A379581 Numerators of the partial alternating sums of the reciprocals of the powerfree part function (A055231).

1, 1, 5, -1, 1, -2, 1, -104, 1, -19, 1, -769, -7687, -4916, -261, -1262, -20453, -57923, -1066503, -5979161, -17475593, -8958244, -201189767, -79457304, -42275159, -87410483, -13046193, -23669663, -612055937, -1025912126, -28568429291, -128848674356, -125809879051

Offset: 1

Author

Amiram Eldar, Dec 26 2024

Keywords

• sign
• easy
• frac

Examples

			Fractions begin with 1, 1/2, 5/6, -1/6, 1/30, -2/15, 1/105, -104/105, 1/105, -19/210, 1/2310, -769/2310, ...
		

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 4.11, pp. 31-32.

Crossrefs

Cf. A055231, A328013, A370900, A370901, A379579, A379582 (denominators).

Programs

Mathematica

f[p_, e_] := If[e==1, p, 1]; powfree[n_] := Times @@ f @@@ FactorInteger[n]; Numerator[Accumulate[Table[(-1)^(n+1)/powfree[n], {n, 1, 50}]]]

PARI

powfree(n) = {my(f = factor(n)); prod(i=1, #f~, if(f[i, 2] == 1, f[i, 1], 1)); }
list(nmax) = {my(s = 0); for(k = 1, nmax, s += (-1)^(k+1) / powfree(k); print1(numerator(s), ", "))};

Formula

a(n) = numerator(Sum_{k=1..n} (-1)^(k+1)/A055231(k)).

a(n)/A379582(n) = A * n^(1/2) + B * n^(1/3) + O(n^(1/5)), where A = ((9-12*sqrt(2))/23) * A328013, and B = ((2^(5/3) - 3*2^(1/3) - 1)/(2^(5/3) - 2^(1/3) + 1)) * (zeta(2/3)/zeta(2)) * Product_{p prime} (1 + (p^(1/3)-1)/(p*(p^(2/3)-p^(1/3)+1))) = 1.42776088919948241359... .