A318319 Numerators of rational valued sequence whose Dirichlet convolution with itself yields A064989.

1, 1, 1, 3, 3, 1, 5, 5, 3, 3, 7, 3, 11, 5, 3, 35, 13, 3, 17, 9, 5, 7, 19, 5, 27, 11, 5, 15, 23, 3, 29, 63, 7, 13, 15, 9, 31, 17, 11, 15, 37, 5, 41, 21, 9, 19, 43, 35, 75, 27, 13, 33, 47, 5, 21, 25, 17, 23, 53, 9, 59, 29, 15, 231, 33, 7, 61, 39, 19, 15, 67, 15, 71, 31, 27, 51, 35, 11, 73, 105, 35, 37, 79, 15, 39, 41, 23, 35, 83, 9, 55, 57

Offset: 1

Author

Antti Karttunen, Aug 24 2018

Keywords

• nonn
• frac
• mult

Comments

Multiplicative because A064989 is.

No negative terms among the first 2^20 terms.

Links

• Antti Karttunen, Table of n, a(n) for n = 1..16384
• Index entries for sequences computed from indices in prime factorization

Crossrefs

Cf. A064989, A317932 (seems to give denominators, see A261179).

Cf. also A318321.

Programs

PARI

up_to = 16384;
A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
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.
v318319aux = DirSqrt(vector(up_to, n, A064989(n)));
A318319(n) = numerator(v318319aux[n]);

Formula

a(n) = numerator of f(n), where f(1) = 1, f(n) = (1/2) * (A064989(n) - Sum_{d|n, d>1, d 1.