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

73, 83, 131, 167, 173, 251, 331, 383, 443, 563, 643, 739, 971, 1123, 1223, 1367, 1579, 1609, 1783, 1867, 1999, 2293, 2539, 2617, 2683, 3083, 3217, 3253, 3343, 3457, 3847, 4003, 4513, 4783, 4813, 4969, 5167, 5233, 5527, 5737, 5779, 5839, 5857, 6199, 6733

Offset: 1

Author

Amarnath Murthy, Sep 26 2002

Keywords

• nonn

Examples

			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
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A052297, A075581, A075580, A059960, A075583, A075584, A075585, A075586, A075587, A075589.

Programs

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

Mathematica

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 *)

Extensions

More terms from Matthew Conroy, Apr 30 2003