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Search: id:A138313
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| A138313 |
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Decimal expansion of constant 'kappa' = limit_{n -> infty)F_n-H_n, where H_n are harmonic numbers, F_n are squarefree totient analogs of H_n. |
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6
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OFFSET
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0,1
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COMMENT
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The squarefree totient analog of the harmonic number F_n is given by F_n = 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.
Conjectured to be equivalent to Mertens' constant B_3 less Euler's constant. B_3-gamma is given by sum_(i=1 to infty)log p_i/(p_i(p_i-1)), p_i is the i^th prime.
sum_(j=2 to infty)mu(j)zeta'(j)/zeta(j), mu(j) is the Mobius function, zeta'(j) is the derivative of zeta(j).
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LINKS
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Dick Boland, An Analog of the Harmonic Numbers Over the Squarefree Integers
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FORMULA
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limit_{n -> infty)((sum_(k=1 to n) mu^2(k)/phi(k))-H_n), mu(k) is the Mobius function, phi(k) is Euler's Totient function and H_n is the nth Harmonic Number.
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EXAMPLE
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0.755366...
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MATHEMATICA
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<< 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]
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CROSSREFS
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Cf. A138316, A138317 = Numerators and Denominators of the Squarefree Totient Analogs of the Harmonic Numbers Cf. A138312 = Mertens' B_3 less Euler's Constant Cf. A083343 = Mertens' B_3. Cf. A001620 = Euler's Constant.
Sequence in context: A109134 A075778 A010510 this_sequence A138312 A098842 A070273
Adjacent sequences: A138310 A138311 A138312 this_sequence A138314 A138315 A138316
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KEYWORD
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cons,more,nonn
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AUTHOR
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Dick Boland (abstract(AT)imathination.org), Mar 13 2008
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