A374873 Smallest primes p_1 where products m of n consecutive primes p_1..p_n are such that only p_1 < m^(1/n).
2, 3, 113, 3229, 15683, 736279, 8332427, 37305713, 4948884397, 6193302809, 316781230427
Offset: 2
Examples
a(2) = 2 since m = 2*3 = 6 and 3 > sqrt(6). a(3) = 3 since m = 3*5*7 = 105 and 5 > 105^(1/3). a(4) = 113 since m = 113 * 127 * 131 * 137 = 257557397 and 127 > 257557397^(1/4), etc.
Programs
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Mathematica
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}] -
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
Extensions
a(10)-a(12) from Amiram Eldar, Sep 23 2024