A318512 -id:A318512 - cp's OEIS frontend

A318653 Numerators of the sequence whose Dirichlet convolution with itself yields A007947, the squarefree kernel of n.

1, 1, 3, 1, 5, 3, 7, 1, 3, 5, 11, 3, 13, 7, 15, 3, 17, 3, 19, 5, 21, 11, 23, 3, -5, 13, 15, 7, 29, 15, 31, 3, 33, 17, 35, 3, 37, 19, 39, 5, 41, 21, 43, 11, 15, 23, 47, 9, -21, -5, 51, 13, 53, 15, 55, 7, 57, 29, 59, 15, 61, 31, 21, 5, 65, 33, 67, 17, 69, 35, 71, 3, 73, 37, -15, 19, 77, 39, 79, 15, 3, 41, 83, 21, 85, 43, 87, 11, 89, 15

Offset: 1

Antti Karttunen, Aug 31 2018

No zeros among the first 2^20 terms.

Antti Karttunen, Table of n, a(n) for n = 1..65537
Vaclav Kotesovec, Graph - the asymptotic ratio (100000 terms)

Cf. A007947, A299150 (denominators).

Cf. also A317935, A318511, A318512, A318649.

rad[n_] := Times @@ (First@# & /@ FactorInteger[n]); f[1] = 1; f[n_] := f[n] = (rad[n] - DivisorSum[n, f[#]*f[n/#] &, 1 < # < n &])/2; a[n_] := Numerator [f[n]]; Array[a, 100] (* Amiram Eldar, Dec 07 2020 *)

up_to = 65537;
A007947(n) = factorback(factorint(n)[, 1]);
DirSqrt(v) = {my(n=#v, u=vector(n)); u[1]=1; for(n=2, n, u[n]=(v[n]/v[1] - sumdiv(n, d, if(d>1&&dA007947(n)));
A318653(n) = numerator(v318653_aux[n]);
for(n=1, 100, print1(numerator(direuler(p=2, n, ((1 + p*X - X)/(1 - X))^(1/2))[n]), ", ")) \\ Vaclav Kotesovec, May 08 2025

a(n) = numerator of f(n), where f(1) = 1, f(n) = (1/2) * (A007947(n) - Sum_{d|n, d>1, d 1.
From Vaclav Kotesovec, May 08 2025: (Start)
Let f(s) = Product_{p prime} (1 + 1/p^(2*s-1) - 1/p^(2*s-2) - 1/p^s).
Sum_{k=1..n} A318653(k)/A299150(k) ~ n^2 * sqrt(Pi*f(2)/(24*log(n))) * (1 - (gamma - 1 + f'(2)/f(2) + 6*zeta'(2)/Pi^2) / (4*log(n))), where
f(2) = A065464 = Product_{p prime} (1 - 2/p^2 + 1/p^3) = 0.4282495056770944402187657075818235461212985133559361440319...
f'(2) = f(2) * Sum_{p prime} (3*p-2)*log(p) / ((p-1)*(p^2+p-1)) = f(2) * 1.469536740824614833203393993450164364663334798759143895712...
and gamma is the Euler-Mascheroni constant A001620. (End)

A318680 a(n) = n * A318653(n).

Original entry on oeis.org

1, 2, 9, 4, 25, 18, 49, 8, 27, 50, 121, 36, 169, 98, 225, 48, 289, 54, 361, 100, 441, 242, 529, 72, -125, 338, 405, 196, 841, 450, 961, 96, 1089, 578, 1225, 108, 1369, 722, 1521, 200, 1681, 882, 1849, 484, 675, 1058, 2209, 432, -1029, -250, 2601, 676, 2809, 810, 3025, 392, 3249, 1682, 3481, 900, 3721, 1922, 1323, 320

Offset: 1

Antti Karttunen, Sep 02 2018

Dirichlet convolution of a(n)/A299150(n) with itself gives A064549 [= n * Product_{primes p|n} p], like gives also the self-convolution of A318511(n)/A318512(n), as it is the same ratio reduced to its lowest terms. However, in contrast to A318511, this sequence is multiplicative as both A000027 and A318653 are multiplicative sequences (also, because A064549 and A299150 are both multiplicative).

A007814 gives the 2-adic valuation of this sequence, because there are no even terms in A318653.

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A007947, A064549, A299150, A318511, A318512, A318653, A318681.

rad[n_] := Times @@ (First@# & /@ FactorInteger[n]); f[1] = 1; f[n_] := f[n] = (rad[n] - DivisorSum[n, f[#]*f[n/#] &, 1 < # < n &])/2; a[n_] := n * Numerator [f[n]]; Array[a, 100] (* Amiram Eldar, Dec 07 2020 *)

up_to = 65537;
A007947(n) = factorback(factorint(n)[, 1]);
DirSqrt(v) = {my(n=#v, u=vector(n)); u[1]=1; for(n=2, n, u[n]=(v[n]/v[1] - sumdiv(n, d, if(d>1&&dA007947(n)));
A318653(n) = numerator(v318653_aux[n]);
A318680(n) = (n*A318653(n));

a(n) = n * A318653(n).

a(n)/A299150(n) = A318511(n)/A318512(n).

A383768 Numerators of the sequence whose Dirichlet convolution with itself yields cubes (A000578).

