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

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

Author

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

Keywords

• frac
• nonn

Comments

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.

Examples

			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).
		

Links

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

Crossrefs

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

Programs

Mathematica

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

PARI

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

Formula

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.