A374873 - OEIS

%I #17 Sep 24 2024 09:28:41

%S 2,3,113,3229,15683,736279,8332427,37305713,4948884397,6193302809,

%T 316781230427

%N Smallest primes p_1 where products m of n consecutive primes p_1..p_n are such that only p_1 < m^(1/n).

%e a(2) = 2 since m = 2*3 = 6 and 3 > sqrt(6).

%e a(3) = 3 since m = 3*5*7 = 105 and 5 > 105^(1/3).

%e a(4) = 113 since m = 113 * 127 * 131 * 137 = 257557397 and 127 > 257557397^(1/4), etc.

%t k = 1; Table[r = Range[0, n - 1]; While[(Set[{p, q, m}, {#[[1]], #[[2]], Times @@ #}]; q < Surd[m, n]) &[Prime[k + r]], k++]; p, {n, 2, 6}]

%o (PARI) a(n) = {my(ps = vector(n, k, prime(k))); forprime(p = prime(n+1), , if(ps[2]^n > vecprod(ps), return(ps[1])); ps = concat(vecextract(ps, "^1"), p));} \\ _Amiram Eldar_, Sep 23 2024

%Y Cf. A372209, A375974, A376136.

%K nonn,hard,more

%O 2,1

%A _Michael De Vlieger_, Sep 19 2024

%E a(10)-a(12) from _Amiram Eldar_, Sep 23 2024