A370903 -id:A370903 - cp's OEIS frontend

A370902 Partial sums of the powerful part function (A057521).

1, 2, 3, 7, 8, 9, 10, 18, 27, 28, 29, 33, 34, 35, 36, 52, 53, 62, 63, 67, 68, 69, 70, 78, 103, 104, 131, 135, 136, 137, 138, 170, 171, 172, 173, 209, 210, 211, 212, 220, 221, 222, 223, 227, 236, 237, 238, 254, 303, 328, 329, 333, 334, 361, 362, 370, 371, 372, 373

Offset: 1

Amiram Eldar, Mar 05 2024

Amiram Eldar, Table of n, a(n) for n = 1..10000
Maurice-Étienne Cloutier, Les parties k-puissante et k-libre d’un nombre, Thèse de doctorat, Université Laval (2018); alternative link.
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.

Cf. A057521, A328013, A370903.

f[p_, e_] := If[e == 1, 1, p^e]; pfp[n_] := Times @@ f @@@ FactorInteger[n]; pfp[1] = 1; Accumulate[Array[pfp[#] &, 100]]

pfp(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i, 2] == 1, 1, f[i, 1]^f[i, 2]));}
lista(kmax) = {my(s = 0); for(k = 1, kmax, s += pfp(k); print1(s, ", "))};

a(n) = Sum_{k=1..n} A057521(k).
a(n) = c_1 * n^(3/2) / 3 + c_2 * n^(4/3) / 4 + O(n^(6/5)), where c_1 = A328013 and c_2 are positive constants (Tóth, 2017).
c_2 = zeta(2/3) * Product_{p prime} (1 + 1/p^(4/3) - 2/p^2 - 1/p^(7/3) + 1/p^3) = -2.59305556147555965163... (László Tóth, personal communication). - Amiram Eldar, Mar 07 2024

A379583 Numerators of the partial sums of the reciprocals of the powerful part function (A057521).

Original entry on oeis.org

1, 2, 3, 13, 17, 21, 25, 51, 467, 539, 611, 629, 701, 773, 845, 1699, 1843, 1859, 2003, 2039, 2183, 2327, 2471, 2489, 62369, 65969, 198307, 201007, 211807, 222607, 233407, 467489, 489089, 510689, 532289, 532889, 554489, 576089, 597689, 600389, 621989, 643589, 665189

Offset: 1

Amiram Eldar, Dec 26 2024

			Fractions begin with 1, 2, 3, 13/4, 17/4, 21/4, 25/4, 51/8, 467/72, 539/72, 611/72, 629/72, ...

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, A191622, A370902, A370903, A379584 (denominators), A379585.

f[p_, e_] := If[e > 1, p^e, 1]; powful[n_] := Times @@ f @@@ FactorInteger[n]; Numerator[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(numerator(s), ", "))};

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

a(n)/A379584(n) = c * n + O(n^(1/2)), where c = A191622 (Cloutier et al., 2014). The error term was improved by Tóth (2017) to O(n^(1/2) * exp(-c1 * log(n)^(3/5) / log(log(n))^(1/5))) unconditionally, and O(n^(2/5) * exp(c2 * log(n) / log(log(n)))) assuming the Riemann hypothesis, where c1 and c2 are positive constants.

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

Original entry on oeis.org

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.

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

Original entry on oeis.org

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

Amiram Eldar, Dec 26 2024

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

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}]]]

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), ", "))};

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

cp's OEIS Frontend

A370902 Partial sums of the powerful part function (A057521).

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A379583 Numerators of the partial sums of the reciprocals of the powerful part function (A057521).

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

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

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Mathematica

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

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Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula