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.

A145651 gives the linking prime for the second and third 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 6*13 + 7*17 = 197, the linking prime for 13 and 17, 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==1, a=b)); if(c>1, print1(a, ", ")); n++)}

(MAGMA) [ n*p+(n+1)*q: 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), A145651.

Sequence in context: A138631 A142115 A141941 this_sequence A142527 A142161 A142659

Adjacent sequences: A145647 A145648 A145649 this_sequence A145651 A145652 A145653

nonn

Enoch Haga (Enokh(AT)comcast.net), Oct 15 2008

Edited by Klaus Brockhaus, Dec 10 2008