A126117 Primes p of the form a^4+b^4+c^4 with a,b,c>=1 such that a^2+b^2+c^2 is another prime < p.

83, 113, 163, 353, 593, 787, 1553, 2593, 3217, 5393, 6563, 6833, 7187, 7793, 8273, 8963, 9043, 11953, 14177, 14723, 16193, 17123, 20753, 21283, 21377, 23603, 25457, 26561, 27763, 28643, 29363, 31153, 35393, 45233, 46273, 50593, 50627, 51137, 52177, 56113, 57203, 59233, 61553, 65267, 65633, 65713

Offset

1,1

Comments

Some terms may occur in more than one way (but are just listed once): for example, a(73)=137633 = 8^4+13^4+18^4 = 9^4+16^4+16^4 with 8^2+13^2+18^2 = 557 and 9^2+16^2+16^2 = 593. - Robert Israel, Oct 26 2020

Links

Robert Israel, Table of n, a(n) for n = 1..10000

Formula

P1=a^4+b^4+c^4; P= a^2+b^2+c^2; P1 and P are primes; P1 > P

Example

5393=1^4+6^4+8^4=1+1296+4096 and 101=1^2+6^2+8^2=1+36+64;
6833= 2^4+4^4+9^4 = 16+256+6561 and 101= 2^2+4^2+9^2 =4+16+81;
339690083=99^4 +103^4 +107^4=96059601+112550881+131079601 and
31859=99^2 +103^2 +107^2 =9801+10609+11499;

Maple

N:= 10^5: # for terms <= N
R:= {}:
for a from 1 to floor((N/3)^(1/4)) do
  for b from a to floor(((N-a^4)/2)^(1/4)) do
    for c from b to floor((N-a^4-b^4)^(1/4)) do
      x:= a^4 + b^4 + c^4;
      if x > 3 and isprime(x) and isprime(a^2+b^2+c^2) then
        R:= R union {x};
      fi
od od od:
sort(convert(R, list)); # Robert Israel, Oct 26 2020

Crossrefs

Sequence in context: A347226 A261449 A288880 * A096279 A340817 A261087

Adjacent sequences: A126114 A126115 A126116 * A126118 A126119 A126120

Keyword

nonn

Author

Tomas Xordan, Mar 05 2007

Extensions

Corrected by Robert Israel, Oct 26 2020

Status

approved