A114409 Length of all-prime chain of prime[n] + successive even pentagonal numbers.

1, 2, 3, 2, 1, 2, 4, 1, 2, 4, 1, 2, 1, 2, 1, 2, 4, 4, 2, 1, 1, 1, 2, 2, 2, 1, 1, 1, 1, 4, 1, 2, 2, 1, 3, 1, 1, 2, 1, 2, 2, 1, 1, 1, 2, 4, 1, 2, 3, 1, 2, 1, 2, 2, 1, 2, 3, 1, 2, 1, 1, 1, 1, 1, 1, 1, 3, 2, 1, 1, 1, 3, 1, 1, 1, 2, 4, 1, 4, 2, 4, 2, 1, 1, 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 pentagonal numbers A014632: 2 prime, 2+1 = 3 prime, 2+5 = 7 prime, 2+35 = 37 prime, 2+51 = 53 prime, but 2+117 = 119 = 7 * 17 nonprime. Pentagonal numbers A000326 = n*(3*n-1)/2 = 0, 1, 5, 12, 22, 35, 51, 70, 92, 117, 145, ... Even pentagonal numbers A014633 = 12, 22, 70, 92, 176, 210, 330, 376, 532, 590, 782, 852, 1080, ...

Links

Table of n, a(n) for n=2..88.

Formula

a(n) = k = length of chain prime[n] + A014633(1) + ... + A014633(k) such that each term in the chain is prime.

Example

a(2) = 1 because prime(2) = 3 is prime, but prime(2) + EvenPent(1) = 3 + 12 = 15 = 3 * 5 is nonprime, giving a chain of just 1 successive prime.
a(3) = 2 because 5 is prime, prime(3) + EvenPent(1) = 5 + 12 = 17 is prime, but prime(3) + EvenPent(2) = 5 + 22 = 27 = 3^3 is nonprime, giving a chain of 2 successive primes.
a(4) = 3 because 7 is prime, 7+12 = 19 is prime, 7+22 = 29 is prime, but 7+70 = 77 = 7*11 is nonprime, for a chain of 3 successive primes.

Mathematica

evp = Select[#*(3*# - 1)/2 &@ Range[200], EvenQ]; a[n_] := Block[{s = Prime@n, c = 1}, While[PrimeQ[s + evp[[c]]], c++]; c]; a /@ Range[2, 90] (* Giovanni Resta, Jun 14 2016 *)

Crossrefs

Cf. A000040, A000326, A014633.

Sequence in context: A105969 A275892 A163530 * A340680 A343897 A193585

Adjacent sequences: A114406 A114407 A114408 * A114410 A114411 A114412

Keyword

easy,nonn

Author

Jonathan Vos Post, Feb 11 2006

Extensions

Corrected and extended by Giovanni Resta, Jun 14 2016

Status

approved