A138313 -id:A138313 - cp's OEIS frontend

A138312 Decimal expansion of Mertens's constant B_3 minus Euler's constant.

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

Offset: 0

Dick Boland (abstract(AT)imathination.org), Mar 13 2008, Mar 14 2008, Mar 27 2008

Arises in the coefficients of the formula for the variance of the average order of omega(n), where omega(n) is the number of distinct prime factors of n - see MathWorld "Distinct Prime Factors" link and Hardy and Wright reference.
Conjectured to be equivalent to 'kappa' = lim_{n->oo} ((Sum_{k=1..n} mu^2(k)/phi(k)) - H_n), where mu(k) is the Mobius function, phi(k) is Euler's Totient and H_n is the n-th harmonic number.
De Koninck and Doyon proved that the asymptotic sum of the index of composition Sum_{k<=x} log(k)/log(rad(k)) = x + c*x/log(x) + O(x/(log(x))^2), where c is this constant and rad(n) in the squarefree kernel of n (A007947). - Amiram Eldar, May 02 2019

			0.755366610831688021159316685988625317796300153102499062981363664872472...

Hardy, G. H. and Wright, E. M., "The Number of Prime Factors of n" and "The Normal Order of omega(n) and Omega(n)." Sections 22.10 and 22.11 in An Introduction to the Theory of Numbers, 5th ed. Oxford, England: Clarendon Press, pp. 354-358, 1979.

Robert G. Wilson v, Table of n, a(n) for n = 0..9999
Dick Boland, An Analog of the Harmonic Numbers Over The Squarefree Integers
David Broadhurst, The Mertens Constant
Jean-Marie De Koninck and Nicolas Doyon, À propos de l’indice de composition des nombres, Monatshefte für Mathematik, Vol. 139, No. 2 (2003), pp. 151-167, alternative link.
Anne-Maria Ernvall-Hytönen, Tapani Matala-aho, and Louna Seppälä, On Mahler's transcendence measure for e, arXiv:1704.01374 [math.NT], 2017. See Theorem 6.4.
Eric Weisstein's World of Mathematics, Mertens Constant
Eric Weisstein's World of Mathematics, Distinct Prime Factors

Cf. A083343 (Mertens' B_3), A001620 (Euler's Constant), A138313 (The constant 'Kappa' conjectured to be equivalent to this sequence), A138316, A138317, A007947.

f[n_] := f[n] = Sum[MoebiusMu[j]* Zeta'[j]/Zeta[j], {j, 2, n}] // RealDigits[#, 10, 105]& // First; f[100]; f[n = 200]; While[f[n] != f[n - 100], n = n + 100]; f[n] (* Jean-François Alcover, Feb 14 2013, from 2nd formula *)

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).

More terms from Jean-François Alcover, Feb 14 2013

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.

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

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A138312 Decimal expansion of Mertens's constant B_3 minus Euler's constant.

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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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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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