A379585 Numerators of the partial alternating sums of the reciprocals of the powerful part function (A057521).

1, 0, 1, 3, 7, 3, 7, 13, 125, 53, 125, 107, 179, 107, 179, 349, 493, 53, 69, 65, 81, 65, 81, 79, 1991, 1591, 43357, 40657, 51457, 40657, 51457, 102239, 123839, 102239, 123839, 123239, 144839, 123239, 144839, 142139, 163739, 142139, 163739, 158339, 160739, 139139

Offset

1,4

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 4.12, p. 33.

Formula

a(n) = numerator(Sum_{k=1..n} (-1)^(k+1)/A057521(k)).

a(n)/A379586(n) = (5/19) * A191622 * n + O(n^(1/2) * exp(-c1 * log(n)^(3/5) / log(log(n))^(1/5))) unconditionally, and with an improved error term O(n^(2/5) * exp(c2 * log(n) / log(log(n)))) assuming the Riemann hypothesis, where c1 and c2 are positive constants.

Example

Fractions begin with 1, 0, 1, 3/4, 7/4, 3/4, 7/4, 13/8, 125/72, 53/72, 125/72, 107/72, ...

Mathematica

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

Prog

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

Crossrefs

Cf. A057521, A191622, A370902, A370903, A379583, A379586 (denominators).

Sequence in context: A016665 A320831 A120124 * A151573 A113832 A115631

Adjacent sequences: A379582 A379583 A379584 * A379586 A379587 A379588

Keyword

nonn,easy,frac

Author

Amiram Eldar, Dec 26 2024

Status

approved