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%I
%S 7,5,2,0,1,0,7,4,2,3
%N Decimal expansion of linear asymptotic constant B in Sum_{k=1..n} 1/A000688(k) = ~B*n + ...
%D Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 5.1 Abelian group enumeration constants, p. 274.
%H Jean-Marie De Koninck and Aleksandar Ivić, <a href="https://www.sciencedirect.com/bookseries/north-holland-mathematics-studies/vol/43">Topics in Arithmetical Functions: Asymptotic Formulae for Sums of Reciprocals of Arithmetical Functions and Related Fields</a>, Amsterdam, Netherlands: North-Holland, 1980. See p. 16.
%H László Tóth, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL20/Toth/toth25.html">Alternating sums concerning multiplicative arithmetic functions</a>, Journal of Integer Sequences, Vol. 20 (2017), Article 17.2.1; <a href="https://arxiv.org/abs/1608.00795">arXiv preprint</a>, arXiv:1608.00795 [math.NT], 2016.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/AbelianGroup.html">Abelian Group</a>.
%F Equals Product_{p prime} (1-Sum_{k >= 2} (1/P(k-1)-1/P(k)/p^k), where P(k) is the unrestricted partition function. - _Jean-François Alcover_, Apr 29 2016
%F Equals lim_{n->oo} (1/n) * Sum_{k=1..n} 1/A000688(k). - _Amiram Eldar_, Oct 16 2020
%e 0.7520107423...
%t 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];
%t 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];
%t s[p_] := Sum[DP[n]/p^n, {n, 2, nmax[p]}] ;
%t 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]]]];
%t A0 = f[m]; RealDigits[A0, 10, digits][[1]] (* _Jean-François Alcover_, Apr 29 2016 *)
%Y Cf. A000688, A021002, A084892, A084893, A272339.
%K nonn,cons,more
%O 0,1
%A _Eric W. Weisstein_, Jun 11 2003
%E Data corrected by _Jean-François Alcover_, Apr 29 2016
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