A096171 -id:A096171 - cp's OEIS frontend

A096172 Largest prime factor of n^4 + 1.

2, 17, 41, 257, 313, 1297, 1201, 241, 193, 137, 7321, 233, 14281, 937, 1489, 65537, 41761, 929, 3833, 160001, 97241, 3209, 139921, 331777, 11489, 26881, 6481, 614657, 353641, 3361, 1129, 61681, 6113, 1336337, 750313, 98801, 10529, 50857, 1156721

Offset: 1

Hugo Pfoertner, Jun 19 2004

nonn

Mabkhout shows that a(n) >= 137 for n > 3. - Charles R Greathouse IV, Apr 07 2014

			a(1)=2 because 1^4 + 1 = 2;
a(2)=17: 2^4 + 1 = 17;
a(8)=241: 8^4 + 1 = 4097 = 17*241.

Mustapha Mabkhout, Minoration de P(x^4+1), Rendiconti del Seminario della Facoltà di Scienze dell'Università di Cagliari 63:2 (1993), pp. 135-148.

Vincenzo Librandi and T. D. Noe, Table of n, a(n) for n = 1..10000 (first 1000 terms from Vincenzo Librandi)
Florian Luca, Primitive divisors of Lucas sequences and prime factors of x^2 + 1 and x^4 + 1, Acta Academiae Paedagogicae Agriensis, Sectio Mathematicae 31 (2004), pp. 19-24.
Marina Mureddu, A lower bound for P(x^4+1), Annales de la Faculté des sciences de Toulouse: Mathématiques, Serie 5, Vol. 8, No. 2 (1986-1987), pp. 109-119.

Cf. A000068, A002523, A014442, A037896, A006530, A069170, A096169, A096171.

FactorInteger[#^4+1][[-1,1]]&/@Range[40] (* Harvey P. Dale, Apr 30 2012 *)

a(n)=my(f=factor(n^4+1)[,1]); f[#f] \\ Charles R Greathouse IV, Apr 07 2014

a(n) = A006530(1+n^4) = A014442(n^2). - R. J. Mathar, Jan 28 2017
From Amiram Eldar, Oct 28 2024: (Start)
a(n) > 113 for n > 3 (Mureddu, 1986-1987).
a(n) >= 233 for n >= 11 (Luca, 2004). (End)

A096169 Odd n such that (n^4+1)/2 is prime.

Original entry on oeis.org

3, 5, 7, 11, 13, 17, 21, 23, 29, 35, 39, 57, 61, 65, 71, 73, 81, 103, 105, 113, 115, 119, 129, 153, 165, 169, 171, 199, 203, 205, 251, 259, 267, 275, 309, 313, 317, 333, 337, 339, 353, 363, 403, 405, 415, 419, 431, 445, 449, 453, 455, 463, 471, 477, 479, 487

Offset: 1

Hugo Pfoertner, Jun 19 2004

nonn

			a(1)=3 because (3^4+1)/2=82/2=41 is prime.

Zak Seidov, Table of n, a(n) for n = 1..1000

Cf. A000068 n^4+1 is prime, A037896 primes of the form n^4+1, A096170 primes of the form (n^4+1)/2, A096171 n^4+1 is an odd semiprime, A096172 largest prime factor of n^4+1.

[ n: n in [0..2500] | IsPrime((n^4+1) div 2) ]; // Vincenzo Librandi, Apr 15 2011

Select[Range[1,501,2],PrimeQ[(#^4+1)/2]&] (* Harvey P. Dale, Jun 04 2011 *)

A096170 Primes of the form (k^4 + 1)/2.

Original entry on oeis.org

41, 313, 1201, 7321, 14281, 41761, 97241, 139921, 353641, 750313, 1156721, 5278001, 6922921, 8925313, 12705841, 14199121, 21523361, 56275441, 60775313, 81523681, 87450313, 100266961, 138461441, 273990641, 370600313, 407865361

Offset: 1

Hugo Pfoertner, Jun 19 2004

nonn

Note that k must be odd. Terms of primitive Pythagorean triples: (k^2, (k^4-1)/2, (k^4+1)/2).

			a(1)=41 because (3^4 + 1)/2 = 82/2 = 41 is prime.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A096169 (n^4+1)/2 is prime, A000068 n^4+1 is prime, A037896 primes of the form n^4+1, A096171 n^4+1 is an odd semiprime, A096172 largest prime factor of n^4+1.

[ a: n in [0..2500] | IsPrime(a) where a is ((n^4+1) div 2) ]; // Vincenzo Librandi, Apr 15 2011

Select[(Range[200]^4+1)/2,PrimeQ] (* Harvey P. Dale, Mar 09 2013 *)

list(lim)=my(v=List(),t); forstep(n=3,sqrtnint(lim\1*2-1,4),2, if(isprime(t=(n^4+1)/2), listput(v,t))); Vec(v) \\ Charles R Greathouse IV, Feb 14 2017

Name edited by Zak Seidov, Apr 14 2011

A256145 Primitive prime factors of the cyclotomic polynomial sequence Phi(8,k) in the order in which they occur.

Original entry on oeis.org

2, 17, 41, 257, 313, 1297, 1201, 241, 193, 73, 137, 7321, 89, 233, 14281, 937, 1489, 65537, 41761, 113, 929, 3833, 160001, 97241, 3209, 139921, 331777, 11489, 26881, 6481, 614657, 353641, 3361, 409, 1129, 61681, 97, 6113, 1336337, 750313, 98801, 10529, 50857

Offset: 1

Robert Price, Mar 16 2015

nonn

Phi(8,k) = k^4 + 1.

Cf. A005529, A096171, A248874, A256144.

prim = {}; Do[prim = Join[prim, Complement[First /@ FactorInteger[Cyclotomic[8, k]], prim]], {k, 1000}]; prim

lista(nn) = {vs = []; for (n=1, nn, vp = factor(polcyclo(8,n))[,1]; for (i=1, #vp, if (!vecsearch(vs, vp[i]), print1(vp[i], ", "); vs = vecsort(concat(vs, vp[i]),,8););););} \\ Michel Marcus, Mar 20 2015

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A096172 Largest prime factor of n^4 + 1.

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A096169 Odd n such that (n^4+1)/2 is prime.

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A096170 Primes of the form (k^4 + 1)/2.

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A256145 Primitive prime factors of the cyclotomic polynomial sequence Phi(8,k) in the order in which they occur.

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