A318649 -id:A318649 - cp's OEIS frontend

A318654 Positions of even terms in A318649.

2, 4, 8, 16, 24, 32, 40, 56, 64, 88, 96, 104, 128, 136, 152, 160, 184, 192, 224, 232, 248, 256, 296, 320, 328, 344, 352, 376, 384, 416, 424, 448, 472, 488, 512, 536, 544, 568, 584, 608, 632, 640, 664, 704, 712, 736, 776, 808, 824, 832, 856, 872, 896, 904, 928, 992, 1016, 1024, 1048, 1088, 1096, 1112, 1184, 1192, 1208, 1216, 1256, 1304

Offset: 1

Antti Karttunen, Sep 01 2018

nonn

Positions of nonzero terms in A318655.

It appears that these are also the positions of even terms in A318511.

Antti Karttunen, Table of n, a(n) for n = 1..21123

Cf. A318511, A318512, A318649, A318652, A318655.

A318655 The 2-adic valuation of A318649, the numerators of "Dirichlet Square Root" of squares.

Original entry on oeis.org

0, 1, 0, 1, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0

Offset: 1

Antti Karttunen, Sep 01 2018

nonn

Probably also the 2-adic valuation of A318511.

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

Cf. A318511, A318649, A318651, A318652, A318654 (the positions of nonzero terms).

A318655(n) = valuation(A318649(n),2); \\ Needs also code from A318649.

a(n) = A007814(A318649(n)).

It seems that for all n >= 1, a(n) <= A007814(A064549(n)) <= A007814(A000290(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).

A318512 Denominators (in their lowest terms) of the sequence whose Dirichlet convolution with itself yields squares (A000290), or equally A064549.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Aug 30 2018

These are also denominators (in their lowest terms) for the sequence whose Dirichlet convolution with itself yields A064549, n * Product_{primes p|n} p.
From Antti Karttunen, Sep 02 2018: (Start)
Proof for the above claim:
This sequence is defined as the denominator (given in the lowest terms) of rational valued function r(1) = 1, r(n) = (1/2) * (A000290(n) - Sum_{d|n, d>1, d 1. Define sequence Ay(n) as the denominator of function s(n), with otherwise similar definition, but with A064549 in place of A000290. Let Ay(n) be the denominator of s(n), reduced also into the lowest terms. (Corresponding numerators are A318649 and A318511 respectively. Note that the denominators in both cases must always be of the form 2^k, with k >= 0).
  By applying the distributive property of Dirichlet Convolution [which says that for any completely multiplicative function f, it doesn't matter whether one multiplies the result of convolution afterwards, or whether one multiplies the operands separately before convolution: f(g * g) = (fg) * (fg)], with A000027 in the role of f in both cases, one obtains a pair of equations:
    A318649(n)     A318681(n)     n*A299149(n)
    ----------  =  ----------  =  ------------
    A318512(n)     A299150(n)      A299150(n)
  and
    A318511(n)     A318680(n)     n*A318653(n)
    ----------  =  ----------  =  ------------
       Ay(n)       A299150(n)      A299150(n)
  where the leftmost ratios are reduced into their lowest terms.
  Sequence A318656 gives the 2-adic valuation of ratio A318649(n)/A318512(n), and because there are no even terms neither in A299149 nor in A318653, it also gives the 2-adic valuation of the latter ratio. As A318511/Ay is given in the lowest terms (not both of A318511(n) and Ay(n) can be even at same n), this implies that Ay must indeed be identical to A318512, and furthermore that A318655(n) = A007814(A318649(n)) = A007814(A318511(n)).
(End)

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

Cf. A000290, A064549, A046644, A299150, A318513, A318651, A318653, A318654, A318655, A318656, A318658, A318680, A318681.

