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A317830
Numerators of rational valued sequence whose Dirichlet convolution with itself yields A175851, the ordinal transform of the nextprime function, A151800.
6
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
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).
Sequence in context: A130875 A370112 A200923 * A317938 A317834 A340144
KEYWORD
sign,frac
AUTHOR
Antti Karttunen, Aug 12 2018
STATUS
approved