This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A037896 #58 Feb 22 2025 02:28:07
%S A037896 2,17,257,1297,65537,160001,331777,614657,1336337,4477457,5308417,
%T A037896 8503057,9834497,29986577,40960001,45212177,59969537,65610001,
%U A037896 126247697,193877777,303595777,384160001,406586897,562448657,655360001,723394817,916636177,1049760001,1416468497
%N A037896 Primes of the form k^4 + 1.
%C A037896 From _Bernard Schott_, Apr 22 2019: (Start)
%C A037896 These primes are the primitive terms which generate the sequence of integers with only one prime factor and whose Euler's totient is a perfect biquadrate: A307690, so this sequence is a subsequence of A078164 and A307690.
%C A037896 If p prime = k^4 + 1, phi(p) = k^4.
%C A037896 The last three Fermat primes in A019434 {17, 257, 65537} belong to this sequence; with F_k = 2^(2^k) + 1 and for k = 2, 3, 4, phi(F_k) = (2^(2^(k-2)))^4. (End)
%H A037896 T. D. Noe, <a href="/A037896/b037896.txt">Table of n, a(n) for n=1..1000</a>
%H A037896 Ernest G. Hibbs, <a href="https://www.proquest.com/openview/4012f0286b785cd732c78eb0fc6fce80">Component Interactions of the Prime Numbers</a>, Ph. D. Thesis, Capitol Technology Univ. (2022), see p. 33.
%H A037896 M. Lal, <a href="http://dx.doi.org/10.1090/S0025-5718-1967-0222007-9">Primes of the form n^4 + 1</a>, Math. Comp. 21 (1967), pp. 245-247.
%F A037896 a(n) = A002523(A000068(n)). - _Elmo R. Oliveira_, Feb 21 2025
%e A037896 6^4 + 1 = 1297 is prime.
%t A037896 Select[Range[200]^4+1,PrimeQ] (* _Harvey P. Dale_, Jul 20 2015 *)
%o A037896 (PARI) j=[]; for(n=1,200, if(isprime(n^4+1),j=concat(j,n^4+1))); j
%o A037896 (PARI) list(lim)=my(v=List([2]),p); forstep(k=2,sqrtnint(lim\1-1,4),2, if(isprime(p=k^4+1), listput(v,p))); Vec(v) \\ _Charles R Greathouse IV_, Mar 31 2022
%o A037896 (Magma) [n^4+1: n in [1..200] | IsPrime(n^4+1)]; // _G. C. Greubel_, Apr 28 2019
%o A037896 (Sage) [n^4+1 for n in (1..200) if is_prime(n^4+1)] # _G. C. Greubel_, Apr 28 2019
%Y A037896 Cf. A000068, A002523, A019434, A078164, A307690.
%Y A037896 Subsequence of A002496, A078164 and A039770.
%K A037896 easy,nonn
%O A037896 1,1
%A A037896 _Donald S. McDonald_, Feb 27 2000
%E A037896 Corrected and extended by _Jason Earls_, Jul 19 2001