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A075584 Primes p such that the number of distinct prime divisors of all composite numbers between p and the next prime is 4. 10

%I #42 Sep 08 2022 08:45:07

%S 13,79,419,461,569,659,857,1019,1049,1091,1229,1289,1301,1319,1427,

%T 1481,1721,1931,1949,2129,2141,2339,2549,2789,2969,3119,3299,3329,

%U 3359,3389,3539,3821,3929,4001,4019,4091,4157,4217,4229,4241,4259,4421,4649,4787

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

%C 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

%H Daria Micovic, <a href="/A075584/b075584.txt">Table of n, a(n) for n = 1..10000</a>

%e 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

%t Select[Prime@ Range@ 650, Length@ Union@ Flatten@ Map[First /@ FactorInteger@ # &, Select[Range[#, NextPrime@ #], CompositeQ]] == 4 &] (* _Michael De Vlieger_, May 27 2016 *)

%t 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 *)

%o (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

%o (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)

%o 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

%o (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

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

%K nonn

%O 1,1

%A _Amarnath Murthy_, Sep 26 2002

%E More terms from _Matthew Conroy_, Apr 30 2003

%E Name edited by _Michel Marcus_, May 28 2016

%E Typo in name fixed by _Daria Micovic_, Jun 01 2016

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Last modified March 29 01:36 EDT 2024. Contains 371264 sequences. (Running on oeis4.)