1,1

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

Carlos Rivera, Puzzle 463

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.

(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++)}

(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) ]; [From Klaus Brockhaus (klaus-brockhaus(AT)t-online.de), Dec 11 2008]

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.

Sequence in context: A023284 A142025 A142387 this_sequence A141570 A031433 A061525

Adjacent sequences: A145648 A145649 A145650 this_sequence A145652 A145653 A145654

nonn

Enoch Haga (Enokh(AT)comcast.net), Oct 15 200

Edited by Klaus Brockhaus, Dec 10 2008