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

Author

Antti Karttunen, Aug 24 2018

Keywords

• nonn
• frac

Comments

The first 2^20 terms are positive.

Links

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

Crossrefs

Cf. A046523, A318324 (gives the denominators).

Programs

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&&dA317937.
v318323_24 = DirSqrt(vector(up_to, n, A046523(n)));
A318323(n) = numerator(v318323_24[n]);

Formula

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