
Showing 1-4 of 4 results.
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