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%I #7 Jan 31 2024 08:07:28
%S 9,8,7,7,1,4,8,4,0,0,4,4,9,3,7,6,3,7,7,4,0,2,3,0,6,8,6,7,0,6,3,9,3,4,
%T 9,3,5,1,9,0,1,0,7,5,6,7,0,3,9,5,6,2,7,1,4,4,9,9,3,6,6,1,2,5,1,9,0,8,
%U 1,8,5,0,7,8,1,8,2,9,8,6,5,2,6,6,0,0,7,6,4,7,5,2,3,9,4,3,1,0,4,3,6,5,9,3,6
%N Decimal expansion of the asymptotic mean of the ratio A000688(k)/A038538(k).
%C The asymptotic mean of the ratio between the number of non-isomorphic abelian groups and the number of non-isomorphic semisimple rings of the same order.
%C The constant A in Kühleitner's paper (1995).
%C The ratio is 1 for all biquadratefree numbers (whose asymptotic density is A215267 = 0.923..., see A046100), and smaller than 1 otherwise.
%H Manfred Kühleitner, <a href="http://dml.cz/dmlcz/129876">Comparing the number of Abelian groups and of semisimple rings of a given order</a>, Mathematica Slovaca, Vol. 45, No. 5 (1995), pp. 509-518.
%F Equals Product_{p prime} (1 - 1/p)*(1 + Sum_{k>=1} A000041(k)/(A004101(k)*p^k)).
%e 0.98771484004493763774023068670639349351901075670395...
%o (PARI) default(realprecision, 120); my(N=512, x='x+O('x^N), v); v = Vec(1/prod(k=1, sqrtint(N)+1, prod(j=1, 1+N\k^2, 1-x^(j*k^2)))); prodeulerrat((1-1/p)*vecsum(vector(N, i, numbpart(i-1)/(v[i]*p^(i-1))))) \\ after _Vaclav Kotesovec_ at A004101
%Y Cf. A000041, A000688, A004101, A038538, A046100, A215267.
%K nonn,cons
%O 0,1
%A _Amiram Eldar_, Jan 31 2024