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A318450
Denominators of the sequence whose Dirichlet convolution with itself yields A001511, the 2-adic valuation of 2n.
4
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
1,3
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) * (A001511(n) - Sum_{d|n, d>1, d<n} f(d) * f(n/d)) for n > 1.
a(n) = 2^A318451(n).
MATHEMATICA
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 *)
PROG
(PARI)
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&&d<n, u[d]*u[n/d], 0)))/2); u}; \\ From A317937.
v318449_51 = DirSqrt(vector(up_to, n, A001511(n)));
A318450(n) = denominator(v318449_51[n]);
CROSSREFS
Cf. A001511, A318449 (numerators), A318451.
Sequence in context: A016443 A327640 A300648 * A120256 A300647 A114811
KEYWORD
nonn,frac
AUTHOR
STATUS
approved