Cf. A318649, A318511 (numerators).

f[1] = 1; f[n_] := f[n] = 1/2 (n*Times @@ FactorInteger[n][[All, 1]] - Sum[f[d] f[n/d], {d, Divisors[n][[2 ;; -2]]}]); Table[Denominator[f[n]], {n, 1, 100}] (* Vaclav Kotesovec, May 10 2025 *)

up_to = 65537;
A064549(n) = { my(f=factor(n)); for (i=1, #f~, f[i, 2]++); 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&&dA064549(n)));
A318512(n) = denominator(v318511_12[n]);

for(n=1, 100, print1(denominator(direuler(p=2, n, 1/(1-p^2*X)^(1/2))[n]), ", ")) \\ Vaclav Kotesovec, May 09 2025

a(n) = denominator of f(n), where f(1) = 1, f(n) = (1/2) * (A000290(n) - Sum_{d|n, d>1, d 1. [Equally, one could use A064549 in place of A000290.]
a(n) = 2^A318513(n).
a(n) = A046644(n)/A318651(n).
a(2n-1) = A046644(2n-1) = A318658(2n-1), for all n >= 1.

The main definition changed, more formulas added by Antti Karttunen, Aug 31 2018

A299149 Numerators of the positive solution to n = Sum_{d|n} a(d) * a(n/d).

Original entry on oeis.org

1, 1, 3, 3, 5, 3, 7, 5, 27, 5, 11, 9, 13, 7, 15, 35, 17, 27, 19, 15, 21, 11, 23, 15, 75, 13, 135, 21, 29, 15, 31, 63, 33, 17, 35, 81, 37, 19, 39, 25, 41, 21, 43, 33, 135, 23, 47, 105, 147, 75, 51, 39, 53, 135, 55, 35, 57, 29, 59, 45, 61, 31, 189, 231, 65, 33

Offset: 1

Gus Wiseman, Feb 03 2018

Dirichlet convolution of a(n)/A046644(n) with itself yields A000265. - Antti Karttunen, Aug 30 2018

			Sequence begins: 1, 1, 3/2, 3/2, 5/2, 3/2, 7/2, 5/2, 27/8, 5/2, 11/2, 9/4, 13/2, 7/2.

Antti Karttunen, Table of n, a(n) for n = 1..65537 (first 1000 terms Andrew Howroyd)
Vaclav Kotesovec, Graph - the asymptotic ratio (65537 terms)
Wikipedia, Dirichlet convolution.

Cf. A000010, A000265, A003958, A007431, A018804, A023900, A029935, A046643, A046644, A165825, A257098, A298971, A299119, A299150 (denominators), A299151, A317848, A318319, A318321, A318649.

nn=50;
sys=Table[n==Sum[a[d]*a[n/d],{d,Divisors[n]}],{n,nn}];
Numerator[Array[a,nn]/.Solve[sys,Array[a,nn]][[2]]]
odd[n_] := n/2^IntegerExponent[n, 2]; f[p_, e_] := odd[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]); numerator(n*prod(i=1, #v, my(e=v[i]); binomial(2*e, e)/4^e))} \\ Andrew Howroyd, Aug 09 2018

\\ 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&&dAndrew Howroyd, Aug 09 2018

for(n=1, 100, print1(numerator(direuler(p=2, n, 1/(1-p*X)^(1/2))[n]), ", ")) \\ Vaclav Kotesovec, May 09 2025

a(n) = numerator(n*A317848(n)/A165825(n)) = A000265(n*A317848(n)). - Andrew Howroyd, Aug 09 2018

Sum_{k=1..n} A299149(k)/A299150(k) ~ n^2 / (2*sqrt(Pi*log(n))) * (1 + (1-gamma) / (4*log(n))), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, May 09 2025

Keyword:mult added by Andrew Howroyd, Aug 09 2018

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

Original entry on oeis.org

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).

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)

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.

cp's OEIS Frontend

A318654 Positions of even terms in A318649.

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A318655 The 2-adic valuation of A318649, the numerators of "Dirichlet Square Root" of squares.

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

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A318512 Denominators (in their lowest terms) of the sequence whose Dirichlet convolution with itself yields squares (A000290), or equally A064549.

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A299149 Numerators of the positive solution to n = Sum_{d|n} a(d) * a(n/d).

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A318681 a(n) = n * A299149(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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A383768 Numerators of the sequence whose Dirichlet convolution with itself yields cubes (A000578).

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