A354682 Interprimes that are products of two successive primes.

6, 15, 667, 1517, 9797, 123197, 233273, 522713, 627239, 826277, 974153, 988027, 1127843, 1162003, 1209991, 2624399, 2637367, 3493157, 4235339, 4384811, 4460543, 6827753, 7784099, 10916407, 11370383, 17065157, 25009997, 26347493, 29964667, 32330587, 32387477, 33419957, 34809991, 35354867, 37088099

Offset: 1

Author

J. M. Bergot and Robert Israel, Jun 03 2022

Keywords

• nonn

Comments

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

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.

Examples

			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.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Intersection of A006094 and A024675. Cf. A174955.

Programs

Maple

R:= NULL: count:= 0:
q:= 2:
while count < 50 do
  p:= q; q:= nextprime(q);
  r:= p*q;
  if prevprime(r)+nextprime(r)=2*r then
    R:= R, r; count:= count+1;
  fi
od:
R;

Mathematica

Select[Table[Prime[n]*Prime[n + 1], {n, 1, 800}], Plus @@ NextPrime[#, {-1, 1}] == 2*# &] (* Amiram Eldar, Jun 03 2022 *)
Select[Times@@@Partition[Prime[Range[1000]],2,1],(NextPrime[#]+NextPrime[#,-1])/2==#&] (* Harvey P. Dale, Nov 03 2024 *)