A137245 -id:A137245 - cp's OEIS frontend

A354952 Decimal expansion of Sum_{p primes} 1/(p*log(p) - 1).

3, 6, 6, 3, 5, 0, 4, 5, 8, 5, 4, 6, 5, 6, 0, 3, 3, 0, 1, 6, 0, 2, 8, 2, 5, 2, 4, 4, 8, 0, 8, 2, 1, 2, 3, 3, 3, 2, 0, 9, 3, 4, 4, 5, 2, 2, 5, 6, 4, 3, 7, 3, 9, 9, 4

Offset: 1

Vaclav Kotesovec, Jun 13 2022

			3.663504585465603301602825244808212333209344522564373994...

Cf. A137245, A319232, A137250.

prec = 60; tot = 0; dif = 10^(-prec); for(s=1, 200, default(realprecision, 200 + 6*s); su = 0; d = 0; k = 0; while(abs(d)>dif || exponent(d)==-oo, k=k+1; d = moebius(k) / ((s-1)! * k^(s+1)) * intnum(x=s*k, [[1], 1], (x-s*k)^(s-1) * log(zeta(x))); su = su + d; ); tot = tot + su; print(tot);); \\ It takes several hours.

Equals Sum_{k>=1} (Sum_{p primes} 1/(p*log(p))^k).

A370112 Decimal expansion of Lichtman constant f(N*(3)).

Original entry on oeis.org

7, 1, 3, 1, 2, 3, 8, 0, 0, 5, 0, 9, 8, 9, 0, 2, 5, 5, 4, 1, 2, 0, 2, 9, 2, 7, 9, 0, 6, 9, 0, 6, 8, 1, 8, 4, 2, 5, 7, 6, 3, 0, 4, 1, 3, 8, 9, 8, 2, 8, 4, 3, 9, 9, 2, 7, 5, 0, 6, 8, 6, 5, 5, 4, 6, 1, 0, 1, 2, 6, 7, 2, 9, 3, 4, 9, 2, 4, 7, 7, 0, 8, 6, 2, 3, 9, 3, 6, 3, 8, 6, 3, 7, 1, 3, 3, 7, 4, 9, 1, 6, 9, 7, 4, 8

Offset: 0

Artur Jasinski, Feb 10 2024

For definition and links see A370093.

			0.71312380050989...

Cf. A001222, A008683, A137245, A143524, A370093, A370113.

A370113 Decimal expansion of Lichtman constant f(N*(4)).

Original entry on oeis.org

6, 5, 2, 8, 1, 2, 9, 0, 9, 8, 5, 5, 4, 0, 6, 2, 5, 6, 9, 8, 8, 3, 5, 4, 2, 4, 5, 6, 5, 3, 1, 8, 0, 9, 6, 0, 2, 0, 4, 5, 6, 9, 9, 7, 4, 9, 1, 4, 4, 5, 6, 8, 0, 5, 0, 4, 6, 9, 6, 6, 7, 7, 4, 7, 0, 3, 8, 9, 9, 9, 7, 9, 1, 5, 1, 9, 5, 9, 1, 4, 9, 4, 4, 4, 5, 9, 8, 3, 0, 2, 5, 0, 6, 0, 2, 3, 8, 6, 7, 9, 0, 5, 0, 2, 9, 1

Offset: 0

Artur Jasinski, Feb 10 2024

For definition and links see A370093.

			0.6528129098554...

Cf. A001222, A008683, A137245, A143524, A370093, A370112.

A363368 Decimal expansion of Sum_{primes p} 1/(plog(p)log(log(p))).

Original entry on oeis.org

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

Offset: 1

Artur Jasinski, Jun 11 2023

Value computed and communicated by Bill Allombert and confirmed by Pascal Sebah.

			1.9069738480349544...

Cf. A077761, A086242, A137245, A138312, A221711, A303493, A319231, A319232, A354887, A354917, A354952, A361972, A362533.

/* 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])

A372765 Decimal expansion of Lichtman constant f(N(2)).

