A138312 -id:A138312 - cp's OEIS frontend

A340839 Decimal expansion of Mertens constant C(5,1).

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

Offset: 1

Artur Jasinski, Jan 23 2021

Data taken from Alessandro Languasco and Alessandro Zaccagnini 2007.

			1.225238438539084580057609774749220527540595509391649938767...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.2 Meissel-Mertens constants (pp. 94-95)

Vaclav Kotesovec, Table of n, a(n) for n = 1..499
Alessandro Languasco and Alessandro Zaccagnini, On the constant in the Mertens product for arithmetic progressions. II: Numerical values, Math. Comp. 78 (2009), 315-326.
Alessandro Languasco and Alessandro Zaccagnini, Computation of the Mertens constants - more than 100 correct digits, (2007), 1-134 (digital data relative to the previous paper). [in this table on page 4, the last correct digit is a(109), beyond the level there certified. - _Vaclav Kotesovec_, Jan 26 2021]
Alessandro Languasco and Alessandro Zaccagnini, Computation of the Mertens constants mod q; 3 <= q <= 100, (2007) (GP-PARI procedure 100 digits accuracy).
Alessandro Languasco and Alessandro Zaccagnini, On the constant in the Mertens product for arithmetic progressions. I. Identities., Funct. Approx. Comment. Math. Volume 42, Number 1 (2010), 17-27.
For other links see A340711.

Cf. A077761, A083343, A091589, A138312, A161529, A230767, A238114, A271971, A340127, A340794, A340866.

A = C(5,1)=1.225238438539084580057609774749220527540595509391649938767...
B = C(5,2)=0.546975845411263480238301287430814037751996324100819295153...
C = C(5,3)=0.805951040448267864057376860278430932081288114939010897934...
D = C(5,4)=1.299364547914977988160840014964265909502574970408329662016...
A*B*C*D = 0.70182435445860646228... = (5/4)*exp(-gamma), where gamma is the Euler-Mascheroni constant A001620.
Formula from the article by Languasco and Zaccagnini, 2010, p.9:
A = ((13*sqrt(5)*Pi^2*exp(-gamma))/(150*log((1+sqrt(5))/2))*A340628/A340808)^(1/4).

Last 11 digits corrected by Vaclav Kotesovec, Jan 25 2021

More digits from Vaclav Kotesovec, Jan 26 2021

A138316 Numerators of the squarefree totient analogs of the harmonic numbers F_n.

Original entry on oeis.org

1, 2, 5, 5, 11, 13, 41, 41, 41, 11, 113, 113, 77, 241, 497, 497, 1009, 1009, 3067, 3067, 3127, 3199, 35549, 35549, 35549, 36209, 36209, 36209, 255443, 262373, 264221, 264221, 266993, 135229, 17048, 17048, 22859, 69347, 139849, 139849, 70271, 35713

Offset: 1

Dick Boland (abstract(AT)imathination.org), Mar 13 2008, Mar 27 2008

F_n-H_n approaches a constant, 'kappa', conjectured to be equivalent to the difference of B_3-gamma, where B_3 is Mertens' 3rd constant and gamma is Euler's constant.

			Numerators of F_n, e.g., F_1 = (1/1), F_2 = (1/1 + 1/1), ... F_11 = (1/1 + 1/1 + 1/2 + 0 + 1/4 + 1/2 + 1/6 + 0 + 0 + 1/4 + 1/10).

G. C. Greubel, Table of n, a(n) for n = 1..10000
Dick Boland, An Analog of the Harmonic Numbers Over the Squarefree Integers

Cf. A138312, A138313, A138312, A138317, A138320, A138321, A083343, A001620.

Table[Numerator[Sum[MoebiusMu[k]^2/EulerPhi[k], {k, 1, n}]], {n, 1, 60}]

a(n) = numerator(sum(k=1, n, if (issquarefree(k), 1/eulerphi(k)))); \\ Michel Marcus, Aug 28 2018

a(n) = numerator[sum(k=1 to n)mu^2(k)/phi(k)] where mu(k) is the Mobius function and phi(k) is Euler's Totient function.

A138317 Denominators of the squarefree totient analogs of the harmonic numbers F_n.

Original entry on oeis.org

1, 1, 2, 2, 4, 4, 12, 12, 12, 3, 30, 30, 20, 60, 120, 120, 240, 240, 720, 720, 720, 720, 7920, 7920, 7920, 7920, 7920, 7920, 55440, 55440, 55440, 55440, 55440, 27720, 3465, 3465, 4620, 13860, 27720, 27720, 13860, 6930, 3465, 3465, 3465, 6930, 79695, 79695

Offset: 1

Dick Boland (abstract(AT)imathination.org), Mar 13 2008, Mar 27 2008

F_n-H_n approaches a constant, 'kappa', conjectured to be equivalent to the difference of B_3-gamma, where B_3 is Mertens' 3rd constant and gamma is Euler's constant.

