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A056127 Minimum m where product_{k=1 to m}[p_k] > (p_{m+1})^n, where p_k is k-th prime. 2

%I #22 Dec 30 2021 00:24:47

%S 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,

%T 34,35,36,38,39,40,41,43,44,45,46,48,49,50,51,52,53,55,56,57,58,59,60,

%U 62,63,64,65,67,68,69,70,72,73,74,75,76,78,79,80,81,82,83,85,86,87,88

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

%H Eric M. Schmidt, <a href="/A056127/b056127.txt">Table of n, a(n) for n = 0..10000</a>

%H Safia Aoudjit and Djamel Berkane, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL24/Berkane/berkane8.html">Explicit Estimates Involving the Primorial Integers and Applications</a>, J. Int. Seq., Vol. 24 (2021), Article 21.7.8.

%e 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.)

%t 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 *)

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

%K nonn

%O 0,2

%A _Leroy Quet_, Aug 30 2000

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Last modified February 21 08:45 EST 2024. Contains 370228 sequences. (Running on oeis4.)