A004831 Numbers that are the sum of at most 2 nonzero 4th powers.

0, 1, 2, 16, 17, 32, 81, 82, 97, 162, 256, 257, 272, 337, 512, 625, 626, 641, 706, 881, 1250, 1296, 1297, 1312, 1377, 1552, 1921, 2401, 2402, 2417, 2482, 2592, 2657, 3026, 3697, 4096, 4097, 4112, 4177, 4352, 4721, 4802, 5392, 6497, 6561, 6562, 6577, 6642

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn

Comments

Apart from 0, 1, 2, there are no three consecutive terms up to 10^16. The first two consecutive terms not of the form n^4, n^4+1 are 3502321 = 25^4 + 42^4, 3502322 = 17^4 + 43^4. - Charles R Greathouse IV, Oct 17 2017

Links

• T. D. Noe, Table of n, a(n) for n = 1..1000

Crossrefs

Subsequences include A003336, A000583 and A002645.

Programs

Haskell

a004831 n = a004831_list !! (n-1)
a004831_list = [x ^ 4 + y ^ 4 | x <- [0..], y <- [0..x]]
-- Reinhard Zumkeller, Jul 15 2013

Mathematica

Reap[For[n = 0, n < 10000, n++, If[MatchQ[ PowersRepresentations[n, 2, 4], {{, },_}], Print[n]; Sow[n]]]][[2, 1]] (* Jean-François Alcover, Oct 30 2017 *)

PARI

is(n)=#thue(thueinit(z^4+1),n) \\ Ralf Stephan, Oct 18 2013

PARI

list(lim)=my(v=List(),t); for(m=0,sqrtnint(lim\=1,4), for(n=0, min(sqrtnint(lim-m^4,4),m), listput(v,n^4+m^4))); Set(v) \\ Charles R Greathouse IV, Sep 28 2015

Formula

Call f(x) the number of terms if this sequence up to x. Then x^(7/16) << f(x) << x^(1/2); in other words, n^2 << a(n) << n^(16/7). The upper bound becomes O(n^2) if A230562 is finite. - Charles R Greathouse IV, Jul 12 2024