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A162583 G.f.: A(x) = exp( 6*Sum_{n>=1} A006519(n)*A038500(n) * x^n/n ).

%I A162583 #7 Jul 04 2018 20:15:53
%S A162583 1,6,24,78,222,570,1356,3030,6432,13074,25608,48558,89502,160854,
%T A162583 282624,486534,822174,1365978,2234400,3602742,5732202,9008034,
%U A162583 13993320,21503730,32711460,49287750,73598280,108968334,160041750,233262786
%N A162583 G.f.: A(x) = exp( 6*Sum_{n>=1} A006519(n)*A038500(n) * x^n/n ).
%C A162583 A006519(n) = highest power of 2 dividing n and
%C A162583 A038500(n) = highest power of 3 dividing n.
%H A162583 G. C. Greubel, <a href="/A162583/b162583.txt">Table of n, a(n) for n = 0..1000</a>
%e A162583 G.f.: A(x) = 1 + 6*x + 24*x^2 + 78*x^3 + 222*x^4 + 570*x^5 + 1356*x^6 + ...
%e A162583 log(A(x))/6 = x + 2*x^2/2 + 3*x^3/3 + 4*x^4/4 + x^5/5 + 6*x^6/6 + x^7/7 + 8*x^8/8 + 9*x^9/9 + ... + A006519(n)*A038500(n)*x^n/n + ...
%t A162583 nmax = 150; a[n_]:= SeriesCoefficient[Series[Exp[Sum[2^(IntegerExponent[k, 2] + 1)*3^(IntegerExponent[k, 3] + 1)*q^k/k, {k, 1, nmax}]], {q, 0, nmax}], n]; Table[a[n], {n, 0, 50}] (* _G. C. Greubel_, Jul 04 2018 *)
%o A162583 (PARI) {a(n)=local(L=sum(m=1,n,6*2^valuation(m,2)*3^valuation(m,3)*x^m/m)+x*O(x^n));polcoeff(exp(L),n)}
%Y A162583 Cf. A006519, A038500.
%K A162583 nonn
%O A162583 0,2
%A A162583 _Paul D. Hanna_, Jul 06 2009