2,2

a(1) is undefined, as prime(1) is the only even prime, for which the length-5 chain is of 2 + successive odd triangular numbers A014493: 2 prime, 2+1 = 3 prime, 2+3 = 5 prime, 2+15 = 17 prime, 2+21 = 23 prime, 2+45 = 47 prime, and then 2+55 = 57 = 3*19 nonprime.

a(n) = k = length of chain prime[n] + A014493(1) + ... + A014493(k) such that each term in the chain is prime. a(n) = k = length of chain A000040(n) + A000217(A014601(1)) + ... + A000217(A014601(k)) such that each term in the chain is in A000040.

a(2) = 1 because prime(2) = 3 is prime, but prime(2) + EvenTriangular(1) = 3 + 6 = 9 = 3^2 is noprime, giving a chain of just 1 successive prime.

a(3) = 2 because prime(3) + EvenTriangular(1) = 5 + 6 = 11 is prime, but prime(3) + EvenTriangular(2) = 5 + 10 = 15 = 3*5 is nonprime, giving a chain of 2 successive primes.

a(4) = 3 because 7 is prime, 7+6 = 13 is prime, 7+10 = 17 is prime, but 7+28 = 35 = 5*7 is nonprime, for a chain of 3 successive primes.

a(6) = 4 because 13 is prime, 13+6 = 19 is prime, 13+10 = 23 is prime, 13+28 = 41 is prime, but 13+36 = 49 = 7^2 is nonprime.

a(11) = 9 because 31 is prime, as is 31+6 = 37; 31+10 = 41; 31+28 = 59; 31+36 = 67; 31+66 = 97; 31+78 = 109; 31+120 = 151; 31+136 = 167; but 31+190 = 221 = 13*17 is nonprime.

a(18) = 11 because of the prime chain 61; 61+6 = 67; 61+10 = 71; 61+28 = 89; 61+36 = 97; 61+66 = 127; 61+78 = 139; 61+120 = 181; 61+136= 197; 61+190 = 251; 61+276 = 337; but 61+300 = 361 = 19^2 is nonprime.

Cf. A000040, A000217, A014601.

Sequence in context: A143422 A071438 A030011 this_sequence A050206 A127237 A066482

Adjacent sequences: A114380 A114381 A114382 this_sequence A114384 A114385 A114386

easy,nonn

Jonathan Vos Post (jvospost3(AT)gmail.com), Feb 11 2006