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A037896
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Primes of the form k^4 + 1.
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33
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2, 17, 257, 1297, 65537, 160001, 331777, 614657, 1336337, 4477457, 5308417, 8503057, 9834497, 29986577, 40960001, 45212177, 59969537, 65610001, 126247697, 193877777, 303595777, 384160001, 406586897, 562448657, 655360001
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OFFSET
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1,1
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COMMENTS
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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.
If p prime = k^4 + 1, phi(p) = k^4.
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)
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LINKS
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EXAMPLE
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6^4 + 1 = 1297 is prime.
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MATHEMATICA
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PROG
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(PARI) j=[]; for(n=1, 200, if(isprime(n^4+1), j=concat(j, n^4+1))); j
(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
(Magma) [n^4+1: n in [1..200] | IsPrime(n^4+1)]; // G. C. Greubel, Apr 28 2019
(Sage) [n^4+1 for n in (1..200) if is_prime(n^4+1)] # G. C. Greubel, Apr 28 2019
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CROSSREFS
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KEYWORD
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easy,nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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