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

2,2

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

Links

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

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.

Example

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.

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 *)

Crossrefs

Cf. A000040, A000217, A014601.

Sequence in context: A353951 A030011 A286000 * A337442 A205689 A050206

Adjacent sequences: A114380 A114381 A114382 * A114384 A114385 A114386

Keyword

nonn,easy

Author

Jonathan Vos Post, Feb 11 2006

Extensions

Corrected and extended by Giovanni Resta, Jun 14 2016

Status

approved