The On-Line Encyclopedia of Integer Sequences (OEIS)

Revision History for A226106

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A226106

G.f.: exp( Sum_{n>=1} A068963(n)*x^n/n ) where A068963(n) = Sum_{d|n} phi(d^3).
(history; published version)

#13 by Vaclav Kotesovec at Sat Mar 31 08:25:38 EDT 2018

STATUS

editing

approved

#12 by Vaclav Kotesovec at Sat Mar 31 08:25:30 EDT 2018

LINKS

Vaclav Kotesovec, <a href="/A226106/a226106.jpg">Graph - The asymptotic ratio</a>

STATUS

approved

editing

#11 by Vaclav Kotesovec at Sat Mar 31 05:22:43 EDT 2018

STATUS

editing

approved

#10 by Vaclav Kotesovec at Sat Mar 31 05:21:59 EDT 2018

MATHEMATICA

nmax = 40; CoefficientList[Series[Exp[Sum[Sum[k*EulerPhi[k] * x^(j*k) / j, {k, 1, Floor[nmax/j] + 1}], {j, 1, nmax}]], {x, 0, nmax}], x] (* Vaclav Kotesovec, Mar 31 2018 *)

STATUS

approved

editing

#9 by Vaclav Kotesovec at Sat Mar 31 04:53:03 EDT 2018

STATUS

editing

approved

#8 by Vaclav Kotesovec at Sat Mar 31 03:21:09 EDT 2018

LINKS

Vaclav Kotesovec, <a href="/A226106/b226106.txt">Table of n, a(n) for n = 0..10000</a>

#7 by Vaclav Kotesovec at Fri Mar 30 10:03:19 EDT 2018

MATHEMATICA

nmax = 40; CoefficientList[Series[Product[1/(1-x^k)^EulerPhi[k^2], {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Mar 30 2018 *)

#6 by Vaclav Kotesovec at Fri Mar 30 10:00:24 EDT 2018

FORMULA

a(n) ~ exp(2^(9/4) * sqrt(Pi) * n^(3/4) / (3 * 5^(1/4)) + 3*Zeta(3) / Pi^2) / (2^(11/8) * 5^(1/8) * Pi^(1/4) * n^(5/8)). - Vaclav Kotesovec, Mar 30 2018

#5 by Vaclav Kotesovec at Fri Mar 30 06:57:05 EDT 2018

MATHEMATICA

nmax = 40; CoefficientList[Series[Product[1/(1-x^k)^(k*EulerPhi[k]), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Mar 30 2018 *)

#4 by Vaclav Kotesovec at Fri Mar 30 06:56:29 EDT 2018

NAME

G.f.: exp( Sum_{n>=1} A068963(n)*x^n/n ) where A068963(n) = Sum_{d|n} phi(d^3).

COMMENTS

Here phi(n) = A000010(n) is the Euler totient function.

Euler transform of A002618. - Vaclav Kotesovec, Mar 30 2018

EXAMPLE

G.f.: A(x) = 1 + x + 3*x^2 + 9*x^3 + 20*x^4 + 52*x^5 + 105*x^6 + 253*x^7 +...

PROG

(PARI) {a(n)=polcoeff(exp(sum(m=1, n+1, sumdiv(m, d, eulerphi(d^3))*x^m/m)+x*O(x^n)), n)}

CROSSREFS

Cf. A068963, A002618, A061255, A299069, A301986.

STATUS

approved

editing