A318669 Numerators of the sequence whose Dirichlet convolution with itself yields A065769 ("Prime cascade").

1, 1, 1, 7, 3, 1, 5, 25, 5, 3, 7, 7, 11, 5, 3, 363, 13, 5, 17, 21, 5, 7, 19, 25, 51, 11, 13, 35, 23, 3, 29, 1335, 7, 13, 15, 35, 31, 17, 11, 75, 37, 5, 41, 49, 15, 19, 43, 363, 115, 51, 13, 77, 47, 13, 21, 125, 17, 23, 53, 21, 59, 29, 25, 9923, 33, 7, 61, 91, 19, 15, 67, 125, 71, 31, 51, 119, 35, 11, 73, 1089, 139, 37, 79, 35, 39, 41, 23

Offset: 1

Author

Antti Karttunen, Sep 03 2018

Keywords

• nonn
• frac
• mult

Comments

Multiplicative because A065769 and A317932 are.

Links

• Antti Karttunen, Table of n, a(n) for n = 1..65537

Crossrefs

Cf. A065769, A317932 (denominators).

Programs

PARI

up_to = 1+(2^16);
A003557(n) = { my(f=factor(n)); for (i=1, #f~, f[i, 2] = max(0,f[i, 2]-1)); factorback(f); };
A065769(n) = { my(f=factor(n>>valuation(n,2))[, 1]~); (A003557(n) * factorback(vector(#f,i,precprime(f[i]-1)))); }; \\ Antti Karttunen, Sep 03 2018
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&&dA065769(n)));
A318669(n) = numerator(v318669_aux[n]);

Formula

a(n) = numerator of f(n), where f(1) = 1, f(n) = (1/2) * (A065769(n) - Sum_{d|n, d>1, d 1.