A318511 -id:A318511 - cp's OEIS frontend

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

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

A318649 Numerators of the sequence whose Dirichlet convolution with itself yields squares, A000290.

Original entry on oeis.org

1, 2, 9, 6, 25, 9, 49, 20, 243, 25, 121, 27, 169, 49, 225, 70, 289, 243, 361, 75, 441, 121, 529, 90, 1875, 169, 3645, 147, 841, 225, 961, 252, 1089, 289, 1225, 729, 1369, 361, 1521, 250, 1681, 441, 1849, 363, 6075, 529, 2209, 315, 7203, 1875, 2601, 507, 2809, 3645, 3025, 490, 3249, 841, 3481, 675, 3721, 961, 11907, 924, 4225, 1089

Offset: 1

Antti Karttunen, Aug 31 2018

Antti Karttunen, Table of n, a(n) for n = 1..65537
Vaclav Kotesovec, Graph - the asymptotic ratio (10000 terms)

Cf. A000290, A318512 (denominators).

Cf. also A046643, A299149, A318511, A318651, A318654 (gives the positions of even terms), A318655 (the 2-adic valuation).

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&&dA318649(n) = numerator(v318649_aux[n]);

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

a(n) = numerator of f(n), where f(1) = 1, f(n) = (1/2) * ((n^2) - Sum_{d|n, d>1, d 1.
a(n) = n*A318512(n)*A299149(n)/A299150(n).
Sum_{k=1..n} A318649(k) / A318512(k) ~ n^3/(3*sqrt(Pi*log(n))) * (1 + (1 - 3*gamma/2) / (6*log(n))), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, May 09 2025

A318654 Positions of even terms in A318649.

Original entry on oeis.org

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)).

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).

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).

A318650 Numerators of the sequence whose Dirichlet convolution with itself yields A057521, the powerful part of n.

Original entry on oeis.org

1, 1, 1, 15, 1, 1, 1, 49, 35, 1, 1, 15, 1, 1, 1, 603, 1, 35, 1, 15, 1, 1, 1, 49, 99, 1, 181, 15, 1, 1, 1, 2023, 1, 1, 1, 525, 1, 1, 1, 49, 1, 1, 1, 15, 35, 1, 1, 603, 195, 99, 1, 15, 1, 181, 1, 49, 1, 1, 1, 15, 1, 1, 35, 14875, 1, 1, 1, 15, 1, 1, 1, 1715, 1, 1, 99, 15, 1, 1, 1, 603, 3235, 1, 1, 15, 1, 1, 1, 49, 1, 35, 1, 15, 1, 1, 1, 2023, 1

Offset: 1

Antti Karttunen, Aug 31 2018

Multiplicative because A046644 and A057521 are.

Antti Karttunen, Table of n, a(n) for n = 1..65537
Vaclav Kotesovec, Graph - the asymptotic ratio (10^8 terms)

Cf. A057521, A046644 (denominators).

Cf. also A317935, A318511, A318649.

ff[p_, e_] := If[e > 1, p^e, 1]; a[1] = 1; a[n_] := Times @@ ff @@@ FactorInteger[n]; f[1] = 1; f[n_] := f[n] = 1/2 (a[n] - Sum[f[d] f[n/d], {d, Divisors[n][[2 ;; -2]]}]); Table[Numerator[f[n]], {n, 1, 100}] (* Vaclav Kotesovec, May 11 2025 *)

up_to = 65537;
A057521(n) = { my(f=factor(n)); prod(i=1, #f~, if(f[i, 2]>1, f[i, 1]^f[i, 2], 1)); }; \\ From A057521
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&&dA057521(n)));
A318650(n) = numerator(v318650_aux[n]);

a(n) = numerator of f(n), where f(1) = 1, f(n) = (1/2) * (A057521(n) - Sum_{d|n, d>1, d 1.
From Vaclav Kotesovec, May 10 2025, simplified May 11 2025: (Start)
Let f(s) = Product_{p prime} (1 - 1/p^(3*s-2) + 1/p^(3*s-3) + 1/p^s).
Sum_{k=1..n} A318650(k) / A046644(k) ~ n^(3/2) * sqrt(2*f(3/2)/(9*Pi*log(n))) * (1 + (2/3 - gamma - f'(3/2)/(2*f(3/2))) / (2*log(n))), where
f(3/2) = Product_{p prime} (1 + 2/p^(3/2) - 1/p^(5/2)) = A328013 = 3.51955505841710664719752940369857817...
f'(3/2)/f(3/2) = Sum_{p prime} (4*p - 3) * log(p) / (1 - 2*p - p^(5/2)) = -3.90914718020692131140714384422938370058563543737256496...
and gamma is the Euler-Mascheroni constant A001620. (End)

cp's OEIS Frontend

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

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A318649 Numerators of the sequence whose Dirichlet convolution with itself yields squares, A000290.

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

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A318650 Numerators of the sequence whose Dirichlet convolution with itself yields A057521, the powerful part of n.

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