Original entry on oeis.org

1, 1, 4, 4, 8, 1, 6, 5, 7, 3, 4, 0, 5, 9, 1, 7, 9, 9, 1, 5, 2, 4, 4, 5, 0, 1, 7, 3, 8, 9, 3, 3, 4, 1, 0, 7, 9, 1, 3, 1, 3, 0, 4, 9, 7, 4, 0, 1, 7, 4, 3, 6, 7, 3, 9, 1, 1, 9, 8, 9, 7, 6, 7, 3, 1, 7, 3, 0, 4, 9, 8, 7, 5, 5, 6, 8, 3, 2, 1, 1, 7, 6, 4, 9, 1, 8, 8, 2, 0, 6, 7, 5, 1, 7, 2, 3, 8, 7, 8, 8, 0, 7, 1, 1, 6

Offset: 1

Artur Jasinski, May 14 2024

Definition:
f(N(k)) = Sum_{n>1 and (big) Omega(n)=k} 1/(n*log(n)), where (big) Omega is number of prime divisors of n counted with multiplicity see A001222.
f(N(k)) = Integral_{s>=1} P_k(s), where P_k(s) = Sum_{n>1 and (big) Omega(n)=k} 1/n^s.
Lichtman constant f(N(1)) see A137245.
Lichtman constant f(N(2)) this sequence.
Lichtman constant f(N(3)) see A372827.
Lichtman constant f(N(4)) see A372828.
Minimal value of f(N(k)) occurs for k=6 f(N(6)) = 0.9887534530145...
For k>=6, 1 > f(N(k+1)) > f(N(k)).
When k -> oo then f(N(k)) -> 1.
Value computed and communicated by Bill Allombert.

			1.1448165734059179915...

Bill Allombert, Results of pari computation of Lichtman constants f(N(k)) with precision 500 decimals for k=1..20, email 20.06.2023
Jared Duker Lichtman, Almost primes and the Banks-Martin conjecture, arXiv:1909.00804 [math.NT], 2019 (Figure 2 left column).

Cf. A001222, A008683, A059956, A117543, A137245, A143524, A370093, A370112, A370113, A372827, A372828.

pz(x)= sum(n=1, max(2, bitprecision(x)/x), my(a=moebius(n)); if(a!=0, a*log(zeta(n*x))/n));
Lichtman(n)=intnum(s=1, [oo, log(2)], exp(sum(i=1, n, pz(i*s)*x^i/i)+O(x^(n+1)))-1)
Lichtman(20)
\\ Bill Allombert, May 14 2024 [via Artur Jasinski]

A361089 a(n) = smallest integer x such that Sum_{k = 2..x} 1/(k*log(log(k))) > n.

Original entry on oeis.org

3, 5, 8, 21, 76, 389, 2679, 23969, 269777, 3717613, 61326301, 1188642478, 26651213526, 682263659097, 19720607003199, 637490095320530, 22857266906194526, 902495758030572213, 38993221443197045348, 1833273720522384358862

Offset: 2

Artur Jasinski, Jun 11 2023

nonn

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.

			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.

Pascal Sebah, Table of n, a(n) for n = 2..35

Cf. A077761, A086242, A137245, A138312, A221711, A303493, A319231, A319232, A354887, A354917, A354952, A361972, A362533, A363368, A363078.

(*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

For n >= 3, a(n) = round(w) where w is the solution of the equation li(log(w)) + 2.7977647035208... = n.

A363078 Decimal expansion of lim_{x->oo} (Sum_{k=2..x} 1 / (k*log(log(k)))) - li(log(x)).

Original entry on oeis.org

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

Offset: 1

Artur Jasinski, Jun 11 2023

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.

			2.7977647035208...

Cf. A077761, A086242, A137245, A138312, A221711, A303493, A319231, A319232, A354887, A354917, A354952, A361972, A362533, A363368, A361089.

A372827 Decimal expansion of Lichtman constant f(N(3)).

Original entry on oeis.org

1, 0, 3, 0, 8, 3, 5, 1, 0, 1, 7, 9, 3, 2, 1, 7, 5, 7, 1, 9, 5, 5, 6, 8, 8, 8, 9, 9, 7, 9, 6, 1, 0, 0, 3, 9, 0, 9, 2, 5, 9, 3, 6, 0, 7, 2, 9, 9, 1, 1, 5, 6, 6, 5, 9, 5, 3, 8, 0, 0, 5, 4, 6, 1, 0, 8, 6, 3, 8, 8, 4, 8, 1, 5, 4, 4, 2, 9, 7, 4, 9, 6, 8, 6, 8, 7, 9, 1, 6, 5, 9, 7, 5, 7, 5, 4, 7, 7, 6, 0, 2, 2, 3, 1, 4

Offset: 1

Artur Jasinski, May 14 2024

For definition and links see A372765.

			1.0308351017932175719...

Cf. A001222, A008683, A059956, A117543, A137245, A143524, A370093, A370112, A370113, A372765, A372828.

A372828 Decimal expansion of Lichtman constant f(N(4)).

