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

Author

Antti Karttunen, Aug 12 2018

Keywords

• sign
• frac

Comments

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

Links

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

Crossrefs

Cf. A078899, A046644 (denominators).

Cf. also A305799, A317833, A317830.

Programs

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[1Jean-François Alcover, Dec 19 2021 *)

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)&&(dA317834aux(d)*A317834aux(n/d),0)))/2);
A317834(n) = numerator(A317834aux(n));

Formula

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