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
KEYWORD
nonn
AUTHOR
J. M. Bergot and Charles R Greathouse IV, Feb 02 2016
STATUS
approved