A299150 -id:A299150 - cp's OEIS frontend

A318681 a(n) = n * A299149(n).

1, 2, 9, 12, 25, 18, 49, 40, 243, 50, 121, 108, 169, 98, 225, 560, 289, 486, 361, 300, 441, 242, 529, 360, 1875, 338, 3645, 588, 841, 450, 961, 2016, 1089, 578, 1225, 2916, 1369, 722, 1521, 1000, 1681, 882, 1849, 1452, 6075, 1058, 2209, 5040, 7203, 3750, 2601, 2028, 2809, 7290, 3025, 1960, 3249, 1682, 3481, 2700

Offset: 1

Antti Karttunen, Sep 02 2018

Dirichlet convolution of a(n)/A299150(n) with itself gives A000290, the squares, like gives also the self-convolution of A318649(n)/A318512(n), as it is the same ratio reduced to its lowest terms. However, in contrast to A318649, this sequence is multiplicative as both A000027 and A299149 are multiplicative sequences (also, because A000290 and A299150 are both multiplicative).

A007814 gives the 2-adic valuation of this sequence, because there are no even terms in A299149.

Antti Karttunen, Table of n, a(n) for n = 1..16384
Wikipedia, Dirichlet convolution

Cf. A000290, A299149, A299150, A318512, A318649, A318680.

up_to = 65537;
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&&dA299149(n) = numerator(v299149_50[n]);
A318681(n) = (n*A299149(n));

a(n) = n * A299149(n).

a(n)/A299150(n) = A318649(n)/A318512(n).

A317929 Numerators of rational valued sequence whose Dirichlet convolution with itself yields A235199, which is a multiplicative permutation of natural numbers.

Original entry on oeis.org

1, 1, 3, 3, 7, 3, 5, 5, 27, 7, 17, 9, 13, 5, 21, 35, 11, 27, 19, 21, 15, 17, 23, 15, 147, 13, 135, 15, 43, 21, 59, 63, 51, 11, 35, 81, 37, 19, 39, 35, 41, 15, 29, 51, 189, 23, 73, 105, 75, 147, 33, 39, 53, 135, 119, 25, 57, 43, 31, 63, 61, 59, 135, 231, 91, 51, 67, 33, 69, 35, 107, 135, 47, 37, 441, 57, 85, 39

Offset: 1

Antti Karttunen, Aug 23 2018

Multiplicative because A235199 is.

Question: Are all terms positive? No negative terms in range 1 .. 2^18. Also checked up to n = 2^18 that the denominators match with A299150.

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A235199, A299150 (seems to give the denominators).

Cf. also A317930.

up_to = 16384;
A235199(n) = if(n<=4,n,my(f = factor(n)); for(i=1, #f~, if(5==f[i,1], f[i,1] += 2, if(7==f[i,1], f[i,1] -= 2, my(k=primepi(f[i,1])); if(k>4, f[i,1] = prime(A235199(k)))))); factorback(f));
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&&dA317937.
v317929aux = DirSqrt(vector(up_to, n, A235199(n)));
A317929(n) = numerator(v317929aux[n]);

a(n) = numerator of f(n), where f(1) = 1, f(n) = (1/2) * (A235199(n) - Sum_{d|n, d>1, d 1.

A318498 Denominators of the sequence whose Dirichlet convolution with itself yields A061389, number of (1+phi)-divisors of n.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Aug 30 2018

The sequence seems to give the denominators of a few other similarly constructed rational valued sequences obtained as "Dirichlet Square Roots" (of possibly A092520 and A293443).

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A061389, A318497 (numerators), A318499.

Cf. also A299150, A046644.

up_to = 65537;
A061389(n) = factorback(apply(e -> (1+eulerphi(e)),factor(n)[,2]));
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&&dA317937.
v318497_98 = DirSqrt(vector(up_to, n, A061389(n)));
A318497(n) = numerator(v318497_98[n]);
A318498(n) = denominator(v318497_98[n]);

a(n) = denominator of f(n), where f(1) = 1, f(n) = (1/2) * (A061389(n) - Sum_{d|n, d>1, d 1.

a(n) = 2^A318499(n).

