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A318454
Denominators of the sequence whose Dirichlet convolution with itself yields A001227, number of odd divisors of n.
4
1, 2, 1, 8, 1, 2, 1, 16, 1, 2, 1, 8, 1, 2, 1, 128, 1, 2, 1, 8, 1, 2, 1, 16, 1, 2, 1, 8, 1, 2, 1, 256, 1, 2, 1, 8, 1, 2, 1, 16, 1, 2, 1, 8, 1, 2, 1, 128, 1, 2, 1, 8, 1, 2, 1, 16, 1, 2, 1, 8, 1, 2, 1, 1024, 1, 2, 1, 8, 1, 2, 1, 16, 1, 2, 1, 8, 1, 2, 1, 128, 1, 2, 1, 8, 1, 2, 1, 16, 1, 2, 1, 8, 1, 2, 1, 256, 1, 2, 1, 8, 1, 2, 1, 16, 1
OFFSET
1,2
COMMENTS
The sequence seems to give the denominators of several other similarly constructed "Dirichlet Square Roots".
LINKS
FORMULA
a(n) = denominator of f(n), where f(1) = 1, f(n) = (1/2) * (A001227(n) - Sum_{d|n, d>1, d<n} f(d) * f(n/d)) for n > 1.
a(n) = 2^A318455(n).
MATHEMATICA
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]]}]);
Table[f[n] // Denominator, {n, 1, 105}] (* Jean-François Alcover, Sep 13 2018 *)
PROG
(PARI)
up_to = 16384;
A001227(n) = numdiv(n>>valuation(n, 2)); \\ From A001227
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.
v318453_54 = DirSqrt(vector(up_to, n, A001227(n)));
A318454(n) = denominator(v318453_54[n]);
CROSSREFS
Cf. A001227.
Cf. A318453 (numerators), A318455.
Sequence in context: A317832 A317928 A011327 * A317926 A318318 A318314
KEYWORD
nonn,frac
AUTHOR
STATUS
approved