Original entry on oeis.org

9, 9, 7, 3, 4, 2, 1, 4, 8, 5, 9, 5, 2, 5, 2, 3, 5, 9, 7, 7, 7, 5, 9, 3, 5, 9, 9, 5, 4, 8, 7, 8, 1, 9, 7, 9, 2, 7, 1, 1, 9, 2, 4, 1, 3, 5, 5, 3, 8, 2, 2, 1, 7, 2, 7, 1, 8, 8, 9, 8, 2, 9, 2, 4, 7, 7, 0, 8, 4, 2, 3, 4, 7, 6, 7, 5, 2, 7, 6, 7, 8, 4, 6, 0, 4, 4, 9, 8, 1, 5, 1, 7, 6, 9, 9, 0, 6, 1, 6, 6, 5, 8, 7, 7, 3

Offset: 0

Artur Jasinski, May 14 2024

For definition and links see A372765.

			0.997342148595252359...

Cf. A001222, A008683, A059956, A117543, A137245, A143524, A370093, A370112, A370113, A372765, A372827.

A137250 Decimal expansion of the constant sum 1/(q*log(q)), summed over prime powers q > 1.

Original entry on oeis.org

2, 0, 0, 6, 6, 6, 6, 4, 5, 2, 8, 3, 1, 0, 6, 8, 7, 5, 6, 4, 3, 2, 2, 9, 6, 9, 9, 9, 4, 7, 1, 3, 5, 8, 2, 0, 8, 4, 8, 8, 6, 8, 3, 5, 4, 1, 4, 7, 5, 0, 4, 5, 7, 8, 0, 5, 9, 0, 5, 4, 9, 8, 2, 7, 8, 2, 7, 4, 7, 8, 2, 1, 9, 2, 1, 6, 4, 7, 0, 5, 5, 0, 3, 1, 8, 4, 3, 8, 1, 7, 5, 9, 2, 0, 1, 5, 6, 1, 0, 1, 3, 0, 7, 9, 6

Offset: 1

R. J. Mathar, Mar 09 2008

Evaluated from Sum_{m,k >= 1} A008683(k)*I(k*m)/k^2, where I(x) = Integral_{t=x..infinity} log zeta(t) dt is Cohen's underivative.

			2.0066664528310687...

Henri Cohen, Number Theory, Volume II: Analytic and Modern Tools, GTM Vol. 240, Springer, 2007; see pp. 208-209.

D. A. Clark, An upper bound of sum 1/(a_i log a_i) for quasi-primitive sequences, Comp. Math. Appl., 35 (1998), 105-109.
H. Cohen, High precision computation of Hardy-Littlewood constants, preprint, 1998.
Henri Cohen, High-precision computation of Hardy-Littlewood constants. [pdf copy, with permission]
R. J. Mathar, Twenty digits of some Integrals of the Prime Zeta Function, arXiv:0811.4739 [math.NT].

Cf. A137245, A221711.

default(realprecision, 200); su = 0; for(s=1, 400, su = su + sum(k=1, 500, moebius(k)/k^2 * intnum(x=s*k,[[1], 1], log(zeta(x))))/s; print(su)); \\ Vaclav Kotesovec, Jun 12 2022

Equals Sum_{n>=2} 1/(A000961(n)*log(A000961(n))).

Equals Sum_{p primes} -log(1-1/p)/log(p). - Vaclav Kotesovec, Jun 12 2022

8 more digits from R. J. Mathar, Dec 04 2008

More terms from Vaclav Kotesovec, Jun 12 2022

cp's OEIS Frontend

A354952 Decimal expansion of Sum_{p primes} 1/(p*log(p) - 1).

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A370112 Decimal expansion of Lichtman constant f(N*(3)).

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A370113 Decimal expansion of Lichtman constant f(N*(4)).

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A363368 Decimal expansion of Sum_{primes p} 1/(p*log(p)*log(log(p))).

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A372765 Decimal expansion of Lichtman constant f(N(2)).

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A361089 a(n) = smallest integer x such that Sum_{k = 2..x} 1/(k*log(log(k))) > n.

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A363078 Decimal expansion of lim_{x->oo} (Sum_{k=2..x} 1 / (k*log(log(k)))) - li(log(x)).

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A372827 Decimal expansion of Lichtman constant f(N(3)).

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A372828 Decimal expansion of Lichtman constant f(N(4)).

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A137250 Decimal expansion of the constant sum 1/(q*log(q)), summed over prime powers q > 1.

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A363368 Decimal expansion of Sum_{primes p} 1/(plog(p)log(log(p))).