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A137245
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Decimal expansion of Sum_{p prime} 1/(p * log p).
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8
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1, 6, 3, 6, 6, 1, 6, 3, 2, 3, 3, 5, 1, 2, 6, 0, 8, 6, 8, 5, 6, 9, 6, 5, 8, 0, 0, 3, 9, 2, 1, 8, 6, 3, 6, 7, 1, 1, 8, 1, 5, 9, 7, 0, 7, 6, 1, 3, 1, 2, 9, 3, 0, 5, 8, 6, 0, 0, 3, 0, 4, 9, 1, 9, 7, 8, 1, 3, 3, 9, 9, 7, 4, 4, 6, 7, 9, 4, 6, 9, 8, 6, 5, 4, 7, 0, 0
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OFFSET
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1,2
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COMMENTS
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Sum_{p prime} 1/(p^s * log p) equals this value here if s=1, equals A221711 if s=2, 0.22120334039... if s=3. See arXiv:0811.4739.
Erdős (1935) proved that for any sequence where no term divides another, the sum of 1/(x log x) is at most some constant C. He conjectures (1989) that C can be taken to be this constant 1.636..., that is, the primes maximize this sum. - Charles R Greathouse IV, Mar 26 2012
Note that sum 1/(p * log p) is almost (a tiny bit less than) 1 + 2/Pi = 1+A060294 = 1.63661977236758... (Why is it so close?) - Daniel Forgues, Mar 26 2012
Sum 1/(p * log p) is quite close to sum 1/n^2 = Pi^2/6 = 1.644934066... (Cf. David C. Ullrich, "Re: What is Sum(1/p log p)?" for why this is so; mentions A115563.) - Daniel Forgues, Aug 13 2012
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REFERENCES
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Henri Cohen, Number Theory, Volume II: Analytic and Modern Tools, GTM Vol. 240, Springer, 2007; see pp. 208-209.
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LINKS
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Table of n, a(n) for n=1..87.
Karim Belabas and Henri Cohen, Computation of sum_{p prime} 1/(p^s log(p)), PARI/GP script, 2020.
Henri Cohen, High-precision computation of Hardy-Littlewood constants, (1998).
Henri Cohen, High-precision computation of Hardy-Littlewood constants. [pdf copy, with permission]
P. Erdős, Note on sequences of integers no one of which is divisible by any other, J. London Math. Soc. 10 (1935), pp. 126-128, [DOI].
P. Erdős, Some problems and results on combinatorial number theory, Graph theory and its applications: East and West (Jinan, 1986), Ann. New York Acad. Sci., 576 , pp. 132-145, New York Acad. Sci., New York, 1989.
J. K. Lichtman, Almost primes and the Banks-Martin conjecture, arXiv:1909.00804 [math.NT], 2019.
R. J. Mathar, Twenty digits of some integrals of the prime zeta function, arXiv:0811.4739 [math.NT], 2008-2009, table in Section 2.4.
David C. Ullrich, Re: What is Sum(1/p log p)?, posting in newsgroup sci.math.research, 11 Feb 2006.
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FORMULA
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Equals Sum_(n>=1} 1/(A000040(n)*log A000040(n)).
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EXAMPLE
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1.63661632335...
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MATHEMATICA
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(* This naive script gives 12 digits in a few seconds *)
digits = 12;
InLogZeta[k_Integer] := NIntegrate[Log[Zeta[t]], {t, k, Infinity}, WorkingPrecision -> digits+10];
f[k_Integer] := With[{mu = MoebiusMu[k]}, If[mu==0, 0, (mu/k^2)* InLogZeta[k]]];
s = NSum[f[k], {k, 1, Infinity}, WorkingPrecision -> digits+10];
RealDigits[s, 10, digits][[1]] (* Jean-François Alcover, Feb 06 2021 *)
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PROG
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(PARI) See Belabas, Cohen link. Run as SumEulerlog(1) after setting the required precision.
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CROSSREFS
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Cf. A221711 (p squared), A115563, A319231 (log squared), A319232 (p and log squared).
Sequence in context: A176715 A229522 A227400 * A060294 A181171 A193025
Adjacent sequences: A137242 A137243 A137244 * A137246 A137247 A137248
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KEYWORD
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cons,nonn
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AUTHOR
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R. J. Mathar, Mar 09 2008
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EXTENSIONS
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More terms from Hugo Pfoertner, Feb 01 2020
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STATUS
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approved
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