A318450 -id:A318450 - cp's OEIS frontend

A318451 The 2-adic valuation of A318450.

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)).

A323885 Sum of A001511 and its Dirichlet inverse.

Original entry on oeis.org

2, 0, 0, 4, 0, 4, 0, 4, 1, 4, 0, 2, 0, 4, 2, 5, 0, 2, 0, 2, 2, 4, 0, 4, 1, 4, 1, 2, 0, 0, 0, 6, 2, 4, 2, 3, 0, 4, 2, 4, 0, 0, 0, 2, 1, 4, 0, 5, 1, 2, 2, 2, 0, 2, 2, 4, 2, 4, 0, 4, 0, 4, 1, 7, 2, 0, 0, 2, 2, 0, 0, 4, 0, 4, 1, 2, 2, 0, 0, 5, 1, 4, 0, 4, 2, 4, 2, 4, 0, 2, 2, 2, 2, 4, 2, 6, 0, 2, 1, 3, 0, 0, 0, 4, 0

Offset: 1

Antti Karttunen, Feb 08 2019

nonn

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

Cf. A001511, A092673.

Cf. also A318449, A318450, A323365, A323882, A323884, A323887, A323896.

A001511(n) = (1+valuation(n,2));
A092673(n) = (moebius(n)-if(n%2,0,moebius(n/2)));
A323885(n) = (A001511(n)+A092673(n));

from sympy import mobius
def A323885(n): return (n&-n).bit_length()+mobius(n)-(0 if n&1 else mobius(n>>1)) # Chai Wah Wu, Jul 13 2022

a(n) = A001511(n) + A092673(n).

A318449 Numerators of the sequence whose Dirichlet convolution with itself yields A001511, the 2-adic valuation of 2n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 3, 1, 5, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 3, 3, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 3, 1, 1, 1, 1, 1, 35, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 3, 3, 3, 1, 1, 1, 1, 1

Offset: 1

Antti Karttunen and Andrew Howroyd, Aug 29 2018

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

Cf. A001511, A318450 (denominators).

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] // Numerator, {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)));
A318449(n) = numerator(v318449_51[n]);

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

Sum_{k=1..n} A318449(k) / A318450(k) ~ n * sqrt(2/(Pi*log(n))) * (1 + (1 - gamma/2 + log(2)/2)/(2*log(n))), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, May 10 2025

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A318451 The 2-adic valuation of A318450.

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A323885 Sum of A001511 and its Dirichlet inverse.

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A318449 Numerators of the sequence whose Dirichlet convolution with itself yields A001511, the 2-adic valuation of 2n.

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Author

Keywords

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PARI

Formula