A268350 Primes p where q = p + 4 is also prime and rad((p+1)(p+2)(p+3)) < pq, where rad(k) is the largest squarefree number dividing k.

7, 13, 79, 97, 223, 349, 673, 1087, 1213, 1663, 3697, 13309, 13687, 16927, 20479, 21139, 25999, 32797, 33613, 78649, 122449, 151549, 263167, 401407, 651247, 1058749, 1656247, 1893373, 2060449, 2146687, 3058873, 3276799, 3733207, 3866623, 3880897, 4070197

Offset: 1

Author

J. M. Bergot and Charles R Greathouse IV, Feb 02 2016

Keywords

• nonn

Comments

Are there any consecutive primes p and q for which rad((p+1)(p+2)...(q-1)) < pq with q - p > 4?

Examples

			79 and 83 are prime, and rad(80*81*82) = rad(2^5*3^4*5*41) = 2*3*5*41 = 1230 < 6557 = 79*83, so 79 is a member of this sequence.
		

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 1..299

Crossrefs

Subsequence of A029710.

Cf. A078637, A007947.

Programs

Maple

rad:= n -> convert(numtheory:-factorset(n),`*`):
select(p -> isprime(p) and isprime(p+4) and rad((p+1)*(p+2)*(p+3)) < p*(p+4), [seq(i,i=7..10^7,6)]); # Robert Israel, Feb 05 2016

Mathematica

p4Q[n_]:=PrimeQ[n+4]&&Select[Divisors[Times@@(n+{1,2,3})],SquareFreeQ][[-1]]<(n(n+4)); Select[Prime[Range[300000]],p4Q] (* Harvey P. Dale, Jul 25 2020 *)

PARI

rad(n)=factorback(factor(n)[,1])
has(p,q)=if(q-p!=4, return(0)); my(t=rad((p+1)/2)*rad((p+3)/2),pq=p*q); 3*t