A073071 Least k such that k! > prime(1)*prime(2)*...*prime(n) where prime(n) is the n-th prime.

2, 3, 4, 5, 6, 7, 8, 10, 11, 12, 14, 15, 16, 17, 19, 20, 21, 23, 24, 25, 27, 28, 29, 31, 32, 33, 34, 36, 37, 38, 40, 41, 42, 44, 45, 46, 47, 49, 50, 51, 53, 54, 55, 57, 58, 59, 60, 62, 63, 64, 66, 67, 68, 70, 71, 72, 73, 75, 76, 77, 79, 80, 81, 83, 84, 85, 86, 88, 89

Offset

0,1

Comments

If the greater than sign in the definition is replaced by >=, we get A048964. - R. J. Mathar, May 01 2008

Links

David Wilson, Table of n, a(n) for n = 0..10000

Formula

a(n) should be asymptotic to C*n (where 1<C<3/2).

Example

n=0: A002110(0) = 1, 2! > 1, a(0) = 2.
n=1: A002110(1) = 2, 3! > 2, a(1) = 3.
n=2: A002110(2) = 6, 4! > 6, a(2) = 4.

Maple

p:= proc(n) option remember; `if`(n<1, 1, p(n-1)*ithprime(n)) end:
a:= proc(n) option remember; local k;
      for k from a(n-1) while k!<=p(n) do od; k
    end: a(-1):=0:
seq(a(n), n=0..68);  # Alois P. Heinz, Mar 10 2026

Mathematica

Module[{nn=100, prmorl, fctorl}, prmolr=FoldList[Times, Prime[Range[ nn]]]; fctorl=Range[nn]!; Table[Position[fctorl, _?(#>prmolr[[n]]&), 1, 1], {n, 70}]]//Flatten (* Harvey P. Dale, Jul 04 2021 *)

Prog

(PARI) a(n) = my(k=1, p=vecprod(primes(n))); while(k! <= p, k++); k; \\ Michel Marcus, Feb 18 2021

Crossrefs

Cf. A002110, A048964, A135667.

Sequence in context: A274374 A007298 A373782 * A190856 A127033 A028826

Adjacent sequences: A073068 A073069 A073070 * A073072 A073073 A073074

Keyword

easy,nonn

Author

Benoit Cloitre, Aug 17 2002

Extensions

Corrected by Peter Pein (petsie(AT)dordos.net), May 01 2008

a(0)=2 prepended by Alois P. Heinz, Mar 10 2026

Status

approved