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A075588 Primes p such that the number of distinct prime divisors of all composite numbers between p and the next prime is 8.

%I A075588 #25 Sep 08 2022 08:45:07
%S A075588 73,83,131,167,173,251,331,383,443,563,643,739,971,1123,1223,1367,
%T A075588 1579,1609,1783,1867,1999,2293,2539,2617,2683,3083,3217,3253,3343,
%U A075588 3457,3847,4003,4513,4783,4813,4969,5167,5233,5527,5737,5779,5839,5857,6199,6733
%N A075588 Primes p such that the number of distinct prime divisors of all composite numbers between p and the next prime is 8.
%H A075588 Amiram Eldar, <a href="/A075588/b075588.txt">Table of n, a(n) for n = 1..10000</a>
%e A075588 For p = 131, the next prime number is 137. The numbers between 131 and 137 and the prime factors are respectively 132 { 2, 3, 11 }, 133 { 7, 19 }, 134 { 2, 67 }, 135 { 3, 5 }, 136 { 2, 17 }. The set of prime divisors is { 2, 3, 5, 7, 11, 17, 19, 67 } and has 8 elements, so 131 is a term. - _Marius A. Burtea_, Sep 26 2019
%t A075588 Select[Partition[Prime[Range[1000]],2,1],Length[Union[ Flatten[ FactorInteger[ Range[ #[[1]]+1,#[[2]]-1]],1][[All,1]]]]==8&][[All,1]] (* _Harvey P. Dale_, Dec 26 2019 *)
%o A075588 (Magma) a:=[]; for p in PrimesInInterval(2,7000) do b:={}; for s in [p..NextPrime(p)-1] do if not IsPrime(s) then b:=b join Set(PrimeDivisors(s)); end if; end for; if #Set(b) eq 8 then Append(~a,p); end if; end for; a; // _Marius A. Burtea_, Sep 26 2019
%Y A075588 Cf. A052297, A075581, A075580, A059960, A075583, A075584, A075585, A075586, A075587, A075589.
%K A075588 nonn
%O A075588 1,1
%A A075588 _Amarnath Murthy_, Sep 26 2002
%E A075588 More terms from _Matthew Conroy_, Apr 30 2003