|
|
A317834
|
|
Numerators of rational valued sequence whose Dirichlet convolution with itself yields A078899 (the ordinal transform of A006530, the largest prime factor of n).
|
|
5
|
|
|
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
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,4
|
|
COMMENTS
|
The first negative term is a(216) = -97.
|
|
LINKS
|
|
|
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;
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]];
|
|
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)));
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
|
|
|
KEYWORD
|
sign,frac
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|