A318449 -id:A318449 - cp's OEIS frontend

A323885 Sum of A001511 and its Dirichlet inverse.

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).

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).

cp's OEIS Frontend

A323885 Sum of A001511 and its Dirichlet inverse.

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