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

1,2

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

Formula

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

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 *)

Prog

(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&&d<n, u[d]*u[n/d], 0)))/2); u}; \\ From A317937.
v318313_15 = DirSqrt(vector(up_to, n, A068068(n)));
A318313(n) = numerator(v318313_15[n]);
A318314(n) = denominator(v318313_15[n]);

Crossrefs

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

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

Sequence in context: A318454 A317926 A318318 * A230369 A341406 A342937

Adjacent sequences: A318311 A318312 A318313 * A318315 A318316 A318317

Keyword

nonn,frac,mult

Author

Antti Karttunen and Andrew Howroyd, Aug 29 2018

Status

approved