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

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

1,3

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.

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

Formula

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.

Example

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).

Mathematica

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

Prog

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

Crossrefs

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

Sequence in context: A226978 A243331 A300218 * A103659 A069947 A051547

Adjacent sequences: A138314 A138315 A138316 * A138318 A138319 A138320

Keyword

frac,nonn

Author

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

Status

approved