A318451 -id:A318451 - 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)).

A318450 Denominators of the sequence whose Dirichlet convolution with itself yields A001511, the 2-adic valuation of 2n.

Original entry on oeis.org

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

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. A001511, A318449 (numerators), A318451.

a1511[n_] := IntegerExponent[2n, 2];
f[1] = 1; f[n_] := f[n] = 1/2 (a1511[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 = 65537;
A001511(n) = 1+valuation(n,2);
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.
v318449_51 = DirSqrt(vector(up_to, n, A001511(n)));
A318450(n) = denominator(v318449_51[n]);

a(n) = denominator of f(n), where f(1) = 1, f(n) = (1/2) * (A001511(n) - Sum_{d|n, d>1, d 1.

a(n) = 2^A318451(n).

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)).

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A318315 The 2-adic valuation of A318314.

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A318450 Denominators of the sequence whose Dirichlet convolution with itself yields A001511, the 2-adic valuation of 2n.

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

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