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A317933
Numerators of rational valued sequence whose Dirichlet convolution with itself yields A034444 (number of unitary divisors of n).
5
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1
OFFSET
1,16
COMMENTS
Multiplicative because A034444 is.
The first 2^20 terms are positive. Is the sequence nonnegative?
Records seem to be A001790, occurring at A000302 (apart from 4).
LINKS
FORMULA
a(n) = numerator of f(n), where f(1) = 1, f(n) = (1/2) * (A034444(n) - Sum_{d|n, d>1, d<n} f(d) * f(n/d)) for n > 1.
PROG
(PARI)
A034444(n) = (2^omega(n));
A317933perA317934(n) = if(1==n, n, (A034444(n)-sumdiv(n, d, if((d>1)&&(d<n), A317933perA317934(d)*A317933perA317934(n/d), 0)))/2);
A317933(n) = numerator(A317933perA317934(n));
(PARI)
up_to = 65537;
\\ Faster:
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.
v317933aux = DirSqrt(vector(up_to, n, A034444(n)));
A317933(n) = numerator(v317933aux[n]);
CROSSREFS
Cf. A001790, A034444, A317934 (denominators).
Sequence in context: A355818 A326538 A333844 * A363329 A370079 A318497
KEYWORD
nonn,frac,mult
AUTHOR
Antti Karttunen, Aug 12 2018
STATUS
approved