OFFSET
1,1
COMMENTS
(2k^3 - 2k^2 + k + 2) / 2 is an integer if and only if k is even.
If (2k^3 - 2k^2 + k + 2) / 2 is prime, k is divisible by 12.
Let S(c0, c1, c2, c3, d) be the set of all prime numbers of the form m = (c0 + c1*k + c2*k^2 + c3*k^3) / d, where c0, c1, c2, and c3, are all integers and d is a positive integer. There are 5^4 * 2 = 1250 sets of these five parameters in which no parameter's absolute value is greater than 2. Of these 1250 sets, there are
. 902 that yield one or more negative values of m,
. 29 others that yield no integer values of m, and
. 69 others that yield no prime values of m, i.e.,
. ----
. 1000 in total that do not strictly yield nonnegative values of m of which at least one is a prime. Of the remaining 250 sets, 98 yield a prime value of m (which never exceeds 17) only at a single value of k, and 12 yield prime values of m (not necessarily distinct) at only two values of k. Additionally, one set, S(2, 0, 0, 0, 1), yields m = 2 for every value of k. Aside from the above, it appears that each of the 139 remaining sets produces an infinite number of primes, the smallest of which, for all but six sets, is a 1- or 2-digit prime. The only set for which the smallest prime is larger than 601 is S(2, 1, -2, 2, 2), which generates this sequence of primes, the smallest of which is a(1) = 585691, making this set an extreme outlier, in terms of the smallest prime it generates, among the sets of parameter values described above.
LINKS
Chai Wah Wu, Table of n, a(n) for n = 1..2598
EXAMPLE
k=84: (2*84^3 - 2*84^2 + 84 + 2)/2 = 585691.
MATHEMATICA
Select[Table[(2 n^3 - 2 n^2 + n + 2) / 2, {n, 0, 1500}], PrimeQ] (* Vincenzo Librandi, Nov 10 2014 *)
Select[Table[(2 n^3-2 n^2+n+2)/2, {n, 0, 1500, 12}], PrimeQ] (* Harvey P. Dale, Jun 12 2022 *)
PROG
(PARI) forstep(k=0, 10^3, 2, my(t=(2*k^3-2*k^2+k+2)/2); if(isprime(t), print1(t, ", "))); \\ Joerg Arndt, Nov 02 2014
(Python)
from sympy import isprime
A249610_list, m = [], [48, -56, 13, 1]
for _ in range(10**5):
....for i in range(3):
........m[i+1]+= m[i]
....if isprime(m[-1]):
........A249610_list.append(m[-1]) # Chai Wah Wu, Nov 09 2014
(Magma) [a: n in [0..2500 by 2] | IsPrime(a) where a is (2*n^3 - 2*n^2 + n + 2) div 2 ]; // Vincenzo Librandi, Nov 10 2014
CROSSREFS
KEYWORD
nonn
AUTHOR
Jon E. Schoenfield, Nov 02 2014
STATUS
approved