cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-2 of 2 results.

A174957 Lesser of twin primes p1 such that p1*p2-4 and p1*p2-6 are twin prime numbers.

Original entry on oeis.org

5, 11, 1031, 2711, 3851, 4421, 5867, 8837, 10067, 12041, 12251, 12611, 17957, 21491, 21521, 22037, 22481, 23537, 32141, 32411, 42641, 48311, 48731, 49367, 50261, 53231, 60167, 72167, 77417, 80147, 80447, 81047, 87641, 88337, 90527, 95231
Offset: 1

Views

Author

Keywords

Comments

5*7=35; 35-4=31; 35-6=29; 29,31 twin primes

Crossrefs

Programs

  • Mathematica
    lst={};Do[p1=Prime[n];p2=p1+2;If[PrimeQ[p2]&&PrimeQ[p1*p2-4]&&PrimeQ[p1*p2-6],(*Print[p1];*)AppendTo[lst,p1]],{n,8!}];lst
    ltp[{a_,b_}]:=b-a==2&&AllTrue[a*b-{4,6},PrimeQ]; Select[Partition[Prime[ Range[ 10000]],2,1],ltp][[All,1]] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Dec 07 2017 *)

A354682 Interprimes that are products of two successive primes.

Original entry on oeis.org

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

Views

Author

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

Keywords

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.
		

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 *)
Showing 1-2 of 2 results.