A096169 -id:A096169 - 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)

A176116 Primes p such that p^4+1 = 2q where q is prime.

Original entry on oeis.org

3, 5, 7, 11, 13, 17, 23, 29, 61, 71, 73, 103, 113, 199, 251, 313, 317, 337, 353, 419, 431, 449, 463, 479, 487, 503, 523, 607, 613, 643, 677, 701, 719, 761, 769, 811, 821, 829, 857, 883, 919, 997, 1013, 1019, 1049, 1087, 1123, 1163, 1259, 1327, 1381, 1483, 1493

Offset: 1

Kevin Batista (kevin762401(AT)yahoo.com), Apr 08 2010

			3^4+1 = 2*41; 5^4+1 = 2*313; 7^4+1 = 2*1201; 11^4+1 = 2*7321.

Jon E. Schoenfield, Table of n, a(n) for n = 1..10000

Subsequence of A096169.

Cf. A277201 (resulting primes).

Select[Prime[Range[250]],PrimeQ[(#^4+1)/2]&] (* Harvey P. Dale, Jul 20 2012 *)

lista(nn) = forprime(p=3, nn, if (isprime((p^4+1)/2), print1(p, ", "));); \\ Michel Marcus, Oct 03 2016

Edited by Ray Chandler, Apr 10 2010

A096171 Numbers k such that k^4+1 is an odd semiprime.

Original entry on oeis.org

8, 10, 12, 14, 18, 22, 26, 30, 32, 36, 38, 40, 42, 50, 52, 58, 62, 68, 72, 78, 84, 86, 92, 94, 98, 100, 102, 108, 112, 114, 116, 120, 122, 124, 128, 130, 138, 146, 148, 152, 158, 162, 166, 170, 172, 176, 184, 186, 200, 212, 214, 216, 218, 222, 224, 226, 234, 250, 252

Offset: 1

Hugo Pfoertner, Jun 19 2004

nonn

			a(1)=8 because 8^4 + 1 = 4097 = 17*241;
a(2)=10: 10^4 + 1 = 10001 = 73*137.

Hugo Pfoertner, Table of n, a(n) for n = 1..10000

Cf. A000068 (n^4+1 is prime), A037896 (primes of the form k^4+1), A096169 ((n^4+1)/2 is prime), A069170 (primes of the form (k^4+1)/2), A096172 (largest prime factor of n^4+1), A046388.

Select[Range[2,300,2],PrimeOmega[#^4+1]==2&] (* Harvey P. Dale, Dec 25 2021 *)

isA096171(n) = {local(m);m=n^4+1;(m%2==1)&&(bigomega(m)==2)} \\ Michael B. Porter, Feb 02 2010

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

A277201 Primes of the form (p^4 + 1)/2, where p is prime.

Original entry on oeis.org

41, 313, 1201, 7321, 14281, 41761, 139921, 353641, 6922921, 12705841, 14199121, 56275441, 81523681, 784119601, 1984563001, 4798962481, 5049019561, 6448958881, 7763701441, 15410832361, 17253574561, 20321481601, 22977034081, 26321586241

Offset: 1

Lechoslaw Ratajczak, Oct 04 2016

nonn

The sequence is a subsequence of A096170.
Conjecture: the sequence consists of the numbers k such that tau(2k) = 4 and tau(2k-1) = 5. tau(82) = 4 and tau(81) = 5, 82/2 = 41 = a(1). tau(626) = 4 and tau(625) = 5, 626/2 = 313 = a(2). tau(2402) = 4 and tau(2401) = 5, 2402/2 = 1201 = a(3). The conjecture was checked for 10^9 consecutive integers.
The above conjecture is true: since tau(2k-1) = 5, 2k-1 must be the 4th power of some prime p, i.e., k = (p^4 + 1)/2 (so p is odd, so p^4 == 1 (mod 16), so k is odd), and since tau(2k) = 4, 2k must be the product of two distinct primes, so k is an odd prime. Thus, the set of numbers k such that tau(2k) = 4 and tau(2k-1) = 5 is the set of primes of the form (p^4 + 1)/2, where p is prime. - Jon E. Schoenfield, Mar 17 2019
Primes of the form a^2 + b^2 such that a^2 - b^2 = p^2, where p is prime. - Thomas Ordowski, Feb 14 2017

			a(1) = 41 because 3 is prime and (3^4 + 1)/2 = 41 is prime.
a(2) = 313 because 5 is prime and (5^4 + 1)/2 = 313 is prime.
a(3) = 1201 because 7 is prime and (7^4 + 1)/2 = 1201 is prime.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A096169, A096170, A176116.

[a: p in PrimesUpTo(1000) | IsPrime(a) where a is (p^4+1) div 2 ]; // Vincenzo Librandi, Nov 07 2016

Select[Map[(#^4 + 1)/2 &, Prime@ Range@ 100], PrimeQ] (* Michael De Vlieger, Oct 04 2016 *)
Select[Table[(p^4+1)/2,{p,Prime[Range[100]]}],PrimeQ] (* Harvey P. Dale, Dec 21 2018 *)

makelist(if primep(k)=true then ((k^4)+1)/2 else 0,k,3,500,1)$ sublist(%,primep);

lista(nn) = {forprime(p=3, nn, if (isprime(q=(p^4+1)/2), print1(q, ", ")););} \\ Michel Marcus, Oct 04 2016

a(n) = (A176116(n)^4 + 1)/2.

cp's OEIS Frontend

A096172 Largest prime factor of n^4 + 1.

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A176116 Primes p such that p^4+1 = 2q where q is prime.

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A096171 Numbers k such that k^4+1 is an odd semiprime.

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

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A277201 Primes of the form (p^4 + 1)/2, where p is prime.

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