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Numbers n such that there exists a k>0 such that all six of n +/- k, n^2 +/- k, and n^3 +/- k are prime.
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%I #13 Nov 02 2024 18:22:38

%S 12,25,29,36,45,55,78,87,105,109,111,130,140,141,155,160,190,196,209,

%T 216,231,245,246,265,274,280,289,294,311,315,329,356,364,385,409,441,

%U 444,465,475,489,494,531,535,572,582,600,624,629,650,665

%N Numbers n such that there exists a k>0 such that all six of n +/- k, n^2 +/- k, and n^3 +/- k are prime.

%C This is similar to A239146; however, here the numbers listed are the n values for which k != 0.

%C It is very likely that k does not exist for most n values since k < n for all n. Thus, only the numbers n with some such k (depending on n) are listed.

%e n = 1,2,3,...11 do not have a k such that n +/- k, n^2 +/- k, and n^3 +/- k are all prime. However, for n = 12, 12 +/- 5 (7 and 17), 12^2 +/- 5 (139 and 149) and 12^3 +/- 5 (1723 and 1733) are all prime. So 12 is a member of this sequence.

%p isA239147 := proc(n)

%p local k ;

%p for k from 1 do

%p if n-k <= 1 then

%p return false;

%p end if;

%p if isprime(n+k) and isprime(n-k) and isprime(n^2+k)

%p and isprime(n^2-k) and isprime(n^3+k) and isprime(n^3-k) then

%p return true;

%p end if;

%p end do;

%p end proc:

%p for n from 1 to 800 do

%p if isA239147(n) then

%p printf("%d,",n) ;

%p end if;

%p end do: # _R. J. Mathar_, Mar 12 2014

%o (Python)

%o import sympy

%o from sympy import isprime

%o def c(n):

%o for k in range(n):

%o if isprime(n+k) and isprime(n-k) and isprime(n**2+k) and isprime(n**2-k) and isprime(n**3+k) and isprime(n**3-k):

%o return k

%o n = 1

%o while n < 10**3:

%o if c(n) != None:

%o print(n)

%o n += 1

%Y Cf. A239146.

%K nonn

%O 1,1

%A _Derek Orr_, Mar 11 2014