A318314 Denominators of the sequence whose Dirichlet convolution with itself yields A068068, number of odd unitary divisors of n.

1, 2, 1, 8, 1, 2, 1, 16, 2, 2, 1, 8, 1, 2, 1, 128, 1, 4, 1, 8, 1, 2, 1, 16, 2, 2, 2, 8, 1, 2, 1, 256, 1, 2, 1, 16, 1, 2, 1, 16, 1, 2, 1, 8, 2, 2, 1, 128, 2, 4, 1, 8, 1, 4, 1, 16, 1, 2, 1, 8, 1, 2, 2, 1024, 1, 2, 1, 8, 1, 2, 1, 32, 1, 2, 2, 8, 1, 2, 1, 128, 8, 2, 1, 8, 1, 2, 1, 16, 1, 4, 1, 8, 1, 2, 1, 256, 1, 4, 2, 16, 1, 2, 1, 16, 1

Offset: 1

Author

Antti Karttunen and Andrew Howroyd, Aug 29 2018

Keywords

• nonn
• frac
• mult

Comments

The sequence seems to give the denominators of several other similarly constructed "Dirichlet Square Roots".

Note that A318314 differs from A318454 at exactly those n where A001227 differs from A068068, the numbers in A038838. - Antti Karttunen, Sep 07 2018

Links

• Antti Karttunen, Table of n, a(n) for n = 1..16384

Crossrefs

Cf. A005187, A011371, A068068, A318313 (numerators), A318315.

Cf. also A006519, A046644, A299150, A317932, A317934, A317940, A318454, A318662.

Programs

Mathematica

a35[n_] := (1 - (-1)^n)/2;
a120[n_] := DigitCount[n, 2, 1];
a[n_] := Product[{p, e} = pe; 2^(((2 - a35[p])*e) - a120[e]), {pe, FactorInteger[n]}];
a /@ Range[100] (* Jean-François Alcover, Sep 19 2019 *)

PARI

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]);
A318314(n) = denominator(v318313_15[n]);

Formula

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

a(n) = 2^A318315(n).

From Antti Karttunen, Sep 03-07 2018: (Start, conjectured formulas)

a(n) = A006519(n) * A317934(n), thus multiplicative with a(2^e) = 2^A005187(e), a(p^e) = 2^A011371(e) for odd primes p.

Equally, multiplicative with a(p^e) = 2^(((2-A000035(p))*e)-A000120(e)) for all primes p.

(End)