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A372209 Primes p_1 where products m of k = 3 consecutive primes p_1..p_k are such that only p_1 < m^(1/k).

%I A372209 #29 Sep 16 2024 12:52:30
%S A372209 3,7,13,23,31,37,47,53,61,67,73,89,97,103,113,131,139,151,157,167,173,
%T A372209 181,193,199,211,223,233,241,251,257,263,271,277,293,307,317,337,359,
%U A372209 367,373,389,409,421,433,449,457,467,479,491,509,523,547,557,563,577,587
%N A372209 Primes p_1 where products m of k = 3 consecutive primes p_1..p_k are such that only p_1 < m^(1/k).
%C A372209 Primes p such that the second differences of p and the next 2 primes is never positive.
%C A372209 Superset of A022005.
%C A372209 Does not intersect A022004.
%H A372209 Michael De Vlieger, <a href="/A372209/b372209.txt">Table of n, a(n) for n = 1..10000</a>
%e A372209 3 is in the sequence since m = 3*5*7 = 105 is such that 3 is less than the cube root of 105, but both 5 and 7 exceed it.
%e A372209 5 is not in the sequence because m = 5*7*11 = 385 is such that both 5 and 7 are less than the cube root.
%e A372209 7 is in the sequence since m = 7*11*13 = 1001 is such that 7 < 1001^(1/3), but both 11 and 13 are larger than 1001^(1/3), etc.
%t A372209 k = 3; 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, 600}] ][[-1, 1]]
%Y A372209 Cf. A022004, A022005, A375008.
%K A372209 nonn,easy
%O A372209 1,1
%A A372209 _Michael De Vlieger_, Sep 11 2024