A145651 - OEIS

%I #11 Oct 03 2024 05:41:01

%S 83,271,1553,2693,5051,10651,23333,34123,219389,230933,312007,338017,

%T 395309,512891,699437,763999,815257,1078127,1208791,1417019,1577561,

%U 1629083,2420609,2787947,2868787,2944429,3038639,3222101,3868201

%N Linking prime for the second and third 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 A145650 gives the linking prime for the first and second 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 7*17 + 8*19 = 271, the linking prime for 17 and 19, 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==2, a=b)); if(c>1, print1(a, ",")); n++)}

%o (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

%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), A145650.

%K nonn

%O 1,1

%A _Enoch Haga_, Oct 15 2008

%E Edited by _Klaus Brockhaus_, Dec 10 2008