OFFSET
1,1
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
LINKS
Daria Micovic, Table of n, a(n) for n = 1..10000
EXAMPLE
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
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 *)
PROG
(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
(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
CROSSREFS
KEYWORD
nonn
AUTHOR
Amarnath Murthy, Sep 26 2002
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
STATUS
approved