A114383 Length of all-prime chain of prime(n) + successive even triangular numbers.

1, 2, 3, 2, 4, 2, 1, 2, 1, 9, 3, 2, 1, 2, 2, 1, 12, 2, 1, 8, 1, 2, 1, 3, 2, 5, 2, 1, 1, 1, 2, 1, 1, 1, 2, 3, 1, 2, 2, 1, 1, 2, 2, 1, 1, 1, 4, 2, 1, 2, 1, 1, 2, 2, 2, 1, 3, 2, 1, 1, 1, 3, 2, 1, 1, 2, 1, 2, 1, 2, 1, 2, 6, 1, 2, 1, 1, 1, 1, 1, 1, 1, 4, 1, 2, 1

Offset: 2

Author

Jonathan Vos Post, Feb 11 2006

Keywords

• nonn
• easy

Comments

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(18) = 12. The next n for which a(n) >= 10 is a(22509) = 10. What is the next for which a(n) >= 12? Such n > 5761440. - Robert Israel, Jun 14 2016

Examples

			a(2) = 1 because prime(2) = 3 is prime, but prime(2) + EvenTriangular(1) = 3 + 6 = 9 = 3^2 is nonprime, 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.
		

Links

• Robert Israel, Table of n, a(n) for n = 2..10000

Crossrefs

Cf. A000040, A000217, A014601.

Programs

Maple

f:= proc(n) local p,k,j,count;
p:= ithprime(n);
count:= -1;
for k from 0 do
  for j in [0,3] do
     count:= count+1;
     if not isprime (p + 1/2*(4*k+j)*(4*k+j+1)) then return count fi;
od od
end proc:
map(f, [$2..100]); # Robert Israel, Jun 14 2016

Mathematica

evt = Select[(# + 1) #/2 &@Range[200], EvenQ]; a[n_] := Block[{s = Prime@n, c = 1}, While[PrimeQ[s + evt[[c]]], c++]; c]; a /@ Range[2, 90] (* Giovanni Resta, Jun 14 2016 *)

Formula

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.

Extensions

Corrected and extended by Giovanni Resta, Jun 14 2016