Original entry on oeis.org

1, 4, 27, 24, 125, 54, 343, 160, 2187, 250, 1331, 324, 2197, 686, 3375, 1120, 4913, 2187, 6859, 1500, 9261, 2662, 12167, 2160, 46875, 4394, 98415, 4116, 24389, 3375, 29791, 8064, 35937, 9826, 42875, 6561, 50653, 13718, 59319, 10000, 68921, 9261, 79507, 15972, 273375

Offset: 1

Vaclav Kotesovec, May 09 2025

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000
Vaclav Kotesovec, Graph - the asymptotic ratio (10000 terms)

Cf. A000578, A299149, A299150, A318649, A318512, A383769 (denominators).

for(n=1, 100, print1(numerator(direuler(p=2, n, 1/(1-p^3*X)^(1/2))[n]), ", "))

Sum_{k=1..n} A383768(k) / A383769(k) ~ n^4/(4*sqrt(Pi*log(n))) * (1 + (1-2*gamma)/(8*log(n))), where gamma is the Euler-Mascheroni constant A001620.

A383769 Denominators of the sequence whose Dirichlet convolution with itself yields cubes (A000578).

Original entry on oeis.org

1, 1, 2, 1, 2, 1, 2, 1, 8, 1, 2, 1, 2, 1, 4, 1, 2, 2, 2, 1, 4, 1, 2, 1, 8, 1, 16, 1, 2, 1, 2, 1, 4, 1, 4, 1, 2, 1, 4, 1, 2, 1, 2, 1, 16, 1, 2, 1, 8, 2, 4, 1, 2, 4, 4, 1, 4, 1, 2, 1, 2, 1, 16, 1, 4, 1, 2, 1, 4, 1, 2, 1, 2, 1, 16, 1, 4, 1, 2, 1, 128, 1, 2, 1, 4, 1, 4

Offset: 1

Vaclav Kotesovec, May 09 2025

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000

Cf. A000578, A299149, A299150, A318649, A318512, A383768 (numerators).

for(n=1, 100, print1(denominator(direuler(p=2, n, 1/(1-p^3*X)^(1/2))[n]), ", "))

A383791 Numerators of the sequence whose Dirichlet convolution with itself yields fourth powers (A000583).

Original entry on oeis.org

1, 8, 81, 96, 625, 324, 2401, 1280, 19683, 2500, 14641, 3888, 28561, 9604, 50625, 17920, 83521, 19683, 130321, 30000, 194481, 58564, 279841, 51840, 1171875, 114244, 2657205, 115248, 707281, 101250, 923521, 258048, 1185921, 334084, 1500625, 236196, 1874161, 521284, 2313441

Offset: 1

Vaclav Kotesovec, May 10 2025

Numerators of Dirichlet g.f.: Sum_{n>=1} a(n)/n^s = zeta(s-4)^(1/2).

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000
Vaclav Kotesovec, Graph - the asymptotic ratio (10000 terms)

Cf. A000583, A383792 (denominators).

Cf. A299149, A299150, A318649, A318512, A383768, A383769.

for(n=1, 100, print1(numerator(direuler(p=2, n, 1/(1-p^4*X)^(1/2))[n]), ", "))

Sum_{k=1..n} A383791(k) / A383792(k) ~ n^5 / (5*sqrt(Pi*log(n))) * (1 + (1/5 - gamma/2)/(2*log(n))), where gamma is the Euler-Mascheroni constant A001620.

A383792 Denominators of the sequence whose Dirichlet convolution with itself yields fourth powers (A000583).

Original entry on oeis.org

1, 1, 2, 1, 2, 1, 2, 1, 8, 1, 2, 1, 2, 1, 4, 1, 2, 1, 2, 1, 4, 1, 2, 1, 8, 1, 16, 1, 2, 1, 2, 1, 4, 1, 4, 1, 2, 1, 4, 1, 2, 1, 2, 1, 16, 1, 2, 1, 8, 1, 4, 1, 2, 2, 4, 1, 4, 1, 2, 1, 2, 1, 16, 1, 4, 1, 2, 1, 4, 1, 2, 1, 2, 1, 16, 1, 4, 1, 2, 1, 128, 1, 2, 1, 4

Offset: 1

Vaclav Kotesovec, May 10 2025

Denominators of Dirichlet g.f.: Sum_{n>=1} a(n)/n^s = zeta(s-4)^(1/2).

First differs from A318658 at n = 54.

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000

Cf. A000583, A318658, A383791 (numerators).

Cf. A299149, A299150, A318649, A318512, A383768, A383769.

for(n=1, 100, print1(denominator(direuler(p=2, n, 1/(1-p^4*X)^(1/2))[n]), ", "))

cp's OEIS Frontend

A318653 Numerators of the sequence whose Dirichlet convolution with itself yields A007947, the squarefree kernel of n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A318680 a(n) = n * A318653(n).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A383768 Numerators of the sequence whose Dirichlet convolution with itself yields cubes (A000578).

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

A383769 Denominators of the sequence whose Dirichlet convolution with itself yields cubes (A000578).

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

A383791 Numerators of the sequence whose Dirichlet convolution with itself yields fourth powers (A000583).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Formula

A383792 Denominators of the sequence whose Dirichlet convolution with itself yields fourth powers (A000583).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI