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

1,1

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.

Links

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

Example

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.

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 *)

Crossrefs

Intersection of A006094 and A024675. Cf. A174955.

Sequence in context: A145558 A145612 A307812 * A280964 A191463 A235943

Adjacent sequences: A354679 A354680 A354681 * A354683 A354684 A354685

Keyword

nonn

Author

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

Status

approved