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%I #10 Sep 08 2022 08:45:38
%S 43,197,1307,2371,4561,9941,22573,33203,214507,227611,306853,332993,
%T 389167,505907,695059,758441,810023,1072657,1202987,1404211,1567487,
%U 1621621,2407309,2773681,2854331,2932511,3013601,3206773,3851423
%N Linking prime for the first and second member of maximal chains of primes that have at least three members.
%C 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.
%C A145651 gives the linking prime for the second and third member of maximal chains of primes that have at least three members.
%C Suggested by _J. M. Bergot_ in Puzzle 463 of Carlos Rivera's Prime Puzzles & Problems Connection
%H Carlos Rivera, <a href="http://www.primepuzzles.net/puzzles/puzz_463.htm">Puzzle 463</a>
%e 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.
%o (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++)}
%o (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
%Y 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.
%K nonn
%O 1,1
%A _Enoch Haga_, Oct 15 2008
%E Edited by _Klaus Brockhaus_, Dec 10 2008
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