cp's OEIS Frontend

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

%I A277201 #41 Sep 08 2022 08:46:17
%S A277201 41,313,1201,7321,14281,41761,139921,353641,6922921,12705841,14199121,
%T A277201 56275441,81523681,784119601,1984563001,4798962481,5049019561,
%U A277201 6448958881,7763701441,15410832361,17253574561,20321481601,22977034081,26321586241
%N A277201 Primes of the form (p^4 + 1)/2, where p is prime.
%C A277201 The sequence is a subsequence of A096170.
%C A277201 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.
%C A277201 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
%C A277201 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
%H A277201 Amiram Eldar, <a href="/A277201/b277201.txt">Table of n, a(n) for n = 1..10000</a>
%F A277201 a(n) = (A176116(n)^4 + 1)/2.
%e A277201 a(1) = 41 because 3 is prime and (3^4 + 1)/2 = 41 is prime.
%e A277201 a(2) = 313 because 5 is prime and (5^4 + 1)/2 = 313 is prime.
%e A277201 a(3) = 1201 because 7 is prime and (7^4 + 1)/2 = 1201 is prime.
%t A277201 Select[Map[(#^4 + 1)/2 &, Prime@ Range@ 100], PrimeQ] (* _Michael De Vlieger_, Oct 04 2016 *)
%t A277201 Select[Table[(p^4+1)/2,{p,Prime[Range[100]]}],PrimeQ] (* _Harvey P. Dale_, Dec 21 2018 *)
%o A277201 (Maxima)
%o A277201 makelist(if primep(k)=true then ((k^4)+1)/2 else 0,k,3,500,1)$ sublist(%,primep);
%o A277201 (PARI) lista(nn) = {forprime(p=3, nn, if (isprime(q=(p^4+1)/2), print1(q, ", ")););} \\ _Michel Marcus_, Oct 04 2016
%o A277201 (Magma) [a: p in PrimesUpTo(1000) | IsPrime(a) where a is (p^4+1) div 2 ]; // _Vincenzo Librandi_, Nov 07 2016
%Y A277201 Cf. A096169, A096170, A176116.
%K A277201 nonn
%O A277201 1,1
%A A277201 _Lechoslaw Ratajczak_, Oct 04 2016