A084911 Decimal expansion of linear asymptotic constant B in Sum_{k=1..n} 1/A000688(k) = ~B*n + ...
7, 5, 2, 0, 1, 0, 7, 4, 2, 3, 7, 7, 0, 2, 9, 1, 6, 1, 5, 2, 0, 6, 3, 6, 0, 7, 7, 4, 5, 5, 4, 3, 2, 5, 7, 6, 5, 6, 0, 7, 1, 8, 1, 4, 6, 9, 5, 9, 1, 2, 8, 5, 2, 6, 6, 9, 6, 3, 9, 9, 7, 9, 8, 3, 2, 6, 7, 2, 3, 5, 0, 5, 6, 8, 4, 6, 4, 7, 9, 7, 3, 7, 8, 6, 3, 9, 4, 7, 3, 6, 3, 7, 8, 0, 8, 6, 5, 4, 3, 7, 1, 0, 1, 3, 6
Offset: 0
Examples
0.7520107423...
References
- Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 5.1 Abelian group enumeration constants, p. 274.
Links
- 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.
- László Tóth, Alternating sums concerning multiplicative arithmetic functions, Journal of Integer Sequences, Vol. 20 (2017), Article 17.2.1; arXiv preprint, arXiv:1608.00795 [math.NT], 2016.
- Eric Weisstein's World of Mathematics, Abelian Group.
Programs
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Mathematica
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]] (* Jean-François Alcover, Apr 29 2016 *) -
PARI
default(realprecision, 120); default(parisize, 10000000); prodeulerrat((1-1/p)*(1 + sum(i = 1, 512, 1/(numbpart(i)*p^i)))) \\ Amiram Eldar, Mar 08 2024
Formula
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, [typo corrected by Vaclav Kotesovec, Mar 05 2024]
Equals lim_{n->oo} (1/n) * Sum_{k=1..n} 1/A000688(k). - Amiram Eldar, Oct 16 2020
Extensions
Data corrected by Jean-François Alcover, Apr 29 2016
a(10) from Vaclav Kotesovec, Mar 07 2024
More terms from Amiram Eldar, Mar 08 2024