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%I #19 Sep 03 2019 17:48:08
%S 1,1,2,1,6,1,4,3,2,1,6,1,4,3,2,1,6,1,12,3,16,1,6,7,4,15,2,1,12,1,6,9,
%T 8,3,6,1,4,3,2,1,30,1,4,3,8,1,6,5,10,3,10,1,6,5,4,15,2,1,42,1,6,21,4,
%U 3,6,1,4,15,2,1,30,1,6,33,8,25,6,1,10,3,16,1,24,5,4,15
%N Least positive number k such that n+k and n+k^2 are both prime.
%C For n > 1, a(n) == n+1 (mod 2). a(n) = 1 for n in A006093. - _Robert Israel_, Feb 02 2018
%H Robert Israel, <a href="/A243145/b243145.txt">Table of n, a(n) for n = 1..10000</a>
%e 8+1 and 8+1^2 (9) isn't prime. 8+2 and 8+2^2 (10 and 12) aren't both prime. But 8+3 and 8+3^2 (11 and 17) are both prime. Thus a(8) = 3.
%p f:= proc(n) local k;
%p for k from (n mod 2)+1 by 2 do
%p if isprime(n+k) and isprime(n+k^2) then return k fi
%p od
%p end proc:
%p f(1):= 1:
%p map(f, [$1..100]); # _Robert Israel_, Feb 02 2018
%t f[n_] := Module[{k}, For[k = Mod[n, 2] + 1, True, k += 2, If[PrimeQ[n + k] && PrimeQ[n + k^2], Return[k]]]]; f[1] = 1; f /@ Range[100] (* _Jean-François Alcover_, Feb 03 2018, after _Robert Israel_ *)
%o (PARI) a(n)=for(k=1,10^6,if(ispseudoprime(n+k)&&ispseudoprime(n+k^2),return(k)))
%o n=1;while(n<100,print1(a(n),", ");n+=1)
%o (Python)
%o from sympy import isprime, nextprime
%o def A243145(n):
%o m = n
%o while True:
%o m = nextprime(m)
%o k = m-n
%o if isprime(n+k**2):
%o return k # _Chai Wah Wu_, Sep 03 2019
%Y Cf. A006093.
%K nonn
%O 1,3
%A _Derek Orr_, May 30 2014
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