A152658 Beginnings of maximal chains of primes.

5, 13, 29, 37, 43, 61, 89, 109, 131, 139, 227, 251, 269, 277, 293, 359, 389, 401, 449, 461, 491, 547, 569, 607, 631, 743, 757, 773, 809, 857, 887, 947, 971, 991, 1069, 1109, 1151, 1163, 1187, 1237, 1289, 1301, 1319, 1373, 1427, 1453, 1481, 1499, 1549, 1601

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.

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).

Links

Klaus Brockhaus, Table of n, a(n) for n=1..10000

Example

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.

Prog

(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) ];

Crossrefs

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).

Cf. A105454 - Zak Seidov, Feb 04 2016

Sequence in context: A207040 A309588 A268614 * A347836 A100877 A261580

Adjacent sequences: A152655 A152656 A152657 * A152659 A152660 A152661

Keyword

nonn

Author

Klaus Brockhaus, Dec 10 2008

Status

approved