A365228 Denominator of Sum_{1<=j<=k<=n, gcd(j,k)=1} 1/(j*k).

1, 2, 1, 3, 4, 20, 10, 210, 168, 504, 105, 1155, 792, 1560, 60060, 180180, 16016, 123760, 510510, 29099070, 21162960, 335920, 3233230, 74364290, 45762640, 187210800, 13385572200, 40156716600, 97349616, 31054527504, 166363540200, 12033629407800, 2831442213600, 1698865328160

Offset: 1

Author

Franz Vrabec, Aug 28 2023

Keywords

• nonn
• frac

Crossrefs

Cf. A365227 (numerator of this sum).

Programs

Maple

A365228 := proc(n)
   local j,k,s; s := 0;
   for j from 1 to n do
      for k from j to n do
         if gcd(j,k) = 1 then s := s + 1/(j*k);
         end if;
      end do;
   end do;
   denom(s);
end proc;
seq(A365228(n), n = 1..30);
# second Maple program:
a:= n-> denom(add(add(`if`(igcd(j, k)=1, 1/j, 0), j=1..k)/k, k=1..n)):
seq(a(n), n=1..45);  # Alois P. Heinz, Aug 28 2023

Mathematica

a[n_Integer]:=Module[{sum,j,k},sum=Sum[If[GCD[j,k]==1,1/(j*k),0],{j,1,n},{k,j,n}]; Denominator[sum]]; Table[a[n],{n,1,34}] (* Robert P. P. McKone, Aug 29 2023 *)

PARI

a(n) = denominator(sum(j=1, n, sum(k=j, n, if (gcd(j,k)==1, 1/(j*k))))); \\ Michel Marcus, Aug 28 2023

Python

from fractions import Fraction
from math import gcd
def A365228(n): return sum(sum(Fraction(1,j) for j in range(1,k+1) if gcd(j,k)==1)/k for k in range(1,n+1)).denominator # Chai Wah Wu, Aug 29 2023