cp's OEIS Frontend

A061255 Euler transform of Euler totient function phi(n), cf. A000010.

%I A061255 #32 Mar 03 2025 12:36:13
%S A061255 1,1,2,4,7,13,21,37,60,98,157,251,392,612,943,1439,2187,3293,4930,
%T A061255 7330,10839,15935,23315,33933,49170,70914,101861,145713,207638,294796,
%U A061255 417061,588019,826351,1157651,1616849,2251623,3126775,4330271,5981190
%N A061255 Euler transform of Euler totient function phi(n), cf. A000010.
%H A061255 Seiichi Manyama, <a href="/A061255/b061255.txt">Table of n, a(n) for n = 0..10000</a> (terms 0..1000 from T. D. Noe)
%H A061255 Alessandro De Luca, Gabriele Fici, and Andrea Frosini, <a href="https://arxiv.org/abs/2502.19926">Digital Convexity and Combinatorics on Words</a>, arXiv:2502.19926 [math.CO], 2025. See p. 11.
%H A061255 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%F A061255 G.f.: Product_{k>=1} (1 - x^k)^(-phi(k)).
%F A061255 a(n) = 1/n*Sum_{k=1..n} a(n-k)*b(k), n>1, a(0)=1, b(k) = Sum_{d|k} d*phi(d), cf. A057660.
%F A061255 Logarithmic derivative yields A057660 (equivalent to above formula). - _Paul D. Hanna_, Sep 05 2012
%F A061255 a(n) ~ exp(3^(4/3) * Zeta(3)^(1/3) * n^(2/3) / (2^(1/3) * Pi^(2/3)) - 1/6) * A^2 * Zeta(3)^(1/9) / (2^(4/9) * 3^(7/18) * Pi^(8/9) * n^(11/18)), where A is the Glaisher-Kinkelin constant A074962. - _Vaclav Kotesovec_, Mar 23 2018
%F A061255 G.f.: exp(Sum_{k>=1} (sigma_2(k^2)/sigma_1(k^2)) * x^k/k). - _Ilya Gutkovskiy_, Apr 22 2019
%t A061255 nn = 20; b = Table[EulerPhi[n], {n, nn}]; CoefficientList[Series[Product[1/(1 - x^m)^b[[m]], {m, nn}], {x, 0, nn}], x] (* _T. D. Noe_, Jun 19 2012 *)
%Y A061255 Cf. A000010, A057660, A006171, A001970, A061256, A061257, A299069, A318975.
%K A061255 easy,nonn
%O A061255 0,3
%A A061255 _Vladeta Jovovic_, Apr 21 2001