A376331
a(n) is the smallest product of n consecutive primes p_1..p_k, where only p_1 < m^(1/n).
Original entry on oeis.org
6, 105, 257557397, 362469273063260281, 15119658537284521518782249, 117383204057701408834470517376101793436427, 23238824136447515117641387686174787861885627837847997511, 139957288120766060385660710153537529132218663535147563443966068820553
Offset: 2
a(2) = 6 since m = 2*3 = 6 and 3 > sqrt(6).
a(3) = 105 since m = 3*5*7 = 105 and 5 > 105^(1/3).
a(4) = 257557397 since m = 113 * 127 * 131 * 137 = 257557397 and 127 > 257557397^(1/4), etc.
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k = 1; Table[r = Range[0, n - 1]; While[(Set[{p, q, m}, {#[[1]], #[[2]], Times @@ #}]; q < Surd[m, n]) &[Prime[k + r]], k++]; m, {n, 2, 6}]
A376440
Smallest primes p_1 where products m of n consecutive primes p_1..p_n are such that only p_n > m^(1/n).
Original entry on oeis.org
2, 2, 107, 1657, 25453, 404819, 1388449, 137414987, 402301129, 87241770523
Offset: 2
a(2) = 2 since m = 2*3 = 6 and 2 < sqrt(6).
a(3) = 2 since m = 2*3*5 = 30 and 3 < 30^(1/3).
a(4) = 107 since m = 107 * 109 * 113 * 127 = 167375713 and 113 < 167375713^(1/4), etc.
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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}]
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a(n) = {my(ps = vector(n, k, prime(k))); forprime(p = prime(n+1), , if(ps[#ps-1]^n < vecprod(ps), return(ps[1])); ps = concat(vecextract(ps, "^1"), p));} \\ Amiram Eldar, Sep 23 2024
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