A145651 Linking prime for the second and third member of maximal chains of primes that have at least three members.

83, 271, 1553, 2693, 5051, 10651, 23333, 34123, 219389, 230933, 312007, 338017, 395309, 512891, 699437, 763999, 815257, 1078127, 1208791, 1417019, 1577561, 1629083, 2420609, 2787947, 2868787, 2944429, 3038639, 3222101, 3868201

Offset: 1

Author

Enoch Haga, Oct 15 2008

Keywords

• nonn

Comments

A sequence of consecutive primes prime(k), ..., prime(k+r), r >= 1, is called a chain of primes if i*prime(i) + (i+1)*prime(i+1)* is prime (the linking prime for prime(i) and prime(i+1), cf. A119487) for i from k to k+r-1. A chain of primes prime(k), ..., prime(k+r) is maximal if it is not part of a longer chain, i.e., if neither (k-1)*prime(k-1) + k*prime(k) nor (k+r)*prime(k+r) + (k+r+1)*prime(k+r+1) is prime.

A145650 gives the linking prime for the first and second member of maximal chains of primes that have at least three members.

Suggested by J. M. Bergot in Puzzle 463 of Carlos Rivera's Prime Puzzles & Problems Connection

Examples

			Primes 13, 17, 19, 23 have prime indices 6, 7, 8, 9. 6*13 + 7*17 = 197 is prime; 7*17 + 8*19 = 271 is prime; 8*19 + 9*23 = 359 is prime. Neither 5*11 + 6*13 = 133 nor 9*23 + 10*29 = 497 is prime, so 13, 17, 19, 23 is maximal. Hence 7*17 + 8*19 = 271, the linking prime for 17 and 19, is in the sequence.
		

Links

• Carlos Rivera, Puzzle 463

Crossrefs

Cf. A152117 (n*(n-th prime) + (n+1)*((n+1)-th prime)), A119487 (primes in A152117, linking primes), A152658 (beginnings of maximal chains of primes), A145650.

Programs

Magma

[ (n+1)*q+(n+2)*r: n in [1..520] | (n eq 1 or not IsPrime((n-1)*PreviousPrime(p)+n*p) ) and IsPrime(n*p+(n+1)*q) and IsPrime((n+1)*q+(n+2)*r) where r is NextPrime(q) where q is NextPrime(p) where p is NthPrime(n) ]; // Klaus Brockhaus, Dec 11 2008

PARI

{n=1; while(n<520, c=0; while(isprime(b=n*prime(n)+(n+1)*prime(n+1)), c++; n++; if(c==2, a=b)); if(c>1, print1(a, ",")); n++)}

Extensions

Edited by Klaus Brockhaus, Dec 10 2008