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

Original entry on oeis.org

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]), ", "))

A317848 Multiplicative with a(p^e) = binomial(2*e, e).

Original entry on oeis.org

1, 2, 2, 6, 2, 4, 2, 20, 6, 4, 2, 12, 2, 4, 4, 70, 2, 12, 2, 12, 4, 4, 2, 40, 6, 4, 20, 12, 2, 8, 2, 252, 4, 4, 4, 36, 2, 4, 4, 40, 2, 8, 2, 12, 12, 4, 2, 140, 6, 12, 4, 12, 2, 40, 4, 40, 4, 4, 2, 24, 2, 4, 12, 924, 4, 8, 2, 12, 4, 8, 2, 120, 2, 4, 12, 12, 4, 8, 2, 140

Offset: 1

Andrew Howroyd, Aug 08 2018

The Dirichlet convolution square of this sequence is A165825.

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

Cf. A000265, A000984, A006519, A037445, A046643, A046644, A165825, A299149, A299150.

f[p_, e_] := Binomial[2*e, e]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Apr 30 2023 *)

a(n)={my(v=factor(n)[,2]); prod(i=1, #v, binomial(2*v[i], v[i]))}

\\ DirSqrt(v) finds u such that v = v[1]*dirmul(u, u).
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&&d

A317848(n) = factorback(apply(e -> binomial(e+e,e),factor(n)[,2])); \\ Antti Karttunen, Sep 17 2018

A037445(n) = A006519(a(n)).
A046643(n) = numerator(a(n)/A165825(n)) = A000265(a(n)).
A046644(n) = denominator(a(n)/A165825(n)) = A165825(n)/A037445(n).
A299149(n) = numerator(n*a(n)/A165825(n)) = A000265(n*a(n)).
A299150(n) = denominator(n*a(n)/A165825(n)) = A165825(n)/(A037445(n) * A006519(n)).

A318656 The 2-adic valuation of ratio A318649(n)/A318512(n); a(n) = 2*A007814(n) - A046645(n).

Original entry on oeis.org

0, 1, -1, 1, -1, 0, -1, 2, -3, 0, -1, 0, -1, 0, -2, 1, -1, -2, -1, 0, -2, 0, -1, 1, -3, 0, -4, 0, -1, -1, -1, 2, -2, 0, -2, -2, -1, 0, -2, 1, -1, -1, -1, 0, -4, 0, -1, 0, -3, -2, -2, 0, -1, -3, -2, 1, -2, 0, -1, -1, -1, 0, -4, 2, -2, -1, -1, 0, -2, -1, -1, -1, -1, 0, -4, 0, -2, -1, -1, 0, -7, 0, -1, -1, -2, 0, -2, 1, -1, -3, -2, 0

Offset: 1

Antti Karttunen, Sep 02 2018

sign

Also the 2-adic valuation of ratio A318681(n)/A299150(n) [which is equal to A318649(n)/A318512(n), but not represented in lowest terms], as well as the 2-adic valuation of A318680(n)/A299150(n) = A318511(n)/A318512(n).

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

Cf. A007814, A046645, A299150, A318440, A318511, A318512, A318513, A318649, A318655, A318680, A318681.

Cf. A318654 (positions of positive terms).

A007814(n) = valuation(n,2);
A005187(n) = { my(s=n); while(n>>=1, s+=n); s; };
A046644(n) = factorback(apply(e -> 2^A005187(e),factor(n)[,2]));
A318656(n) = ((2*A007814(n))-A007814(A046644(n)));

a(n) = A318655(n) - A318513(n).
a(n) = A007814(n) - A318440(n).
a(n) = 2*A007814(n) - A046645(n) = A007814(n^2) - A046645(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.

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A318681 a(n) = n * A299149(n).

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A317929 Numerators of rational valued sequence whose Dirichlet convolution with itself yields A235199, which is a multiplicative permutation of natural numbers.

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A318498 Denominators of the sequence whose Dirichlet convolution with itself yields A061389, number of (1+phi)-divisors of n.

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

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

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

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

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A317848 Multiplicative with a(p^e) = binomial(2*e, e).

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