A318321 Numerators of rational valued sequence whose Dirichlet convolution with itself yields A003961.

1, 3, 5, 27, 7, 15, 11, 135, 75, 21, 13, 135, 17, 33, 35, 2835, 19, 225, 23, 189, 55, 39, 29, 675, 147, 51, 625, 297, 31, 105, 37, 15309, 65, 57, 77, 2025, 41, 69, 85, 945, 43, 165, 47, 351, 525, 87, 53, 14175, 363, 441, 95, 459, 59, 1875, 91, 1485, 115, 93, 61, 945, 67, 111, 825, 168399, 119, 195, 71, 513, 145, 231, 73

Offset: 1

Author

Antti Karttunen, Aug 24 2018

Keywords

• nonn
• frac
• mult

Comments

Multiplicative because A003961 is.

Links

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

Crossrefs

Cf. A003961, A046644 (denominators).

Cf. also A318319.

Programs

PARI

up_to = 16384;
A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ From A003961
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.
v318321aux = DirSqrt(vector(up_to, n, A003961(n)));
A318321(n) = numerator(v318321aux[n]);

Formula

a(n) = numerator of f(n), where f(1) = 1, f(n) = (1/2) * (A003961(n) - Sum_{d|n, d>1, d 1.