A370901 -id:A370901 - cp's OEIS frontend

A370900 Partial sums of the powerfree part function (A055231).

1, 3, 6, 7, 12, 18, 25, 26, 27, 37, 48, 51, 64, 78, 93, 94, 111, 113, 132, 137, 158, 180, 203, 206, 207, 233, 234, 241, 270, 300, 331, 332, 365, 399, 434, 435, 472, 510, 549, 554, 595, 637, 680, 691, 696, 742, 789, 792, 793, 795, 846, 859, 912, 914, 969, 976, 1033

Offset: 1

Amiram Eldar, Mar 05 2024

Steven R. Finch, Mathematical Constants II, Cambridge University Press, 2018, p. 52.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eckford Cohen, An elementary method in the asymptotic theory of numbers, Duke Mathematical Journal, Vol. 28, No. 2 (1961), pp. 183-192.
Eckford Cohen, Some asymptotic formulas in the theory of numbers, Transactions of the American Mathematical Society, Vol. 112, No. 2 (1964), pp. 214-227.
László Tóth, Alternating Sums Concerning Multiplicative Arithmetic Functions, Journal of Integer Sequences, Vol. 20 (2017), Article 17.2.1.

Cf. A055231, A191622, A370901.

f[p_, e_] := If[e == 1, p, 1]; 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, f[i, 1], 1));}
lista(kmax) = {my(s = 0); for(k = 1, kmax, s += pfp(k); print1(s, ", "))};

a(n) = Sum_{k=1..n} A055231(k).

a(n) = c * n^2 / 2 + O(R(n)), where c = Product_{p prime} (1 - (p^2+p-1)/(p^3*(p+1))) = 0.649606699337... (A191622), R(n) = x^(3/2) * exp(-c_1 * log(n)^(3/5) / log(log(n))^(1/5)) unconditionally, or x^(7/5) * exp(c_2 * log(n) / log(log(n))) assuming the Riemann hypothesis, and c_1 and c_2 are positive constants (Tóth, 2017).

A379579 Numerators of the partial sums of the reciprocals of the powerfree part function (A055231).

Original entry on oeis.org

1, 3, 11, 17, 91, 16, 117, 152, 187, 381, 4261, 13553, 178499, 90322, 30441, 35446, 607587, 1300259, 24875091, 25521737, 77027101, 38733998, 895731799, 932913944, 1044460379, 2097501253, 2320594123, 2352464533, 68444564327, 11443370128, 355822756173, 389249504528

Offset: 1

Amiram Eldar, Dec 26 2024

			Fractions begin with 1, 3/2, 11/6, 17/6, 91/30, 16/5, 117/35, 152/35, 187/35, 381/70, 4261/770, 13553/2310, ...

D. Suryanarayana and P. Subrahmanyam, The maximal k-full divisor of an integer, Indian J. Pure Appl. Math., Vol. 12, No. 2 (1981), pp. 175-190.

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 4.11, pp. 31-32.

Cf. A055231, A328013, A370900, A370901, A379580 (denominators), A379581.

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

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 / powfree(k); print1(numerator(s), ", "))};

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

a(n)/A379580(n) = A * n^(1/2) + B * n^(1/3) + O(n^(1/5)), where A = A328013, and B = (zeta(2/3)/zeta(2)) * Product_{p prime} (1 + (p^(1/3)-1)/(p*(p^(2/3)-p^(1/3)+1))) = -2.59305556147555965163... .

A379580 Denominators of the partial sums of the reciprocals of the powerfree part function (A055231).

Original entry on oeis.org

1, 2, 6, 6, 30, 5, 35, 35, 35, 70, 770, 2310, 30030, 15015, 5005, 5005, 85085, 170170, 3233230, 3233230, 9699690, 4849845, 111546435, 111546435, 111546435, 223092870, 223092870, 223092870, 6469693230, 1078282205, 33426748355, 33426748355, 9116385915, 18232771830

Offset: 1

Amiram Eldar, Dec 26 2024

D. Suryanarayana and P. Subrahmanyam, The maximal k-full divisor of an integer, Indian J. Pure Appl. Math., Vol. 12, No. 2 (1981), pp. 175-190.

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 4.11, pp. 31-32.

Cf. A055231, A370900, A370901, A379579 (numerators), A379582.

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

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

a(n) = denominator(Sum_{k=1..n} 1/A055231(k)).

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

Original entry on oeis.org

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

Amiram Eldar, Dec 26 2024

			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, ...

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.

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

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}]]]

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), ", "))};

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... .

A379582 Denominators of the partial alternating sums of the reciprocals of the powerfree part function (A055231).

Original entry on oeis.org

1, 2, 6, 6, 30, 15, 105, 105, 105, 210, 2310, 2310, 30030, 15015, 1001, 1001, 17017, 34034, 646646, 3233230, 9699690, 4849845, 111546435, 37182145, 37182145, 74364290, 74364290, 74364290, 2156564410, 3234846615, 100280245065, 100280245065, 100280245065, 200560490130

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 4.11, pp. 31-32.

Cf. A055231, A370900, A370901, A379580, A379581 (numerators).

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

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

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

cp's OEIS Frontend

A370900 Partial sums of the powerfree part function (A055231).

Views

Author

Keywords

References

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A379579 Numerators of the partial sums of the reciprocals of the powerfree part function (A055231).

Views

Author

Keywords

Examples

References

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A379580 Denominators of the partial sums of the reciprocals of the powerfree part function (A055231).

Views

Author

Keywords

References

Links

Crossrefs

Programs

Mathematica

PARI

Formula

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

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A379582 Denominators of the partial alternating sums of the reciprocals of the powerfree part function (A055231).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula