A318324 Denominators of rational valued sequence whose Dirichlet convolution with itself yields A046523, smallest number with same prime signature as n.

1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 8, 1, 2, 1, 2, 1, 1, 1, 1, 2, 1, 2, 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, 1, 1, 1, 2, 16, 1, 1, 1, 2, 1, 1, 1, 4, 1, 1, 2, 2, 1, 1, 1, 8, 8, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 4, 1, 2, 2, 4, 1, 1, 1, 1, 1

Offset

1,4

Links

Antti Karttunen, Table of n, a(n) for n = 1..16384

Formula

a(n) = denominator of f(n), where f(1) = 1, f(n) = (1/2) * (A046523(n) - Sum_{d|n, d>1, d<n} f(d) * f(n/d)) for n > 1.

Prog

(PARI)
up_to = 16384;
A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); };  \\ From A046523
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.
v318323_24 = DirSqrt(vector(up_to, n, A046523(n)));
A318324(n) = denominator(v318323_24[n]);

Crossrefs

Cf. A046523, A318323 (numerators).

Sequence in context: A363265 A347456 A294874 * A317934 A353376 A279848

Adjacent sequences: A318321 A318322 A318323 * A318325 A318326 A318327

Keyword

nonn,frac

Author

Antti Karttunen, Aug 24 2018

Status

approved