A056127 Minimum m where product_{k=1 to m}[p_k] > (p_{m+1})^n, where p_k is k-th prime.

1, 2, 4, 5, 7, 8, 10, 11, 13, 14, 15, 16, 18, 19, 20, 21, 23, 24, 25, 26, 28, 29, 30, 32, 33, 34, 35, 36, 38, 39, 40, 41, 43, 44, 45, 46, 48, 49, 50, 51, 52, 53, 55, 56, 57, 58, 59, 60, 62, 63, 64, 65, 67, 68, 69, 70, 72, 73, 74, 75, 76, 78, 79, 80, 81, 82, 83, 85, 86, 87, 88

Offset: 0

Author

Leroy Quet, Aug 30 2000

Keywords

• nonn

Examples

			a(2) = 4, since 2*3*5 < 7^2, but 2*3*5*7 > 11^2. (The product of the first 4 primes is greater than the 5th prime squared.)
		

Links

• Eric M. Schmidt, Table of n, a(n) for n = 0..10000
• Safia Aoudjit and Djamel Berkane, Explicit Estimates Involving the Primorial Integers and Applications, J. Int. Seq., Vol. 24 (2021), Article 21.7.8.

Crossrefs

Cf. A002110, A060797, A127600, A127601, A127602, A127603, A127604.

Programs

Mathematica

a = {}; Do[x = 1; While[Prime[x + 1] >= (Product[Prime[x], {x, 1, x}])^(1/n), x++ ]; AppendTo[a, x], {n, 1, 100}]; a (* Artur Jasinski, May 11 2007 *)