A317833 Numerators of rational valued sequence whose Dirichlet convolution with itself yields A078898 (the ordinal transform of A020639, the smallest prime factor of n).

1, 1, 1, 7, 1, 5, 1, 25, 7, 9, 1, 31, 1, 13, 5, 363, 1, 55, 1, 55, 7, 21, 1, 101, 7, 25, 33, 79, 1, 41, 1, 1335, 11, 33, 5, 305, 1, 37, 13, 177, 1, 59, 1, 127, 47, 45, 1, 1371, 7, 175, 17, 151, 1, 309, 7, 253, 19, 57, 1, 187, 1, 61, 67, 9923, 9, 95, 1, 199, 23, 113, 1, 927, 1, 73, 87, 223, 5, 113, 1, 2379, 715, 81, 1, 265, 11

Offset: 1

Author

Antti Karttunen, Aug 10 2018

Keywords

• sign
• frac

Comments

The first negative term is a(840) = -445.

Links

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

Crossrefs

Cf. A046644, A078898.

Cf. also A305798, A305803, A305804, A317830, A317834.

Programs

Mathematica

lpf[n_] := If[n == 1, 1, FactorInteger[n][[1, 1]]];
b[_] = 1;
A078898[n_] := A078898[n] = If[n == 0, 0, With[{t = lpf[n]}, b[t]++]];
f[n_] := f[n] = If[n == 1, 1, (1/2)(A078898[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

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; };
A020639(n) = if(n>1, if(n>n=factor(n, 0)[1, 1], n, factor(n)[1, 1]), 1); \\ From A020639
v078898 = ordinal_transform(vector(up_to,n,A020639(n)));
A078898(n) = v078898[n];
A317833aux(n) = if(1==n,n,(A078898(n)-sumdiv(n,d,if((d>1)&&(dA317833aux(d)*A317833aux(n/d),0)))/2);
A317833(n) = numerator(A317833aux(n));

Formula

a(n) = numerator of f(n), where f(1) = 1, f(n) = (1/2) * (A078898(n) - Sum_{d|n, d>1, d 1.