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A272169
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Decimal expansion of A_0 (so named by S. Finch), a constant related to the asymptotic expression of the sum of the reciprocals of the number of abelian groups of a given order.
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1
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OFFSET
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0,1
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REFERENCES
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Jean-Marie De Koninck and Aleksandar Ivić, Topics in Arithmetical Functions: Asymptotic Formulae for Sums of Reciprocals of Arithmetical Functions and Related Fields, Amsterdam, Netherlands: North-Holland, 1980. See p. 16.
Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 5.1 Abelian group enumeration constants, p. 274.
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LINKS
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Table of n, a(n) for n=0..9.
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FORMULA
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A_0 = Product_{p prime} (1-Sum_{k >= 2} (1/P(k-1)-1/P(k)/p^k), where P(k) is the unrestricted partition function.
Equals lim_{n->oo} (1/n) * Sum_{k=1..n} 1/A000688(k). - Amiram Eldar, Oct 16 2020
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EXAMPLE
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0.7520107423...
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MATHEMATICA
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digits = 10; m0 (* initial number of primes *) = 10^6; dm = 2*10^5; PP = PartitionsP; DP[n_] := DP[n] = (1/PP[n - 1] - 1 /PP[n]) // N[#, digits + 5]&; pmax = Prime[1000];
nmax[p_ /; p <= pmax] := nmax[p] = Module[{n}, For[n = 2, n < 1000, n++, If[Abs[1/PP[n - 1] - 1 /PP[n]]/p^n < 10^-100, Return[n]]]]; nmax[p_ /; p > pmax] := nmax[pmax];
s[p_] := Sum[DP[n]/p^n, {n, 2, nmax[p]}] ;
f[m_] := f[m] = Product[1 - s[p], {p, Prime[Range[m]]}]; f[m0]; f[m = m0 + dm]; While[RealDigits[f[m], 10, digits + 2][[1]] != RealDigits[f[m - dm], 10, digits + 2][[1]], m = m + dm; Print[m, " ", RealDigits[f[m]]]];
A0 = f[m]; RealDigits[A0, 10, digits][[1]]
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CROSSREFS
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Cf. A000688, A021002, A084892, A084893, A272339.
Sequence in context: A316334 A196486 A216853 * A073742 A084911 A071876
Adjacent sequences: A272166 A272167 A272168 * A272170 A272171 A272172
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KEYWORD
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nonn,cons,more,changed
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AUTHOR
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Jean-François Alcover, Apr 29 2016
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STATUS
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approved
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