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

31, 47, 67, 103, 109, 163, 193, 277, 313, 349, 379, 397, 457, 463, 487, 877, 1087, 1093, 1279, 1303, 1567, 1873, 2269, 2347, 2473, 2797, 3697, 4447, 4789, 4999, 5077, 5413, 5503, 5923, 6007, 6217, 6469, 6997, 7603, 7639, 7723, 7933, 8779, 9277, 10159

Offset: 1

Author

Amarnath Murthy, Sep 26 2002

Keywords

• nonn

Comments

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

Examples

			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
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• XIAO Gang, Factoris, a program that factorizes huge integers

Crossrefs

Cf. A052297, A075581, A075580, A059960, A075583, A075584, A075585, A075587, A075588, A075589.

Programs

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

Mathematica

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 *)

Extensions

More terms from Sam Alexander, Oct 20 2003