A143524 -id:A143524 - cp's OEIS frontend

A370093 Decimal expansion of Lichtman constant f(N*(2)).

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

Offset: 0

Artur Jasinski, Feb 09 2024

Definition:
f(N*(k)) = Integral_{s>=1} P_k*(s), where P_k*(s) = Sum_{n>1 and (big) Omega(n)=k} mu(n)^2/n^s, where mu is Möbius (or Moebius) Mu function see A008683, and (big) Omega is number of prime divisors of n counted with multiplicity see A001222.
Lichtman constant f(N*(1)) see A137245.
Lichtman constant f(N*(2)) this sequence.
Lichtman constant f(N*(3)) see A370112.
Lichtman constant f(N*(4)) see A370113.
Limit_{k->oo} f(N*(k)) = 6/Pi^2 = 0.607927101854... see A059956.
Value computed and communicated by Bill Allombert.

			0.890925479476318332...

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 right column).

Cf. A001222, A008683, A059956, A117543, A137245, A143524, A370112, A370113.

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, Feb 14 2014 [via Artur Jasinski]

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.

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]

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.

A240953 Constant in Sathe's theorem: Product_{p prime} (1 - 1/p)*e^(1/p).

Original entry on oeis.org

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

Offset: 0

Charles R Greathouse IV, Aug 04 2014

Sathe proved that pi_k(x), the count of numbers <= x with exactly k prime factors, satisfies pi_k(x) ~ f(k/log log x) * x/log x * (log log x)^(k-1)/(k-1)! where f(x) = c/gamma(x+1) * Product_{p prime} 1 + x*exp(-x/p)/p and c is this constant. This holds uniformly for k < (2 - eps)log log x for any fixed eps > 0. - Charles R Greathouse IV, Aug 02 2016

			0.72926474425711901885361531693130012817754597103784361867476691287655...

L. G. Sathe, On a problem of Hardy on the distribution of integers having a given number of prime factors. I., J. Indian Math. Soc. (N.S.) 17 (1953), pp. 63-82.
L. G. Sathe, On a problem of Hardy on the distribution of integers having a given number of prime factors. II., J. Indian Math. Soc. (N.S.) 17 (1953), pp. 83-141.
Atle Selberg, Note on a paper by L. G. Sathe, J. Indian Math. Soc. (N.S.) 18 (1954), pp. 83-87.

Cf. A001222, A143524.

digits = 103; S = E^-NSum[PrimeZetaP[ n]/n, {n, 2, Infinity}, WorkingPrecision -> digits + 10, NSumTerms -> 3*digits]; RealDigits[S, 10, digits] // First (* Jean-François Alcover, Sep 11 2015 *)

/* Helper functions and a function f to compute a k-th order approximation of the constant using the primes up to lim. */
eps(x=1.)=my(p=if(x,precision(x),default(realprecision))); precision(2. >> (32 * ceil(p * 38539962 / 371253907)), 9);
primezeta(s)=my(lm=s*log(2));lm=lambertw(lm/eps())\lm;sum(k=1,lm,moebius(k)/k*log(abs(zeta(k*s))));
f(lim,k)=my(t=0.);forprime(p=2,lim,t+=log(1-1/p)+sum(i=1,k,1/i/p^i));exp(t-sum(i=2,k,primezeta(i)/i));
f(1e8, 9)

Equals e^A143524. - Jon Maiga, Nov 17 2018

More digits from Jean-François Alcover, Sep 11 2015

A352081 Numbers of the form k*p^k, where k>1 and p is a prime.

Original entry on oeis.org

8, 18, 24, 50, 64, 81, 98, 160, 242, 324, 338, 375, 384, 578, 722, 896, 1029, 1058, 1215, 1682, 1922, 2048, 2500, 2738, 3362, 3698, 3993, 4374, 4418, 4608, 5618, 6591, 6962, 7442, 8978, 9604, 10082, 10240, 10658, 12482, 13778, 14739, 15309, 15625, 15842, 18818

Offset: 1

Amiram Eldar, Apr 16 2022

nonn

Each term in this sequence has a single presentation in the form k*p^k.

			8 is a term since 8 = 2*2^2.
18 is a term since 18 = 2*3^2.
24 is a term since 24 = 3*2^3.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Peter Lindqvist and Jaak Peetre, On the remainder in a series of Mertens, Expositiones Mathematicae, Vol. 15 (1997), pp. 467-478. See eq. (3).
Eric Weisstein's World of Mathematics, Mertens Constant. See eq. (5).

Subsequences: A036289 \ {0, 2}, A036290 \ {0, 3}, A036291 \ {0, 5}, A036293 \ {0, 7}, A073113 \ {2}, A079704, A100042, A104126.

Cf. A001620, A077761, A143524.

addP[p_, n_] := Module[{k = 2, s = {}, m}, While[(m = k*p^k) <= n, k++; AppendTo[s, m]]; s]; seq[max_] := Module[{m = Floor[Sqrt[max/2]], s = {}, ps}, ps = Select[Range[m], PrimeQ]; Do[s = Join[s, addP[p, max]], {p, ps}]; Sort[s]]; seq[2*10^4]

Sum_{n>=1} 1/a(n) = -A143524 = gamma - B_1, where gamma is Euler's constant (A001620), and B_1 is Mertens's constant (A077761).

A154943 Decimal expansion of the negated value of the sum_q [log(1-1/q)+1/q] over the semiprimes q.

Original entry on oeis.org

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

Offset: 0

R. J. Mathar, Jan 17 2009

The semiprime analog of A143524. Taylor expansion of the logarithm shows that the value is sum_{s=2,3,..,infinity} P_2(s)/s, where P_2(s) are the semiprime zeta functions in Table 3 of the preprint arXiv:0803.0900. P_2(2)=A117543 and P_2(3)=0.023806..., P_2(4)=0.004994... etc.

			Equals 0.079848040306232691897402254...

Equals the negative of Sum_{i>=1} ( log(1-1/A001358(i)) +1/A001358(i)).

cp's OEIS Frontend

A370093 Decimal expansion of Lichtman constant f(N*(2)).

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

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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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A240953 Constant in Sathe's theorem: Product_{p prime} (1 - 1/p)*e^(1/p).

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A352081 Numbers of the form k*p^k, where k>1 and p is a prime.

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A154943 Decimal expansion of the negated value of the sum_q [log(1-1/q)+1/q] over the semiprimes q.

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