A317834 Numerators of rational valued sequence whose Dirichlet convolution with itself yields A078899 (the ordinal transform of A006530, the largest prime factor of n).

1, 1, 1, 7, 1, 3, 1, 17, 11, 3, 1, 19, 1, 3, 5, 139, 1, 23, 1, 19, 5, 3, 1, 39, 19, 3, 45, 19, 1, 13, 1, 263, 5, 3, 9, 77, 1, 3, 5, 55, 1, 13, 1, 19, 43, 3, 1, 387, 27, 47, 5, 19, 1, 59, 9, 71, 5, 3, 1, 43, 1, 3, 51, 995, 9, 13, 1, 19, 5, 25, 1, 87, 1, 3, 59, 19, 13, 13, 1, 707, 467, 3, 1, 59, 9, 3, 5, 71, 1, 53, 13, 19, 5, 3, 9, 1069, 1

Offset

1,4

Comments

The first negative term is a(216) = -97.

Links

Antti Karttunen, Table of n, a(n) for n = 1..16384

Formula

a(n) = numerator of f(n), where f(1) = 1, f(n) = (1/2) * (A078899(n) - Sum_{d|n, d>1, d<n} f(d) * f(n/d)) for n > 1.

Mathematica

gpf[n_] := If[n == 1, 1, FactorInteger[n][[-1, 1]]];
b[_] = 1;
A078899[n_] := A078899[n] = With[{t = gpf[n]}, b[t]++];
f[n_] := f[n] = If[n == 1, 1, (1/2)(A078899[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)
up_to = 16384;
ordinal_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), pt); for(i=1, length(invec), if(mapisdefined(om, invec[i]), pt = mapget(om, invec[i]), pt = 0); outvec[i] = (1+pt); mapput(om, invec[i], (1+pt))); outvec; };
A006530(n) = if(n>1, vecmax(factor(n)[, 1]), 1);
v078899 = ordinal_transform(vector(up_to, n, A006530(n)));
A078899(n) = v078899[n];
A317834aux(n) = if(1==n, n, (A078899(n)-sumdiv(n, d, if((d>1)&&(d<n), A317834aux(d)*A317834aux(n/d), 0)))/2);
A317834(n) = numerator(A317834aux(n));

Crossrefs

Cf. A078899, A046644 (denominators).

Cf. also A305799, A317833, A317830.

Sequence in context: A200923 A317830 A317938 * A340144 A340141 A342377

Adjacent sequences: A317831 A317832 A317833 * A317835 A317836 A317837

Keyword

sign,frac

Author

Antti Karttunen, Aug 12 2018

Status

approved