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

1,1

Links

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

Example

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

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 *)

Prog

(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

Crossrefs

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

Sequence in context: A345804 A064667 A217237 * A039483 A104998 A033247

Adjacent sequences: A075585 A075586 A075587 * A075589 A075590 A075591

Keyword

nonn

Author

Amarnath Murthy, Sep 26 2002

Extensions

More terms from Matthew Conroy, Apr 30 2003

Status

approved