A369763 Decimal expansion of the asymptotic mean of the ratio A000688(k)/A038538(k).

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, 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, 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

Offset: 0

Author

Amiram Eldar, Jan 31 2024

Keywords

• nonn
• cons

Comments

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.

The constant A in Kühleitner's paper (1995).

The ratio is 1 for all biquadratefree numbers (whose asymptotic density is A215267 = 0.923..., see A046100), and smaller than 1 otherwise.

Examples

			0.98771484004493763774023068670639349351901075670395...
		

Links

• Manfred Kühleitner, Comparing the number of Abelian groups and of semisimple rings of a given order, Mathematica Slovaca, Vol. 45, No. 5 (1995), pp. 509-518.

Crossrefs

Cf. A000041, A000688, A004101, A038538, A046100, A215267.

Programs

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

Formula

Equals Product_{p prime} (1 - 1/p)*(1 + Sum_{k>=1} A000041(k)/(A004101(k)*p^k)).