A379586 Denominators of the partial alternating sums of the reciprocals of the powerful part function (A057521).

1, 1, 1, 4, 4, 4, 4, 8, 72, 72, 72, 72, 72, 72, 72, 144, 144, 16, 16, 16, 16, 16, 16, 16, 400, 400, 10800, 10800, 10800, 10800, 10800, 21600, 21600, 21600, 21600, 21600, 21600, 21600, 21600, 21600, 21600, 21600, 21600, 21600, 21600, 21600, 21600, 21600, 1058400

Offset: 1

Author

Amiram Eldar, Dec 26 2024

Keywords

• nonn
• easy
• frac

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.

Crossrefs

Cf. A057521, A370902, A370903, A379584, A379585 (numerators).

Programs

Mathematica

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

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(denominator(s), ", "))};

Formula

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