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

1,4

Links

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

Formula

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

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 *)

Prog

(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)&&(d<n), A317830aux(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)&&(d<n), A317830aux(d)*A317830aux(n/d), 0)))/2); mapput(memo317830, n, v); (v)));

Crossrefs

Cf. A151800, A175851, A046644 (denominators).

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

Sequence in context: A130875 A370112 A200923 * A317938 A317834 A340144

Adjacent sequences: A317827 A317828 A317829 * A317831 A317832 A317833

Keyword

sign,frac

Author

Antti Karttunen, Aug 12 2018

Status

approved