Search: a354887 -id:a354887
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A363368
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Decimal expansion of Sum_{primes p} 1/(p*log(p)*log(log(p))).
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+10
3
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1, 9, 0, 6, 9, 7, 3, 8, 4, 8, 0, 3, 4, 9, 5, 4, 4, 1, 7, 7, 8, 7, 5, 7, 9, 6, 6, 9, 6, 5, 1, 9, 6, 4, 0, 3, 3, 6, 1, 8, 9, 3, 8, 3, 5, 2, 2, 9, 4, 8, 5, 3, 6, 6, 0, 5, 5, 9, 5, 2, 4, 2, 9, 4, 7, 1, 4, 5, 6, 7, 8, 3, 1, 2, 9, 2, 5, 2, 2, 4, 4, 1, 0, 9, 2, 3, 1, 8, 7, 1, 9, 4, 1, 3, 3, 4, 1, 6, 4, 8, 2, 2, 4, 2, 3
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OFFSET
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1,2
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COMMENTS
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Value computed and communicated by Bill Allombert and confirmed by Pascal Sebah.
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LINKS
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EXAMPLE
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1.9069738480349544...
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PROG
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(PARI) /* author Bill Allombert */
\p150
pz(x, ex=0)=
{
my(s=bitprecision(x));
my(B=s/real(polcoef(x, 0))+ex);
sum(n=1, B, my(a=moebius(n)); if(a!=0, a*log(zeta(n*x))/n));
}
my(P=primes([2, 61])); intnum(x=1, [oo, log(67)], (pz(x)-vecsum([p^-x|p<-P]))*intnum(s=0, [oo, 1], (x-1)^s/gamma(1+s))) + vecsum([1/p/log(p)/log(log(p))|p<-P])
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CROSSREFS
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Cf. A077761, A086242, A137245, A138312, A221711, A303493, A319231, A319232, A354887, A354917, A354952, A361972, A362533.
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KEYWORD
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AUTHOR
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STATUS
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approved
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A361089
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a(n) = smallest integer x such that Sum_{k = 2..x} 1/(k*log(log(k))) > n.
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+10
2
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3, 5, 8, 21, 76, 389, 2679, 23969, 269777, 3717613, 61326301, 1188642478, 26651213526, 682263659097, 19720607003199, 637490095320530, 22857266906194526, 902495758030572213, 38993221443197045348, 1833273720522384358862
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OFFSET
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2,1
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COMMENTS
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Because lim_{x->oo} (Sum_{k=2..x} 1 / (k*log(log(k)))) - li(log(x)) = 2.7977647035208... (see A363078) then a(n) = round(w) where w is the solution of the equation li(log(w)) + 2.7977647035208... = n.
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LINKS
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FORMULA
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For n >= 3, a(n) = round(w) where w is the solution of the equation li(log(w)) + 2.7977647035208... = n.
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EXAMPLE
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a(2) = 3 because Sum_{k=2..3} 1/(k*log(log(k))) = 2.18008755... > 2 and Sum_{k=2..2} 1/(k*log(log(k))) = -1.364208386450... < 2.
a(7) = 389 because Sum_{k=2..389} 1/(k*log(log(k))) = 7.000345... > 7 and Sum_{k=2..388} 1/(k*log(log(k))) = 6.99890560988... < 7.
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MATHEMATICA
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(*slow procedure*)
lim = 2; sum = 0; aa = {}; Do[sum = sum + N[1/(k Log[Log[k]]), 100];
If[sum >= lim, AppendTo[aa, k]; Print[{lim, sum, k}];
lim = lim + 1], {k, 2, 269777}]; aa
(*quick procedure *)
aa = {3}; cons = 2.79776470352080492766050456553352884330850083202326989577856315;
Do[ww = w /. NSolve[LogIntegral[Log[w]] + cons == n, w];
AppendTo[aa, Round[ww][[1]]], {n, 3, 21}]; aa
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CROSSREFS
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Cf. A077761, A086242, A137245, A138312, A221711, A303493, A319231, A319232, A354887, A354917, A354952, A361972, A362533, A363368, A363078.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A363078
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Decimal expansion of lim_{x->oo} (Sum_{k=2..x} 1 / (k*log(log(k)))) - li(log(x)).
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+10
2
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2, 7, 9, 7, 7, 6, 4, 7, 0, 3, 5, 2, 0, 8, 0, 4, 9, 2, 7, 6, 6, 0, 5, 0, 4, 5, 6, 5, 5, 3, 3, 5, 2, 8, 8, 4, 3, 3, 0, 8, 5, 0, 0, 8, 3, 2, 0, 2, 3, 2, 6, 9, 8, 9, 5, 7, 7, 8, 5, 6, 3, 1, 5, 0, 0, 5, 0, 6, 4, 3, 2, 8, 9, 3, 6, 2, 4, 5, 4, 5, 9, 4, 8, 3, 6, 8, 6, 8, 2, 5, 4, 8, 1, 8, 2, 9, 5, 4, 1, 9, 2, 5, 5, 0, 8
(list;
constant;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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1,1
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COMMENTS
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Value computed and communicated by Pascal Sebah.
For the smallest integer x such that Sum_{k = 2..x} 1/(k*log(log(k))) > n see A361089.
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LINKS
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EXAMPLE
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2.7977647035208...
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CROSSREFS
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Cf. A077761, A086242, A137245, A138312, A221711, A303493, A319231, A319232, A354887, A354917, A354952, A361972, A362533, A363368, A361089.
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KEYWORD
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AUTHOR
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STATUS
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approved
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A366249
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Decimal expansion of lim_{x->oo} (Sum_{primes p<=x} 1/(p*log(log(p)))) - log(log(log(x))).
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+10
0
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OFFSET
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1,1
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COMMENTS
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Value computed and communicated by Pascal Sebah.
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LINKS
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MAPLE
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2.93832901...
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CROSSREFS
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Cf. A077761, A086242, A137245, A138312, A221711, A303493, A319231, A319232, A354887, A354917, A354952, A361972, A362533, A363078, A363368.
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KEYWORD
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AUTHOR
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STATUS
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approved
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