			Denominators of F_n, e.g., - F_1 = (1/1), F_2 = (1/1 + 1/1), ... F_11 = (1/1 + 1/1 + 1/2 + 0 + 1/4 + 1/2 + 1/6 + 0 + 0 + 1/4 + 1/10).

G. C. Greubel, Table of n, a(n) for n = 1..10000
Dick Boland, An Analog of the Harmonic Numbers Over the Squarefree Integers

Cf. A138312, A138313, A138312, A138316, A138320, A138321, A083343, A001620.

Table[Denominator[Sum[MoebiusMu[k]^2/EulerPhi[k], {k, 1, n}]], {n, 1, 60}]

a(n) = denominator(sum(k=1, n, if (issquarefree(k), 1/eulerphi(k)))); \\ Michel Marcus, Aug 28 2018

a(n)=Denominator[sum(k=1 to n)mu^2(k)/phi(k)] where mu(k) is the Mobius function and phi(k) is Euler's Totient function.

A138313 Decimal expansion of constant 'kappa' = lim_{n -> infinity} (F_n - H_n), where H_n are harmonic numbers, F_n are squarefree totient analogs of H_n.

Original entry on oeis.org

7, 5, 5, 3, 6, 6

Offset: 0

Dick Boland (abstract(AT)imathination.org), Mar 13 2008

The squarefree totient analog of the harmonic number F_n is given by F_n = Sum_{k=1..n} mu^2(k)/phi(k) where mu(k) is the Möbius function and phi(k) is Euler's totient function.

Conjectured to be equivalent to Mertens's constant B_3 minus Euler's constant (A138312). B_3 - gamma is given by Sum_{i>=1} log p_i/(p_i*(p_i-1)), where p_i is the i^th prime = Sum_{j>=2} mu(j)*zeta'(j)/zeta(j), mu(j) is the Mobius function, zeta'(j) is the derivative of zeta(j).

			0.755366...

Dick Boland, An Analog of the Harmonic Numbers Over the Squarefree Integers, 2008.

Cf. A138316, A138317 (numerators and denominators of the squarefree totient analogs of the harmonic numbers).
Cf. A138312 (Mertens's B_3 minus Euler's constant), A083343 (Mertens's B_3), A001620 (Euler's constant).
Cf. A000010, A008683, A001008, A002805.

<< NumberTheory`NumberTheoryFunctions` prl = 100000; ts = 0; f = 1; While[f < 100000000000, If[SquareFreeQ[f], ts += N[1/EulerPhi[f], 15]; If[f > prl, Print[{f, ts, hn = N[HarmonicNumber[f], 15], N[ts - hn, 10]}]; prl += 100000]]; f += 1]

Limit_{n -> infinity} ((Sum_{k=1..n} mu^2(k)/phi(k)) - H_n), where mu(k) is the Möbius function, phi(k) is Euler's totient function and H_n is the n-th harmonic number.

A138320 Numerators of the difference between the squarefree totient analogs of the harmonic numbers and the harmonic numbers: F_n-H_n.

Original entry on oeis.org

0, 1, 2, 5, 7, 4, 173, 587, 1481, 1859, 20701, 18391, 241393, 275713, 148367, 548423, 2342059, 241321, 41436061, 19263077, 40604659, 43779103, 1009564739, 1907583043, 9002492327, 9603126977, 27322095131, 25887926681, 752184042199

Offset: 1

Dick Boland (abstract(AT)imathination.org), Mar 14 2008, Mar 27 2008

F_n-H_n approaches a constant, 'kappa', conjectured to be equivalent to the difference of B_3-gamma, where B_3 is Mertens' 3rd constant and gamma is Euler's constant.

			Numerators of F_n - H_n, e.g. - F_1 - H_1 = (1/1-1/1), F_2 = ((1/1-1/1) + (1/1-1/2)),...
F_11 = ((1/1-1/1) +(1/1-1/2) +(1/2-1/3) +(0-1/4) +(1/4-1/5) +(1/2-1/6) +(1/6-1/7) +(0-1/8) +(0-1/9) +(1/4-1/10) +(1/10-1/11)).

G. C. Greubel, Table of n, a(n) for n = 1..1000
Dick Boland, An Analog of the Harmonic Numbers Over the Squarefree Integers

Cf. A138312, A138313, A138312, A138316, A138317, A138321, A083343, A001620.

