A336536 Numbers n that can be written as both the sum of two nonzero fourth powers and the sum of three nonzero fourth powers.

4802, 57122, 76832, 260642, 388962, 617057, 913952, 1229312, 1847042, 1957682, 3001250, 3502322, 3748322, 3959297, 4170272, 4626882, 6223392, 6837602, 6959682, 9872912, 11529602, 14623232, 19668992, 21112002, 27691682, 29552672, 31322912, 31505922, 35701250, 40127377, 40302242, 46712801, 48020000, 48355137

Offset: 1

Author

Robert Israel, Jul 24 2020

Keywords

• nonn

Comments

The fourth powers are not necessarily distinct.

If n is in the sequence, then so is k^4*n for every k.

The sum of two nonzero fourth powers is never a fourth power (a case of Fermat's last theorem).

Examples

			a(3) = 76832 is in the sequence because 76832 = 14^4 + 14^4 = 6^4 + 10^4 + 16^4.
a(6) = 617057 is in the sequence because 617057 = 7^4 + 28^4 = 3^4 + 20^4 + 26^4.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..1072

Crossrefs

Cf. A000583, A003294, A047714.

Intersection of A003336 and A003337.

Programs

Maple

N:= 10^8: # for terms <= N
F1:= {seq(i^4,i=1..floor(N^(1/4)))}: n1:= nops(F1):
F2:= select(`<=`,{seq(seq(F1[i]+F1[j],i=1..j),j=1..nops(F1))},N):
F3:= select(`<=`,{seq(seq(s+t,s=F1),t=F2)},N):
sort(convert(F3 intersect F2,list));

Python

def aupto(lim):
  p1 = set(i**4 for i in range(1, int(lim**.25)+2) if i**4 <= lim)
  p2 = set(a+b for a in p1 for b in p1 if a+b <= lim)
  p3 = set(apb+c for apb in p2 for c in p1 if apb+c <= lim)
  return sorted(p3 & p2)
print(aupto(5*10**7)) # Michael S. Branicky, Mar 18 2021