A318323 Numerators of rational valued sequence whose Dirichlet convolution with itself yields A046523, smallest number with same prime signature as n.

1, 1, 1, 3, 1, 2, 1, 5, 3, 2, 1, 5, 1, 2, 2, 35, 1, 5, 1, 5, 2, 2, 1, 4, 3, 2, 5, 5, 1, 9, 1, 63, 2, 2, 2, 35, 1, 2, 2, 4, 1, 9, 1, 5, 5, 2, 1, 55, 3, 5, 2, 5, 1, 4, 2, 4, 2, 2, 1, 9, 1, 2, 5, 231, 2, 9, 1, 5, 2, 9, 1, 43, 1, 2, 5, 5, 2, 9, 1, 55, 35, 2, 1, 9, 2, 2, 2, 4, 1, 9, 2, 5, 2, 2, 2, 49, 1, 5, 5, 35, 1, 9, 1, 4, 9

Offset

1,4

Comments

The first 2^20 terms are positive.

Links

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

Formula

a(n) = numerator 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)));
A318323(n) = numerator(v318323_24[n]);

Crossrefs

Cf. A046523, A318324 (gives the denominators).

Sequence in context: A279965 A285121 A295935 * A366803 A347396 A135261

Adjacent sequences: A318320 A318321 A318322 * A318324 A318325 A318326

Keyword

nonn,frac

Author

Antti Karttunen, Aug 24 2018

Status

approved