Table[Numerator[Sum[MoebiusMu[k]^2/EulerPhi[k], {k, 1, n}]-HarmonicNumber[n]], {n, 1, 60}]

for(n=1,60, print1(numerator(sum(k=1,n, moebius(k)^2/eulerphi(k)) - sum(j=1,n,1/j)), ", ")) \\ G. C. Greubel, Aug 31 2018

a(n) = Numerator[(Sum_{k=1..n} mu^2(k)/phi(k)) - H_n] where mu(k) is the Mobius function, phi(k) is Euler's Totient function and H_n is the n-th Harmonic Number.

A138321 Denominators of the difference between the squarefree totient analogs of the harmonic numbers and the harmonic numbers: F_n - H_n.

Original entry on oeis.org

1, 2, 3, 12, 15, 5, 210, 840, 2520, 2520, 27720, 27720, 360360, 360360, 180180, 720720, 3063060, 340340, 58198140, 29099070, 58198140, 58198140, 1338557220, 2677114440, 13385572200, 13385572200, 40156716600, 40156716600, 1164544781400, 582272390700, 18050444111700, 144403552893600

Offset: 1

Dick Boland (abstract(AT)imathination.org), Mar 14 2008, Mar 27 2008

F_n - H_n approaches a constant, 'kappa', conjectured to be equivalent to the difference of B_3-gamma, where B_3 is Mertens's 3rd constant and gamma is Euler's constant.

Original data was given as {1, 1, 12, 24, 240, 80, 560, 3360, 30240, 7560, 831600, 831600, 93600, 21621600, 6177600, 12355200, 2940537600, 980179200, 55870214400, 2234808576, 3724680960, 177365760, 49597067520, 29758240512, 3719780064000} which is in error for this sequence. - G. C. Greubel, Sep 14 2018

			Denominators of F_n - H_n, e.g., -F_1 - H_1 = (1/1 - 1/1), F_2 = ((1/1 - 1/1) + (1/1 - 1/2)), ...
F_11 = ((1/1 - 1/1) + (1/1 - 1/2) + (1/2 - 1/3) + (0 - 1/4) + (1/4 - 1/5) + (1/2 - 1/6) + (1/6 - 1/7) + (0 - 1/8) + (0 - 1/9) + (1/4 - 1/10) + (1/10 - 1/11)).

G. C. Greubel, Table of n, a(n) for n = 1..1000
Dick Boland, An Analog of the Harmonic Numbers Over the Squarefree Integers

Cf. A138312, A138313, A138312, A138316, A138317, A138320, A083343, A001620.

Table[Denominator[Sum[MoebiusMu[k]^2/EulerPhi[k], {k, 1, n}]-HarmonicNumber[n]], {n, 1, 60}]

for(n=1, 60,print1(denominator(sum(k=1,n, moebius(k)^2/eulerphi(k) ) - sum(j=1, n, 1/j)), ", ")) \\ G. C. Greubel, Sep 14 2018

a(n) = denominator( (Sum_{k=1..n} mu(k)^2/phi(k)) - H_n) where mu(k) is the Mobius function, phi(k) is Euler's Totient function and H_n is the n-th Harmonic Number.

Data replaced by G. C. Greubel, Sep 14 2018

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])

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.

A345308 Decimal expansion of Sum_{p primes} log(p) / (p-1)^2.

Original entry on oeis.org

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

Offset: 1

Vaclav Kotesovec, Jun 13 2021

			1.226968805653470005965662568745762562988257454901426311714794620109...

Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 156 (constant C10 + 1).
Eric Weisstein's World of Mathematics, Prime Factor, formula (13)-(14), (constant U).

Cf. A138312, A306016, A345364.

ratfun = 1/((p - 1)^2); zetas = 0; ratab = Table[konfun = Simplify[ratfun + c/(p^power - 1)] // Together; coefs = CoefficientList[Numerator[konfun], p]; sol = Solve[Last[coefs] == 0, c][[1]]; zetas = zetas + c*Zeta'[power]/Zeta[power] /. sol; ratfun = konfun /. sol, {power, 2, 25}]; Do[Print[N[Sum[Log[p]*ratfun /. p -> Prime[k], {k, 1, m}] + zetas, 110]], {m, 1000, 5000, 1000}]

cp's OEIS Frontend

A340839 Decimal expansion of Mertens constant C(5,1).

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A138316 Numerators of the squarefree totient analogs of the harmonic numbers F_n.

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A138317 Denominators of the squarefree totient analogs of the harmonic numbers F_n.

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A138313 Decimal expansion of constant 'kappa' = lim_{n -> infinity} (F_n - H_n), where H_n are harmonic numbers, F_n are squarefree totient analogs of H_n.

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A138320 Numerators of the difference between the squarefree totient analogs of the harmonic numbers and the harmonic numbers: F_n-H_n.

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A138321 Denominators of the difference between the squarefree totient analogs of the harmonic numbers and the harmonic numbers: F_n - H_n.

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

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