A379583 -id:A379583 - cp's OEIS frontend

A379584 Denominators of the partial 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, 144, 144, 144, 144, 144, 144, 144, 3600, 3600, 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

Amiram Eldar, Dec 26 2024

Amiram Eldar, Table of n, a(n) for n = 1..1000
Maurice-Étienne Cloutier, Les parties k-puissante et k-libre d'un nombre, Thèse de doctorat, Université Laval, Québec (2018).
Maurice-Étienne Cloutier, Jean-Marie De Koninck, and Nicolas Doyon, On the powerful and squarefree parts of an integer, Journal of Integer Sequences, Vol. 17 (2014), Article 14.6.6.
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.

Cf. A057521, A370902, A370903, A379583 (numerators), A379586.

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

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 / powerful(k); print1(denominator(s), ", "))};

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

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

Original entry on oeis.org

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

Amiram Eldar, Dec 26 2024

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

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.

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

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}]]]

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), ", "))};

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.

cp's OEIS Frontend

A379584 Denominators of the partial sums of the reciprocals of the powerful part function (A057521).

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