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

1,1

Comments

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

Links

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

Example

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.

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 *)

Prog

(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<pq && rad(p+2)*t<pq
p=2; forprime(q=3, 1e10, if(has(p, q), print1(p", ")); p=q)

Crossrefs

Subsequence of A029710.

Cf. A078637, A007947.

Sequence in context: A174878 A177198 A177163 * A061521 A362246 A073220

Adjacent sequences: A268347 A268348 A268349 * A268351 A268352 A268353

Keyword

nonn

Author

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

Status

approved