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A318498
Denominators of the sequence whose Dirichlet convolution with itself yields A061389, number of (1+phi)-divisors of n.
4
1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 8, 1, 2, 1, 2, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 8, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 8, 2, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 16, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 1, 8, 8, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 8, 1, 2, 2, 4, 1, 1, 1, 1, 1
OFFSET
1,4
COMMENTS
The sequence seems to give the denominators of a few other similarly constructed rational valued sequences obtained as "Dirichlet Square Roots" (of possibly A092520 and A293443).
LINKS
FORMULA
a(n) = denominator of f(n), where f(1) = 1, f(n) = (1/2) * (A061389(n) - Sum_{d|n, d>1, d<n} f(d) * f(n/d)) for n > 1.
a(n) = 2^A318499(n).
PROG
(PARI)
up_to = 65537;
A061389(n) = factorback(apply(e -> (1+eulerphi(e)), factor(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.
v318497_98 = DirSqrt(vector(up_to, n, A061389(n)));
A318497(n) = numerator(v318497_98[n]);
A318498(n) = denominator(v318497_98[n]);
CROSSREFS
Cf. A061389, A318497 (numerators), A318499.
Cf. also A299150, A046644.
Sequence in context: A370077 A370080 A372331 * A093997 A157196 A300410
KEYWORD
nonn,frac
AUTHOR
Antti Karttunen, Aug 30 2018
STATUS
approved