A317830 Numerators of rational valued sequence whose Dirichlet convolution with itself yields A175851, the ordinal transform of the nextprime function, A151800.

1, 1, 1, 7, 1, 3, 1, 9, 11, 7, 1, 3, 1, 3, 5, 171, 1, -1, 1, -5, 5, 7, 1, -1, 11, 7, 29, 35, 1, -7, 1, -41, 5, 7, 9, 93, 1, 3, 5, 11, 1, -3, 1, -5, 3, 7, 1, -61, 11, 7, 9, 27, 1, -29, 5, -1, 9, 11, 1, -29, 1, 3, 3, 771, 9, 9, 1, -5, 5, -3, 1, -73, 1, 3, 3, 19, 9, 9, 1, -141, -45, 7, 1, -53, 5, 7, 9, 43, 1, -63, 5, 11, 9, 11, 13, 1597, 1

Offset: 1

Author

Antti Karttunen, Aug 12 2018

Keywords

• sign
• frac

Links

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

Crossrefs

Cf. A151800, A175851, A046644 (denominators).

Cf. also A305791, A305805, A305806, A317833, A317834, A317847.

Programs

Mathematica

A175851[n_] := If[!CompositeQ[n], 1, n - NextPrime[n, -1] + 1];
f[n_] := f[n] = If[n == 1, 1, (1/2)(A175851[n] - Sum[If[1 < d < n, f[d]* f[n/d], 0], {d, Divisors[n]}])];
a[n_] := Numerator[f[n]];
Array[a, 100] (* Jean-François Alcover, Dec 19 2021 *)

PARI

A175851(n) = if(1==n,n,1 + n - precprime(n));
A317830aux(n) = if(1==n,n,(A175851(n)-sumdiv(n,d,if((d>1)&&(dA317830aux(d)*A317830aux(n/d),0)))/2);
A317830(n) = numerator(A317830aux(n));

PARI

\\ Memoized implementation:
memo317830 = Map();
A317830aux(n) = if(1==n,n,if(mapisdefined(memo317830,n),mapget(memo317830,n),my(v = (A175851(n)-sumdiv(n,d,if((d>1)&&(dA317830aux(d)*A317830aux(n/d),0)))/2); mapput(memo317830,n,v); (v)));

Formula

a(n) = numerator of f(n), where f(1) = 1, f(n) = (1/2) * (A175851(n) - Sum_{d|n, d>1, d 1.