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Numbers x such that 1 + 3*x*(x-1) is a ("cuban") prime (cf. A002407).
(Formerly M0522 N0188)
8

%I M0522 N0188 #63 Feb 22 2024 13:34:13

%S 2,3,4,5,7,10,11,12,14,15,18,24,25,26,28,29,31,33,35,38,39,42,43,46,

%T 49,50,53,56,59,63,64,67,68,75,81,82,87,89,91,92,94,96,106,109,120,

%U 124,126,129,130,137,141,143,148,154,157,158,159,165,166,171,172

%N Numbers x such that 1 + 3*x*(x-1) is a ("cuban") prime (cf. A002407).

%C Equivalently, positive integers x such that x^3 - (x-1)^3 is prime. - _Rémi Guillaume_, Oct 24 2023

%D A. J. C. Cunningham, On quasi-Mersennian numbers, Mess. Math., 41 (1912), 119-146.

%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H T. D. Noe, <a href="/A002504/b002504.txt">Table of n, a(n) for n = 1..1000</a>

%F From _Rémi Guillaume_, Dec 07 2023: (Start)

%F a(n) = ceiling(sqrt(A002407(n)/3)).

%F a(n) = A111251(n) + 1.

%F a(n) = (A121259(n) + 1)/2. (End)

%e From _Rémi Guillaume_, Dec 07 2023: (Start)

%e 1 + 3*7*6 = 127 = A002407(5) is the 5th prime of this form, so a(5) = 7.

%e 1 + 3*10*9 = 271 = A002407(6) is the 6th prime of this form, so a(6) = 10.

%e (End)

%t Select[Range[500], PrimeQ[1 + 3 # (# - 1)] &] (* _T. D. Noe_, Jan 30 2013 *)

%o (PARI) for(k=1,999,isprime(3*k*(k-1)+1)&print1(k",")) \\ _M. F. Hasler_, Nov 28 2007

%Y Cf. A002407 (resulting primes), A111251, A121259.

%K nonn

%O 1,1

%A _N. J. A. Sloane_

%E Edited, updated (1 is no longer regarded as a prime) and extended by _M. F. Hasler_, Nov 28 2007