A307690 -id:A307690 - cp's OEIS frontend

A037896 Primes of the form k^4 + 1.

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, 723394817, 916636177, 1049760001, 1416468497

Offset: 1

Donald S. McDonald, Feb 27 2000

From Bernard Schott, Apr 22 2019: (Start)
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)

			6^4 + 1 = 1297 is prime.

T. D. Noe, Table of n, a(n) for n=1..1000
Ernest G. Hibbs, Component Interactions of the Prime Numbers, Ph. D. Thesis, Capitol Technology Univ. (2022), see p. 33.
M. Lal, Primes of the form n^4 + 1, Math. Comp. 21 (1967), pp. 245-247.

Cf. A000068, A002523, A019434, A078164, A307690.

Subsequence of A002496, A078164 and A039770.

[n^4+1: n in [1..200] | IsPrime(n^4+1)]; // G. C. Greubel, Apr 28 2019

Select[Range[200]^4+1,PrimeQ] (* Harvey P. Dale, Jul 20 2015 *)

j=[]; for(n=1,200, if(isprime(n^4+1),j=concat(j,n^4+1))); j

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

[n^4+1 for n in (1..200) if is_prime(n^4+1)] # G. C. Greubel, Apr 28 2019

a(n) = A002523(A000068(n)). - Elmo R. Oliveira, Feb 21 2025

Corrected and extended by Jason Earls, Jul 19 2001

A013776 a(n) = 2^(4*n+1).

Original entry on oeis.org

2, 32, 512, 8192, 131072, 2097152, 33554432, 536870912, 8589934592, 137438953472, 2199023255552, 35184372088832, 562949953421312, 9007199254740992, 144115188075855872, 2305843009213693952

Offset: 0

N. J. A. Sloane

a(n) ~ -Pi*E(2*n)/B(2*n), E(n) Euler number, B(n) Bernoulli number. - Peter Luschny, Oct 28 2012
Equivalently, powers of 2 with final digit 2. - Muniru A Asiru, Mar 15 2019
As phi(a(n)) = (2^n)^4 is a perfect biquadrate (where phi is the Euler totient A000010), this is a subsequence of A078164 and A307690. - Bernard Schott, Mar 28 2022

			G.f. = 2 + 32*x + 512*x^2 + 8192*x^3 + 131072*x^4 + 2097152*x^5 + ...

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Tanya Khovanova, Recursive Sequences
Index to divisibility sequences
Index entries for linear recurrences with constant coefficients, signature (16).

Subsequence of A307690.
Cf. A000010, A004171, A016813, A264960, A307690.
Intersection of A000079 and A078164.

List([0..20],n->2^(4*n+1)); # Muniru A Asiru, Mar 15 2019

[2^(4*n+1): n in [0..20]]; // Vincenzo Librandi, Jun 27 2011

[2^(4*n+1)$n=0..20]; # Muniru A Asiru, Apr 10 2019

2^(4*Range[0,20]+1) (* G. C. Greubel, Mar 15 2019 *)
NestList[16#&,2,20] (* Harvey P. Dale, Jul 28 2019 *)

a(n)=2<<(4*n) \\ Charles R Greathouse IV, Apr 07 2012

[2^(4*n+1) for n in (0..20)] # G. C. Greubel, Mar 15 2019

From Philippe Deléham, Nov 23 2008: (Start)
a(n) = 16*a(n-1), n > 0, a(0) = 2.
G.f.: 2/(1 - 16*x). (End)
From Peter Bala, Nov 29 2015: (Start)
a(n) = Sum_{k = 0..n} binomial(2*k,k)*binomial(4*n + 2 - 2*k, 2*n + 1 - k).
Bisection of A264960. (End)
a(n) = A000079(A016813(n)). - Michel Marcus, Nov 30 2015
a(n) = Sum_{k = 0..2*n} binomial(4*n + 2, 2*k + 1) = A004171(2*n). - Peter Bala, Nov 25 2016
E.g.f.: 2*exp(16*x). - G. C. Greubel, Jun 30 2019
From Bernard Schott, Apr 15 2022: (Start)
Sum_{n>=0} 1/a(n) = 8/15.
Sum_{n>=0} (-1)^n/a(n) = 8/17. (End)

Wrong comment deleted by Kevin Ryde, Apr 16 2022

A013806 a(n) = 17^(4*n+1).

Original entry on oeis.org

17, 1419857, 118587876497, 9904578032905937, 827240261886336764177, 69091933913008732880827217, 5770627412348402378939569991057, 481968572106750915091411825223071697, 40254497110927943179349807054456171205137

Offset: 0

N. J. A. Sloane

As phi(a(n)) = (2*17^n)^4 is a perfect biquadrate (where phi is the Euler totient A000010), this is a subsequence of A078164 and A307690. - Bernard Schott, Mar 29 2022

Vincenzo Librandi, Table of n, a(n) for n = 0..100
Tanya Khovanova, Recursive Sequences.
Index entries for linear recurrences with constant coefficients, signature (83521).

Cf. A000010, A013776, A307690.

Intersection of A001026 and A078164.

[17^(4*n+1): n in [0..15]]; // Vincenzo Librandi, Jul 06 2011

17^(4Range[0,10]+1) (* or *) NestList[83521#&,17,20] (* Harvey P. Dale, May 21 2013 *)

a(0)=17, a(n)=83521*a(n-1). - Harvey P. Dale, May 21 2013
Sum_{n>=0} 1/a(n) = 4913/83520. - Bernard Schott, Mar 29 2022
Sum_{n>=0} (-1)^n/a(n) = 4913/83522. - Bernard Schott, Apr 08 2022

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A037896 Primes of the form k^4 + 1.

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A013776 a(n) = 2^(4*n+1).

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A013806 a(n) = 17^(4*n+1).

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