A078560 Denominator of Product_{i=1..n} (p_i+1)/(p_i-1). Numerators are in A078559.

1, 1, 1, 1, 5, 5, 10, 1, 11, 11, 11, 11, 55, 5, 115, 1495, 8671, 43355, 476905, 16691675, 16691675, 43398355, 254190365, 559218803, 559218803, 13980470075, 1075420775, 56997301075, 1036314565, 1036314565, 1036314565, 6123676975

Offset: 1

Author

Labos Elemer, Dec 06 2002

Keywords

• nonn
• frac

Comments

According to Koninck (2009), a(8) is the largest value of this sequence known to be 1 (meaning that the product is an integer). [Alonso del Arte, Aug 23 2011]

References

• R. K. Guy, Unsolved Problems in Number Theory, B48.
• J.-M. De Koninck, Those Fascinating Numbers, Amer. Math. Soc., 2009, p. 6.

Links

• Robert Israel, Table of n, a(n) for n = 1..1703
• F. Smarandache, Only Problems, Not Solutions!.

Crossrefs

Cf. A000203, A002110, A000005, A005867, A054640, A020492, A078558, A078559.

Programs

Maple

Q:= 1: p:= 1:
for n from 1 to 100 do
  p:= nextprime(p);
  Q:= Q * (p+1)/(p-1);
  A[n]:= denom(Q);
od: seq(A[i], i=1..100); #

Mathematica

Denominator[Table[Product[(Prime[i] + 1)/(Prime[i] - 1), {i, n}], {n, 30}]] (* Alonso del Arte, Aug 23 2011 *)

PARI

a(n) = denominator(prod(i=1, n, (prime(i)+1)/(prime(i)-1))); \\ Michel Marcus, May 11 2018

Formula

a(n) = A005867(n)/A078558(n).

Extensions

Improved definition from Franklin T. Adams-Watters, Dec 02 2005