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

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

Author

Lechoslaw Ratajczak, Oct 04 2016

Keywords

• nonn

Comments

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

Examples

			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.
		

Links

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

Crossrefs

Cf. A096169, A096170, A176116.

Programs

Magma

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

Mathematica

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 *)

Maxima

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

PARI

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

Formula

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