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A243734 Primes p for which p + 4, p^2 + 4 and p^3 + 4 are primes.

%I A243734 #26 Sep 21 2024 02:28:32
%S A243734 3,7,103,277,487,967,4783,5503,5923,8233,21013,26317,27943,41593,
%T A243734 55213,78307,78853,86197,89653,94723,99013,123727,148153,157177,
%U A243734 166627,172867,177883,179107,185893,192883,194713,203767,204517,223633,225217,227593,236893
%N A243734 Primes p for which p + 4, p^2 + 4 and p^3 + 4 are primes.
%C A243734 This is a subset of the sequences:
%C A243734 A023200: Primes p such that p + 4 is also prime.
%C A243734 A243583: Primes p for which p + 4 and p^3 + 4 are primes.
%C A243734 p is either 2 mod 5 or 3 mod 5, hence p^4 + 4 is 0 mod 5.
%H A243734 Abhiram R Devesh, <a href="/A243734/b243734.txt">Table of n, a(n) for n = 1..1000</a>
%e A243734 p = 3 is in this sequence because p + 4 = 7, p^2 + 4 = 13 and p^3 + 4 = 31 are all primes.
%e A243734 p  : p+4,  p^2+4,     p^3+4
%e A243734 7  :  11,     53,       347
%e A243734 103: 107,  10613,   1092731
%e A243734 277: 281,  76733,  21253937
%e A243734 487: 491, 237173, 115501307
%o A243734 (Python)
%o A243734 import sympy.ntheory as snt
%o A243734 n=2
%o A243734 while n > 1 and n < 10**6:
%o A243734     n1=n+4
%o A243734     n2=((n**2)+4)
%o A243734     n3=((n**3)+4)
%o A243734     ##Check if n1, n2 and n3 are also primes.
%o A243734     if snt.isprime(n1)== True and snt.isprime(n2)== True and snt.isprime(n3)== True:
%o A243734         print(n, end=', ')
%o A243734     n=snt.nextprime(n)
%o A243734 (PARI) s=[]; forprime(p=2, 200000, if(isprime(p+4) && isprime(p^2+4) && isprime(p^3+4), s=concat(s, p))); s \\ _Colin Barker_, Jun 11 2014
%Y A243734 Cf. A023200, A243583.
%K A243734 nonn,easy
%O A243734 1,1
%A A243734 _Abhiram R Devesh_, Jun 09 2014