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A075584
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Primes p such that the number of distinct prime divisors of all composite numbers between p and the next prime is 4.
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10
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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
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OFFSET
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1,1
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COMMENTS
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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
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LINKS
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EXAMPLE
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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
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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