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A075586 Primes p such that the number of distinct prime divisors of all composite numbers between p and the next prime is 6.

%I A075586 #20 Sep 08 2022 08:45:07
%S A075586 31,47,67,103,109,163,193,277,313,349,379,397,457,463,487,877,1087,
%T A075586 1093,1279,1303,1567,1873,2269,2347,2473,2797,3697,4447,4789,4999,
%U A075586 5077,5413,5503,5923,6007,6217,6469,6997,7603,7639,7723,7933,8779,9277,10159
%N A075586 Primes p such that the number of distinct prime divisors of all composite numbers between p and the next prime is 6.
%C A075586 For very large n, the probability of a(n) not being a twin prime is extremely small, unless the twin primes conjecture is false. - _Sam Alexander_, Oct 20 2003
%H A075586 Amiram Eldar, <a href="/A075586/b075586.txt">Table of n, a(n) for n = 1..10000</a>
%H A075586 XIAO Gang, <a href="http://wims.unice.fr/wims/wims.cgi?session=DV2995E06F.4&amp;+lang=en&amp;+module=tool%2Falgebra%2Ffactor">Factoris, a program that factorizes huge integers</a>
%e A075586 Between 31 and the next prime 37, there are 5 composite numbers whose prime divisors are respectively for 32: {2}, 33: {3,11}, 34: {2,17}, 35: {5,7} and 36: {2,3}; hence, these distinct prime divisors are {2,3,5,7,11,17}, the number of these distinct prime divisors is 6, so 31 is a term. - _Bernard Schott_, Sep 26 2019
%t A075586 Select[Partition[Prime[Range[1250]],2,1],Length[Union[Flatten[ FactorInteger/@ Range[ #[[1]]+1,#[[2]]-1],1][[All,1]]]]==6&][[All,1]] (* _Harvey P. Dale_, May 25 2020 *)
%o A075586 (Magma) a:=[]; for k in PrimesInInterval(2,10000) do b:={}; for s in [k..NextPrime(k)-1] do if not IsPrime(s) then b:=b join Set(PrimeDivisors(s)); end if; end for; if #Set(b) eq 6 then  Append(~a,k); end if; end for; a; // _Marius A. Burtea_, Sep 26 2019
%Y A075586 Cf. A052297, A075581, A075580, A059960, A075583, A075584, A075585, A075587, A075588, A075589.
%K A075586 nonn
%O A075586 1,1
%A A075586 _Amarnath Murthy_, Sep 26 2002
%E A075586 More terms from _Sam Alexander_, Oct 20 2003