A318453 -id:A318453 - cp's OEIS frontend

A318454 Denominators of the sequence whose Dirichlet convolution with itself yields A001227, number of odd divisors of n.

1, 2, 1, 8, 1, 2, 1, 16, 1, 2, 1, 8, 1, 2, 1, 128, 1, 2, 1, 8, 1, 2, 1, 16, 1, 2, 1, 8, 1, 2, 1, 256, 1, 2, 1, 8, 1, 2, 1, 16, 1, 2, 1, 8, 1, 2, 1, 128, 1, 2, 1, 8, 1, 2, 1, 16, 1, 2, 1, 8, 1, 2, 1, 1024, 1, 2, 1, 8, 1, 2, 1, 16, 1, 2, 1, 8, 1, 2, 1, 128, 1, 2, 1, 8, 1, 2, 1, 16, 1, 2, 1, 8, 1, 2, 1, 256, 1, 2, 1, 8, 1, 2, 1, 16, 1

Offset: 1

Antti Karttunen and Andrew Howroyd, Aug 29 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. A001227.

Cf. A318453 (numerators), A318455.

f[1] = 1; f[n_] := f[n] = 1/2 (Sum[Mod[d, 2], {d, Divisors[n]}] - Sum[f[d] f[n/d], {d, Divisors[n][[2 ;; -2]]}]);
Table[f[n] // Denominator, {n, 1, 105}] (* Jean-François Alcover, Sep 13 2018 *)

up_to = 16384;
A001227(n) = numdiv(n>>valuation(n, 2)); \\ From A001227
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.
v318453_54 = DirSqrt(vector(up_to, n, A001227(n)));
A318454(n) = denominator(v318453_54[n]);

a(n) = denominator of f(n), where f(1) = 1, f(n) = (1/2) * (A001227(n) - Sum_{d|n, d>1, d 1.
a(n) = 2^A318455(n).
Sum_{k=1..n} A318453(k) / a(k) ~ n/sqrt(2). - Vaclav Kotesovec, May 09 2025

A318455 The 2-adic valuation of A318454(n).

Original entry on oeis.org

0, 1, 0, 3, 0, 1, 0, 4, 0, 1, 0, 3, 0, 1, 0, 7, 0, 1, 0, 3, 0, 1, 0, 4, 0, 1, 0, 3, 0, 1, 0, 8, 0, 1, 0, 3, 0, 1, 0, 4, 0, 1, 0, 3, 0, 1, 0, 7, 0, 1, 0, 3, 0, 1, 0, 4, 0, 1, 0, 3, 0, 1, 0, 10, 0, 1, 0, 3, 0, 1, 0, 4, 0, 1, 0, 3, 0, 1, 0, 7, 0, 1, 0, 3, 0, 1, 0, 4, 0, 1, 0, 3, 0, 1, 0, 8, 0, 1, 0, 3, 0, 1, 0, 4, 0

Offset: 1

Antti Karttunen and Andrew Howroyd, Aug 29 2018

nonn

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

Cf. A001227, A007814, A318453, A318454.

Cf. also A046645, A317946, A318315, A318451.

f[1] = 1; f[n_] := f[n] = 1/2 (Sum[Mod[d, 2], {d, Divisors[n]}] - Sum[f[d] f[n/d], {d, Divisors[n][[2 ;; -2]]}]);
a[n_] := IntegerExponent[Denominator[f[n]], 2];
Array[a, 105] (* Jean-François Alcover, Sep 13 2018 *)

A318455(n) = valuation(A318454(n),2); \\ Needs also program from A318454.

a(n) = A007814(A318454(n)).

A318313 Numerators of the sequence whose Dirichlet convolution with itself yields A068068, number of odd unitary divisors of n.

Original entry on oeis.org

1, 1, 1, 3, 1, 1, 1, 5, 1, 1, 1, 3, 1, 1, 1, 35, 1, 1, 1, 3, 1, 1, 1, 5, 1, 1, 1, 3, 1, 1, 1, 63, 1, 1, 1, 3, 1, 1, 1, 5, 1, 1, 1, 3, 1, 1, 1, 35, 1, 1, 1, 3, 1, 1, 1, 5, 1, 1, 1, 3, 1, 1, 1, 231, 1, 1, 1, 3, 1, 1, 1, 5, 1, 1, 1, 3, 1, 1, 1, 35, 3, 1, 1, 3, 1, 1, 1, 5, 1, 1, 1, 3, 1, 1, 1, 63, 1, 1, 1, 3, 1, 1, 1, 5, 1

Offset: 1

Antti Karttunen and Andrew Howroyd, Aug 29 2018

Antti Karttunen, Table of n, a(n) for n = 1..65537
Vaclav Kotesovec, Graph - the asymptotic ratio (16384 terms)

Cf. A068068, A318314 (denominators).

Differs from A318453 for the first time at n=81, where a(81) = 3, while A318453(81) = 1.

up_to = 16384;
A068068(n) = (2^omega(n>>valuation(n, 2))); \\ From A068068
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.
v318313_15 = DirSqrt(vector(up_to, n, A068068(n)));
A318313(n) = numerator(v318313_15[n]);

a(n) = numerator of f(n), where f(1) = 1, f(n) = (1/2) * (A068068(n) - Sum_{d|n, d>1, d 1.

Sum_{k=1..n} A318313(k) / A318314(k) ~ 2*n/Pi. - Vaclav Kotesovec, May 10 2025

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A318454 Denominators of the sequence whose Dirichlet convolution with itself yields A001227, number of odd divisors of n.

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A318455 The 2-adic valuation of A318454(n).

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A318313 Numerators of the sequence whose Dirichlet convolution with itself yields A068068, number of odd unitary divisors of n.

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