A342706 Primes of the form (p^2 - p*q + q^2)/3, where p and q are consecutive primes.
13, 31, 79, 109, 151, 1201, 3271, 3469, 3889, 4111, 12289, 16879, 17791, 25951, 27673, 108301, 126079, 134857, 138679, 169957, 174259, 186019, 231877, 245389, 259309, 355009, 367501, 371737, 397489, 412939, 461017, 477619, 524197, 544429, 565069, 602401, 741031, 833191, 904303, 961069, 1267501
Offset: 1
Keywords
Examples
For n = 5, p = 19 and q = 23 are consecutive primes and a(5) = (19^2-19*23+23^2)/3 = 151 is prime.
Links
- Robert Israel, Table of n, a(n) for n = 1..10000
Programs
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Maple
R:= NULL: q:= 2: count:= 0: while count<100 do p:= q; q:= nextprime(p); r:= (p^2-p*q+q^2)/3; if r::integer and isprime(r) then count:= count+1; R:= R, r; fi; od: R; -
Mathematica
cpQ[{a_,b_}]:=Module[{c=(a^2-a*b+b^2)/3},If[PrimeQ[c],c,Nothing]]; cpQ/@Partition[Prime[ Range[ 500]],2,1] (* Harvey P. Dale, Dec 31 2023 *) -
Python
from sympy import isprime, nextprime def aupto(limit): p, q, num, alst = 3, 5, 7, [] while num//3 <= limit: if num%3 == 0 and isprime(num//3): alst.append(num//3) p, q, num = q, nextprime(q), p**2 - p*q + q**2 return alst print(aupto(1267501)) # Michael S. Branicky, Mar 18 2021