|
|
A075586
|
|
Primes p such that the number of distinct prime divisors of all composite numbers between p and the next prime is 6.
|
|
10
|
|
|
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
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,1
|
|
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
|
|
LINKS
|
|
|
EXAMPLE
|
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
|
|
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 *)
|
|
PROG
|
(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
|
|
CROSSREFS
|
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
|
|
EXTENSIONS
|
|
|
STATUS
|
approved
|
|
|
|