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A038133 From a subtractive Goldbach conjecture: odd primes that are not cluster primes.

%I A038133 #30 Sep 03 2025 15:13:23
%S A038133 97,127,149,191,211,223,227,229,251,257,263,269,293,307,331,337,347,
%T A038133 349,367,373,379,383,397,409,419,431,457,479,487,499,521,541,547,557,
%U A038133 563,569,587,593,599,631,641,673,691,701,709,719,727,733,739,743,751
%N A038133 From a subtractive Goldbach conjecture: odd primes that are not cluster primes.
%C A038133 Erdős asks if there are infinitely many primes p such that every even number <= p-3 can be expressed as the difference between two primes each <= p. Sequence gives primes not having this property.
%D A038133 R. K. Guy, Unsolved Problems In Number Theory, section C1.
%H A038133 T. D. Noe, <a href="/A038133/b038133.txt">Table of n, a(n) for n = 1..10000</a>
%H A038133 Richard Blecksmith, Paul Erdős and J. L. Selfridge, <a href="http://www.jstor.org/stable/2589585">Cluster Primes</a>, Amer. Math. Monthly, 106 (1999), 43-48.
%H A038133 Thomas Bloom, <a href="https://www.erdosproblems.com/17">Are there infinitely many primes p such that every even number n <= p-3 can be written as a difference of primes n = q_1 - q_2 where q_1, q_2 <= p?</a>, Erdős Problems.
%H A038133 Terence Tao, <a href="https://github.com/teorth/erdosproblems/blob/main/README.md#table">Erdős problem database</a>, see entry no. 17.
%H A038133 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ClusterPrime.html">Cluster Prime</a>.
%H A038133 Wikipedia, <a href="https://en.wikipedia.org/wiki/Cluster_prime">Cluster prime</a>.
%H A038133 <a href="/index/Go#Goldbach">Index entries for sequences related to Goldbach conjecture</a>
%t A038133 m=1000; lst={}; n=PrimePi[m]-1; p=Table[Prime[i+1], {i, n}]; d=Table[0, {m/2}]; For[i=2, i<=n, i++, For[j=1, j<i, j++, diff=p[[i]]-p[[j]]; d[[diff/2]]++ ]; c=Count[Take[d, (p[[i]]-3)/2], 0]; If[c>0, AppendTo[lst, p[[i]]]]]; lst
%Y A038133 Cf. A038134, A039506, A039507, A072325.
%K A038133 nonn,easy,nice,changed
%O A038133 1,1
%A A038133 _N. J. A. Sloane_
%E A038133 More terms from _Christian G. Bower_, Feb 15 1999