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%I #23 Jul 25 2020 12:52:54
%S 7,13,79,97,223,349,673,1087,1213,1663,3697,13309,13687,16927,20479,
%T 21139,25999,32797,33613,78649,122449,151549,263167,401407,651247,
%U 1058749,1656247,1893373,2060449,2146687,3058873,3276799,3733207,3866623,3880897,4070197
%N 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.
%C Are there any consecutive primes p and q for which rad((p+1)(p+2)...(q-1)) < pq with q - p > 4?
%H Charles R Greathouse IV, <a href="/A268350/b268350.txt">Table of n, a(n) for n = 1..299</a>
%e 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.
%p rad:= n -> convert(numtheory:-factorset(n),`*`):
%p 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
%t 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 *)
%o (PARI) rad(n)=factorback(factor(n)[,1])
%o 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
%o p=2;forprime(q=3,1e10,if(has(p,q), print1(p", "));p=q)
%Y Subsequence of A029710.
%Y Cf. A078637, A007947.
%K nonn
%O 1,1
%A _J. M. Bergot_ and _Charles R Greathouse IV_, Feb 02 2016
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