A318512 -id:A318512 - cp's OEIS frontend

A318651 a(n) = A046644(n)/A318512(n).

1, 2, 1, 8, 1, 4, 1, 16, 1, 4, 1, 16, 1, 4, 1, 128, 1, 4, 1, 16, 1, 4, 1, 32, 1, 4, 1, 16, 1, 4, 1, 256, 1, 4, 1, 16, 1, 4, 1, 32, 1, 4, 1, 16, 1, 4, 1, 256, 1, 4, 1, 16, 1, 4, 1, 32, 1, 4, 1, 16, 1, 4, 1, 1024, 1, 4, 1, 16, 1, 4, 1, 64, 1, 4, 1, 16, 1, 4, 1, 256, 1, 4, 1, 16, 1, 4, 1, 32, 1, 4, 1, 16, 1, 4, 1, 512, 1, 4, 1, 16, 1, 4, 1, 32, 1

Offset: 1

Antti Karttunen, Aug 31 2018

nonn

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

Cf. A046644, A318512, A318652.

A318651(n) = A046644(n)/A318512(n); \\ Needs also code from those two respective entries.

a(n) = A046644(n)/A318512(n).

a(n) = 2^A318652(n).

A318513 The 2-adic valuation of A318512.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Aug 30 2018

nonn

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

Cf. A318512.

A318513(n) = valuation(A318512(n),2); \\ Needs also code from A318512.

a(n) = A007814(A318512(n)).

A318652 The 2-adic valuation of A046644(n)/A318512(n) (A318651).

Original entry on oeis.org

0, 1, 0, 3, 0, 2, 0, 4, 0, 2, 0, 4, 0, 2, 0, 7, 0, 2, 0, 4, 0, 2, 0, 5, 0, 2, 0, 4, 0, 2, 0, 8, 0, 2, 0, 4, 0, 2, 0, 5, 0, 2, 0, 4, 0, 2, 0, 8, 0, 2, 0, 4, 0, 2, 0, 5, 0, 2, 0, 4, 0, 2, 0, 10, 0, 2, 0, 4, 0, 2, 0, 6, 0, 2, 0, 4, 0, 2, 0, 8, 0, 2, 0, 4, 0, 2, 0, 5, 0, 2, 0, 4, 0, 2, 0, 9, 0, 2, 0, 4, 0, 2, 0, 5, 0

Offset: 1

Antti Karttunen, Aug 31 2018

nonn

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

Cf. A046644, A318512, A318651.

A318652(n) = valuation(A046644(n)/A318512(n),2); \\ Needs also code from those two respective entries.

a(n) = A007814(A318651(n)).

a(n) = A046645(n) - A318513(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).

A299150 Denominators of the positive solution to n = Sum_{d|n} a(d) * a(n/d).

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Feb 03 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 from Andrew Howroyd)

Cf. A000010, A003958, A005187, A006519, A007431, A011371, A018804, A023900, A029935, A037445, A046643, A046644, A165825, A257098, A298971, A299119, A299149 (numerators), A299151, A317848, A317929, A317932, A318314, A318512, A318653, A318681.

Cf. A318440 (the 2-adic valuation).

nn=50;
sys=Table[n==Sum[a[d]*a[n/d],{d,Divisors[n]}],{n,nn}];
Denominator[Array[a,nn]/.Solve[sys,Array[a,nn]][[2]]]
f[p_, e_] := 2^((1 + Mod[p, 2])*e - DigitCount[e, 2, 1]); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Apr 28 2023 *)

a(n)={my(v=factor(n)[,2]); denominator(n*prod(i=1, #v, my(e=v[i]); binomial(2*e, e)/4^e))} \\ Andrew Howroyd, Aug 09 2018

A299150(n) = { my(f = factor(n), m=1); for(i=1, #f~, m *= 2^(((1+(f[i,1]%2))*f[i,2]) - hammingweight(f[i,2]))); (m); }; \\ Antti Karttunen, Sep 03 2018

for(n=1, 100, print1(denominator(direuler(p=2, n, 1/(1-p*X)^(1/2))[n]), ", ")) \\ Vaclav Kotesovec, May 08 2025

a(n) = denominator(n*A317848(n)/A165825(n)) = A165825(n)/(A037445(n) * A006519(n)). - Andrew Howroyd, Aug 09 2018
a(n) = A046644(n)/A006519(n). - Andrew Howroyd and Antti Karttunen, Aug 30 2018
From Antti Karttunen, Sep 03 2018: (Start)
a(n) = 2^A318440(n).
Multiplicative with a(2^e) = 2^A011371(e), a(p^e) = 2^A005187(e) for odd primes p.
Multiplicative with a(p^e) = 2^(((1+A000035(p))*e)-A000120(e)) for all primes p.
(End)

Keyword:mult added by Andrew Howroyd, Aug 09 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

A318511 Numerators of the sequence whose Dirichlet convolution with itself yields A064549, n * Product_{primes p|n} p.

