A075584 Primes p such that the number of distinct prime divisors of all composite numbers between p and the next prime is 4.

13, 79, 419, 461, 569, 659, 857, 1019, 1049, 1091, 1229, 1289, 1301, 1319, 1427, 1481, 1721, 1931, 1949, 2129, 2141, 2339, 2549, 2789, 2969, 3119, 3299, 3329, 3359, 3389, 3539, 3821, 3929, 4001, 4019, 4091, 4157, 4217, 4229, 4241, 4259, 4421, 4649, 4787

Offset: 1

Author

Amarnath Murthy, Sep 26 2002

Keywords

• nonn

Comments

It seems that for n > 2, a(n) + 2 is prime. Any counterexample p must have p > 3^1000000 and p+4 prime, and {p+1, p+2, p+3} must contain a power of 2 or 3. (The case where p+1 and p+3 are 3-smooth case can be ruled out via Catalan's conjecture/Mihăilescu's theorem.) In particular known Mersenne factorizations rule out the Fermat case below 2^144115188075855872 - 3, GIMPS rules out the Mersenne case below 2^36046457 - 1, and the exponents in A014224 rule out the remaining case below 3^1000000 - 2. - Charles R Greathouse IV, Jun 01 2016

Examples

			For p = 79, the next prime number is 83. The numbers between 79 and 83 and the prime divisors are respectively  80 { 2, 5 }, 81 { 3 }, 82 { 2, 41 }. The set of prime divisors is { 2, 3, 5, 41 } and has 4 elements, so 79 is a term. - _Marius A. Burtea_, Sep 26 2019
		

Links

• Daria Micovic, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A052297, A075581, A075580, A059960, A075583, A075585, A075586, A075587, A075588, A075589.

Programs

Magma

a:=[]; for p in PrimesInInterval(2,4800) 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 4 then Append(~a,p); end if; end for; a; // Marius A. Burtea, Sep 26 2019

Mathematica

Select[Prime@ Range@ 650, Length@ Union@ Flatten@ Map[First /@ FactorInteger@ # &, Select[Range[#, NextPrime@ #], CompositeQ]] == 4 &] (* Michael De Vlieger, May 27 2016 *)
Join[{13,79},Select[Prime[Range[23,650]],PrimeQ[#+2]&&PrimeNu[#+1]==4&]] (* This program assumes the correctness of the conjecture by Charles R. Greathouse, IV, in the Comments. *) (* Harvey P. Dale, Jun 07 2019 *)

PARI

lista(nn)=forprime(p=2, nn, allp = []; forcomposite (c = p+1, nextprime(p+1), allp = Set(concat(allp, (factor(c)[,1])~));); if (#allp == 4, print1(p, ", "));); \\ Michel Marcus, May 28 2016

PARI

is(n)=if(!isprime(n), return(0)); if(isprime(n+2), return(omega(n+1)==4)); if(isprime(n+4), omega(n+1)+omega(n+2)+omega(n+3)==5, 0)
list(lim)=my(v=List(),t,p); lim\=1; for(e=4,logint(lim+2,3), p=precprime(3^e); if(isprime(p+4) && is(p), listput(v,p))); for(e=4,logint(lim+3,2), p=precprime(2^e); if(isprime(p+4) && is(p), listput(v,p))); p=2; forprime(q=3,lim+2, if(q-p==2 && omega(p+1)==4, listput(v,p)); p=q); Set(v) \\ Charles R Greathouse IV, Jun 01 2016

Extensions

More terms from Matthew Conroy, Apr 30 2003

Name edited by Michel Marcus, May 28 2016

Typo in name fixed by Daria Micovic, Jun 01 2016