A379369 Numerators of the partial alternating sums of the reciprocals of the squarefree kernel function (A007947).

1, 1, 5, 1, 8, 11, 107, 1, 12, 17, 257, 193, 3664, 5183, 479, -261, -3436, -37633, -612925, -2017297, -1786352, -4013599, -82613087, -19965872, -12529443, -27919051, -9392863, -12664034, -255710551, -242359181, -5356570201, -19391659278, -55136182529, -116171203003

Offset: 1

Author

Amiram Eldar, Dec 21 2024

Keywords

• sign
• easy
• frac

Examples

			Fractions begin with 1, 1/2, 5/6, 1/3, 8/15, 11/30, 107/210, 1/105, 12/35, 17/70, 257/770, 193/1155, ...
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..1000
• László Tóth, Alternating Sums Concerning Multiplicative Arithmetic Functions, Journal of Integer Sequences, Vol. 20 (2017), Article 17.2.1. See section 4.6, pp. 24-26.

Crossrefs

Cf. A007947, A073355, A370896, A379367, A379368, A379370 (denominators).

Programs

Mathematica

rad[n_] := Times @@ FactorInteger[n][[;;, 1]]; Numerator[Accumulate[Table[(-1)^(n+1)/rad[n], {n, 1, 50}]]]

PARI

rad(n) = vecprod(factor(n)[, 1]);
list(nmax) = {my(s = 0); for(k = 1, nmax, s += (-1)^(k+1) / rad(k); print1(numerator(s), ", "))};

Formula

a(n) = numerator(Sum_{k=1..n} (-1)^(k+1)/A007947(k)).

a(n)/A379370(n) ~ -A379367(n)/A379368(n).