Original entry on oeis.org

1, 2, 9, 2, 25, 9, 49, 4, 27, 25, 121, 9, 169, 49, 225, 6, 289, 27, 361, 25, 441, 121, 529, 18, -125, 169, 405, 49, 841, 225, 961, 12, 1089, 289, 1225, 27, 1369, 361, 1521, 50, 1681, 441, 1849, 121, 675, 529, 2209, 27, -1029, -125, 2601, 169, 2809, 405, 3025, 98, 3249, 841, 3481, 225, 3721, 961, 1323, 20, 4225, 1089, 4489, 289, 4761, 1225, 5041, 27

Offset: 1

Antti Karttunen, Aug 30 2018

No zeros among the first 2^20 terms.

For odd primes p, it seems that a(p) = p^2.

Antti Karttunen, Table of n, a(n) for n = 1..16384
Vaclav Kotesovec, Graph - the asymptotic ratio (100000 terms)

Cf. A064549, A318512 (denominators).

Cf. also A317935.

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[Numerator[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)));
A318511(n) = numerator(v318511_12[n]);

a(n) = numerator of f(n), where f(1) = 1, f(n) = (1/2) * (A064549(n) - Sum_{d|n, d>1, d 1.
From Vaclav Kotesovec, May 10 2025: (Start)
Let f(s) = Product_{p prime} (1 + p^(3-2*s) - p^(4-2*s) - p^(1-s)).
Sum_{k=1..n} A318511(k) / A318512(k) ~ n^3 * sqrt(Pi*f(3)/(54*log(n))) * (1 + (1/3 - gamma/2 - f'(3)/(2*f(3)) - 3*zeta'(2)/Pi^2) / (2*log(n))), where
f(3) = Product_{p prime} (1 - 2/p^2 + 1/p^3) = A065464 = 0.428249505677094440218765707581823546121298513355936144...
f'(3)/f(3) = Sum_{p prime} (3*p - 2) * log(p) / (p^3 - 2*p + 1) = 1.469536740824614833203393993450164364663334798759143895712...
and gamma is the Euler-Mascheroni constant A001620. (End)

A318658 Denominators of the sequence whose Dirichlet convolution with itself yields A087003, a(2n) = 0 and a(2n+1) = moebius(2n+1).

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

Offset: 1

Antti Karttunen, Aug 31 2018

The sequence seems to give the denominators of several other similarly constructed "Dirichlet Square Roots".

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

Cf. A005187, A087003, A318657 (numerators), A318659.

up_to = 65537;
A087003(n) = ((n%2)*moebius(n)); \\ I.e. a(n) = A000035(n)*A008683(n).
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&&dA087003(n)));
A318657(n) = numerator(v318657_18[n]);
A318658(n) = denominator(v318657_18[n]);

a(n) = denominator of f(n), where f(1) = 1, f(n) = (1/2) * (A087003(n) - Sum_{d|n, d>1, d 1.
a(n) = 2^A318659(n).
a(2n) = 1, a(2n-1) = A046644(2n-1) = A318512(2n-1), for all n >= 1.

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.

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).

cp's OEIS Frontend

A318651 a(n) = A046644(n)/A318512(n).

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

A318513 The 2-adic valuation of A318512.

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

A318652 The 2-adic valuation of A046644(n)/A318512(n) (A318651).

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Formula

A299150 Denominators of the positive solution to n = Sum_{d|n} a(d) * a(n/d).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI

PARI

Formula

Extensions

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

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

PARI

Formula

A318511 Numerators of the sequence whose Dirichlet convolution with itself yields A064549, n * Product_{primes p|n} p.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A318658 Denominators of the sequence whose Dirichlet convolution with itself yields A087003, a(2n) = 0 and a(2n+1) = moebius(2n+1).

Views

Author

Keywords

Comments

Links

Crossrefs