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A137245
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Decimal expansion of Sum_{p prime} 1/(p * log p).
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22
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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, 4, 0, 3, 8, 5, 2, 5, 5, 8, 4, 7, 9, 8, 9, 8, 9, 4, 4
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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 [The conjecture has been proved by Lichtman 2022. - Pontus von Brömssen, Jun 23 2022]
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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FORMULA
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EXAMPLE
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1.63661632335...
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MATHEMATICA
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digits = 105;
precision = digits + 10;
tmax = 500; (* integrand considered negligible beyond tmax *)
kmax = 500; (* f(k) considered negligible beyond kmax *)
InLogZeta[k_] := NIntegrate[Log[Zeta[t]], {t, k, tmax}, WorkingPrecision -> precision, MaxRecursion -> 20];
f[k_] := With[{mu = MoebiusMu[k]}, If[mu == 0, 0, (mu/k^2)*InLogZeta[k]]];
s = 0;
Do[s = s + f[k]; Print[k, " ", s], {k, 1, kmax}];
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PROG
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(PARI) See Belabas, Cohen link. Run as SumEulerlog(1) after setting the required precision.
(PARI) default(realprecision, 200); s=0; for(k=1, 500, s = s + moebius(k)/k^2 * intnum(x=k, [[1], 1], log(zeta(x))); print(s)); \\ Vaclav Kotesovec, Jun 12 2022
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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