A354682 - OEIS

%I #12 Nov 03 2024 10:40:37

%S 6,15,667,1517,9797,123197,233273,522713,627239,826277,974153,988027,

%T 1127843,1162003,1209991,2624399,2637367,3493157,4235339,4384811,

%U 4460543,6827753,7784099,10916407,11370383,17065157,25009997,26347493,29964667,32330587,32387477,33419957,34809991,35354867,37088099

%N Interprimes that are products of two successive primes.

%C Numbers that are both the average of two successive primes and the product of two successive primes.

%C Includes p*(p+2) where p, p+2,p^2+2*p-6 and p^2+2*p+6 are all primes but p^2+2*p-2, p^2+2*p-4, p^2+2*p+2 and p^2+2*p+4 are composite. The Generalized Bunyakovsky Conjecture implies there are infinitely many such terms.

%H Robert Israel, <a href="/A354682/b354682.txt">Table of n, a(n) for n = 1..10000</a>

%e a(3) = 667 is a term because 667 = (661 + 673)/2 = 23*29 where 661 and 673 are successive primes and 23 and 29 are successive primes.

%p R:= NULL: count:= 0:

%p q:= 2:

%p while count < 50 do

%p   p:= q; q:= nextprime(q);

%p   r:= p*q;

%p   if prevprime(r)+nextprime(r)=2*r then

%p     R:= R, r; count:= count+1;

%p   fi

%p od:

%p R;

%t Select[Table[Prime[n]*Prime[n + 1], {n, 1, 800}], Plus @@ NextPrime[#, {-1, 1}] == 2*# &] (* _Amiram Eldar_, Jun 03 2022 *)

%t Select[Times@@@Partition[Prime[Range[1000]],2,1],(NextPrime[#]+NextPrime[#,-1])/2==#&] (* _Harvey P. Dale_, Nov 03 2024 *)

%Y Intersection of A006094 and A024675. Cf. A174955.

%K nonn

%O 1,1

%A _J. M. Bergot_ and _Robert Israel_, Jun 03 2022