%I #12 Sep 08 2022 08:45:39

%S 5,13,29,37,43,61,89,109,131,139,227,251,269,277,293,359,389,401,449,

%T 461,491,547,569,607,631,743,757,773,809,857,887,947,971,991,1069,

%U 1109,1151,1163,1187,1237,1289,1301,1319,1373,1427,1453,1481,1499,1549,1601

%N Beginnings of maximal chains of primes.

%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 A chain of primes has two or more members; a prime is called secluded if it is not member of a chain of primes (cf. A152657).

%H Klaus Brockhaus, <a href="/A152658/b152658.txt">Table of n, a(n) for n=1..10000</a>

%e 3*prime(3) + 4*prime(4) = 3*5 + 4*7 = 43 is prime and 4*prime(4) + 5*prime(5) = 4*7 + 5*11 = 83 is prime, so 5, 7, 11 is a chain of primes. 2*prime(2) + 3*prime(3) = 2*3 + 3*5 = 21 is not prime and 5*prime(5) + 6*prime(6) = 5*11 + 6*13 = 133 is not prime, hence 5, 7, 11 is maximal and prime(3) = 5 is the beginning of a maximal chain.

%o (Magma) [ p: n in [1..253] | (n eq 1 or not IsPrime((n-1)*PreviousPrime(p) +n*p) ) and IsPrime((n)*p+(n+1)*NextPrime(p)) where p is NthPrime(n) ];

%Y Cf. A152117 (n*(n-th prime) + (n+1)*((n+1)-th prime)), A152657 (secluded primes), A119487 (primes of the form i*(i-th prime) + (i+1)*((i+1)-th prime), linking primes).

%Y Cf. A105454 - _Zak Seidov_, Feb 04 2016

%K nonn

%O 1,1

%A _Klaus Brockhaus_, Dec 10 2008