A376261 -id:A376261 - cp's OEIS frontend

A376136 Primes p_1 where products m of k = 5 consecutive primes p_1..p_k are such that only p_1 < m^(1/k).

3229, 3271, 4759, 6173, 6803, 6917, 8389, 8971, 9439, 10433, 11743, 12011, 12853, 12983, 13967, 14107, 14593, 15683, 16033, 16141, 18013, 18097, 19183, 19333, 21283, 21347, 21529, 22573, 22817, 23633, 23719, 25261, 27701, 27919, 28229, 29537, 30593, 31397, 31699

Offset: 1

Michael De Vlieger, Sep 17 2024

nonn

Primes p_1 are such that the difference p_2-p_1 is larger than the sum of the differences p_(j+1)-p_j for j < k.

Does not intersect A022006 or A022007.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A022006, A022007, A376261.

k = 5; 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[Prime[i]] ], {i, 32000}]][[-1, 1]]

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

Michael De Vlieger, Sep 20 2024

Proper subset of A120944, since squarefree m is the smallest number in the sequence of numbers that have m as squarefree kernel, and since more than 1 prime is a factor.

			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.

Cf. A120944, A374873, A375008, A375975, A376261.

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}]

A374873(n) = lpf(a(n)) = A020639(a(n)).

a(n) = Product_{k=0..n-1} k + pi(A374873(n)), where pi = A000720.

cp's OEIS Frontend

A376136 Primes p_1 where products m of k = 5 consecutive primes p_1..p_k are such that only p_1 < m^(1/k).

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