A318455 -id:A318455 - cp's OEIS frontend

A318315 The 2-adic valuation of A318314.

0, 1, 0, 3, 0, 1, 0, 4, 1, 1, 0, 3, 0, 1, 0, 7, 0, 2, 0, 3, 0, 1, 0, 4, 1, 1, 1, 3, 0, 1, 0, 8, 0, 1, 0, 4, 0, 1, 0, 4, 0, 1, 0, 3, 1, 1, 0, 7, 1, 2, 0, 3, 0, 2, 0, 4, 0, 1, 0, 3, 0, 1, 1, 10, 0, 1, 0, 3, 0, 1, 0, 5, 0, 1, 1, 3, 0, 1, 0, 7, 3, 1, 0, 3, 0, 1, 0, 4, 0, 2, 0, 3, 0, 1, 0, 8, 0, 2, 1, 4, 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. A318314.

Cf. also A046645, A317946, A318451, A318455.

A318315(n) = valuation(A318314(n),2); \\ Needs also code from A318314.

a(n) = A007814(A318314(n)).

A318451 The 2-adic valuation of A318450.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen and Andrew Howroyd, Aug 29 2018

nonn

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

Cf. A007814, A318450.

Cf. also A046645, A317946, A318315, A318455.

A318451(n) = valuation(A318450(n),2); \\ Needs also code from A318450.

a(n) = A007814(A318450(n)).

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

Original entry on oeis.org

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

cp's OEIS Frontend

A318315 The 2-adic valuation of A318314.

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A318451 The 2-adic valuation of A318450.

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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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