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 A145650 Linking prime for the first and second member of maximal chains of primes that have at least three members. 1
 43, 197, 1307, 2371, 4561, 9941, 22573, 33203, 214507, 227611, 306853, 332993, 389167, 505907, 695059, 758441, 810023, 1072657, 1202987, 1404211, 1567487, 1621621, 2407309, 2773681, 2854331, 2932511, 3013601, 3206773, 3851423 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 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. 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 LINKS Carlos Rivera, Puzzle 463 EXAMPLE 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. PROG (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) ]; // Klaus Brockhaus, Dec 11 2008 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), A145651. Sequence in context: A141941 A197887 A342509 * A142527 A142161 A193144 Adjacent sequences:  A145647 A145648 A145649 * A145651 A145652 A145653 KEYWORD nonn AUTHOR Enoch Haga, Oct 15 2008 EXTENSIONS Edited by Klaus Brockhaus, Dec 10 2008 STATUS approved

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Last modified April 18 14:03 EDT 2021. Contains 343088 sequences. (Running on oeis4.)