A005580 a(n) = smallest number k such that Product_{i=2..k+1} prime(i)/(prime(i)-1) > n.

0, 1, 3, 8, 21, 54, 141, 372, 995, 2697, 7397, 20502, 57347, 161658, 458788, 1309626, 3757383, 10828011, 31326513, 90945528

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• more

Comments

Laatsch (1986) proved that a(n) gives the smallest number of distinct prime factors in odd numbers having an abundancy index > n.

The abundancy index of a number k is sigma(k)/k. - T. D. Noe, May 08 2006

References

• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• R. K. Guy, Letter to N. J. A. Sloane, 1988-04-12 (annotated scanned copy)
• Richard Laatsch, Measuring the abundancy of integers, Mathematics Magazine 59 (2) (1986) 84-92.

Crossrefs

Cf. A005579.

Programs

Mathematica

prod=1; k=1; Table[While[prod<=n, k++; prod=prod*Prime[k]/(Prime[k]-1)]; k-1, {n,0,12}] (* T. D. Noe, May 08 2006 *)

Formula

For n > 0, a(n) = A005579(2n)-1. - T. D. Noe, May 08 2006

Extensions

Edited by T. D. Noe, May 08 2006

a(14)-a(19) from the data at A005579 added by Amiram Eldar, Mar 21 2019

Edited and a(0)-a(1) prepended by Max Alekseyev, Jan 25 2025