A375974 -id:A375974 - cp's OEIS frontend

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

Michael De Vlieger, Sep 19 2024

			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.

Cf. A372209, A375974, A376136.

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}]

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

a(10)-a(12) from Amiram Eldar, Sep 23 2024

A375975 Products m of k = 4 consecutive primes p_1..p_k, where only p_1 < m^(1/k).

Original entry on oeis.org

257557397, 490995677, 1314423991, 2445956099, 8756100193, 14406533983, 34491476237, 168268429891, 453178561051, 526847565721, 588771800473, 673542175381, 874245022517, 1129796633837, 1267153039517, 1385645583389, 1742522070781, 2638237130051, 3021997659211, 3389753359877

Offset: 1

Michael De Vlieger, Sep 12 2024

nonn

In other words, products m of k = 4 consecutive primes p_1..p_k, where floor(log_p_1 m) >= k but floor(log_p_j m) = k-1, j > 1.
a(n) = m is such that floor(log_p_1 m) = k but floor(log_p_j m) = k-1 for j > 1.
Does not intersect A138637, since for m in A138637, both p_1 and p_2 are smaller than m^(1/k).

Michael De Vlieger, Table of n, a(n) for n = 1..65536

Cf. A138637, A375974.

k = 4; s = {1}~Join~Prime[Range[k - 1]]; Reap[Do[s = Append[Rest[s], Prime[i + k - 1]]; r = Surd[Times @@ s, k]; If[Count[s, _?(# < r &)] == 1, Sow[Times @@ s] ], {i, 120}] ][[-1, 1]]

cp's OEIS Frontend

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).

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

PARI

Extensions