A096170 -id:A096170 - cp's OEIS frontend

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

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 *)

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

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

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