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Primes p for which p + 4, p^2 + 4 and p^3 + 4 are primes.
1

%I #26 Sep 21 2024 02:28:32

%S 3,7,103,277,487,967,4783,5503,5923,8233,21013,26317,27943,41593,

%T 55213,78307,78853,86197,89653,94723,99013,123727,148153,157177,

%U 166627,172867,177883,179107,185893,192883,194713,203767,204517,223633,225217,227593,236893

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

%C This is a subset of the sequences:

%C A023200: Primes p such that p + 4 is also prime.

%C A243583: Primes p for which p + 4 and p^3 + 4 are primes.

%C p is either 2 mod 5 or 3 mod 5, hence p^4 + 4 is 0 mod 5.

%H Abhiram R Devesh, <a href="/A243734/b243734.txt">Table of n, a(n) for n = 1..1000</a>

%e p = 3 is in this sequence because p + 4 = 7, p^2 + 4 = 13 and p^3 + 4 = 31 are all primes.

%e p : p+4, p^2+4, p^3+4

%e 7 : 11, 53, 347

%e 103: 107, 10613, 1092731

%e 277: 281, 76733, 21253937

%e 487: 491, 237173, 115501307

%o (Python)

%o import sympy.ntheory as snt

%o n=2

%o while n > 1 and n < 10**6:

%o n1=n+4

%o n2=((n**2)+4)

%o n3=((n**3)+4)

%o ##Check if n1, n2 and n3 are also primes.

%o if snt.isprime(n1)== True and snt.isprime(n2)== True and snt.isprime(n3)== True:

%o print(n, end=', ')

%o n=snt.nextprime(n)

%o (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 Cf. A023200, A243583.

%K nonn,easy

%O 1,1

%A _Abhiram R Devesh_, Jun 09 2014