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A317941
Numerators of rational valued sequence whose Dirichlet convolution with itself yields A037445, number of infinitary divisors (or i-divisors) of n.
3
1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, -5, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 3, 1, 1, 1, 1, 15, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, -5, 1, 1, 1, 1, 1, 3, 1, 3, 1, 1, 1, 1, 1, 1, 1, -11, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, -5, -5, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 15, 1, 1, 1, 1, 1, 1, 1, 3, 1
OFFSET
1,8
COMMENTS
Multiplicative because A037445 is.
LINKS
FORMULA
a(n) = numerator of f(n), where f(1) = 1, f(n) = (1/2) * (A037445(n) - Sum_{d|n, d>1, d<n} f(d) * f(n/d)) for n > 1.
PROG
(PARI)
up_to = 1+(2^16);
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.
A037445(n) = factorback(apply(a -> 2^hammingweight(a), factorint(n)[, 2])) \\ From A037445
v317941aux = DirSqrt(vector(up_to, n, A037445(n)));
A317941(n) = numerator(v317941aux[n]);
CROSSREFS
Cf. A037445, A317934 (denominators).
Cf. also A317933, A317940.
Sequence in context: A204129 A204114 A204131 * A318667 A365426 A030598
KEYWORD
sign,frac,mult
AUTHOR
Antti Karttunen, Aug 22 2018
STATUS
approved