A152657 -id:A152657 - cp's OEIS frontend

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

Klaus Brockhaus, Dec 10 2008

nonn

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).

			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.

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

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

[ 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) ];

A152735 Count of links in n-th maximal chain of primes.

Original entry on oeis.org

2, 3, 1, 1, 2, 4, 3, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 1, 1, 2, 1, 2, 4, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 2, 2, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 6, 1, 4, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 2, 2, 2, 2, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 2, 1

Offset: 1

Klaus Brockhaus, Dec 16 2008

nonn

One less than count of members of n-th maximal chain of primes. For definitions see A152658.

			The consecutive primes 5, 7, 11 form the first maximal chain of primes (see example in A152658); it has three members, two links. Therefore a(1) = 2.

Klaus Brockhaus, Table of n, a(n) for n=1..1000

Cf. A152658 (beginnings of maximal chains of primes), A152657 (secluded primes), A119487 (primes of the form i*(i-th prime) + (i+1)*((i+1)-th prime), linking primes).

{n=1; while(n<560, c=0; while(isprime(n*prime(n)+(n+1)*prime(n+1)), c++; n++); if(c>0, print1(c, ",")); n++)}

cp's OEIS Frontend

A152658 Beginnings of maximal chains of primes.

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