A347773 Square array read by antidiagonals downwards: T(n,k) is the smallest positive integer whose n-th power is the sum of k n-th powers of positive integers, or 0 if no such number exists.

1, 2, 1, 3, 5, 1, 4, 3, 0, 1, 5, 2, 6, 0, 1, 6, 4, 7, 422481, 0, 1, 7, 3, 4, 353

Offset: 1

Author

Eric Chen, Sep 15 2021

Keywords

• nonn
• tabl
• more
• hard

Comments

a(26) = T(5,3) is conjectured to be 0, but this has not been proved.

By Fermat's last theorem, T(n,2) = 0 for n > 2.

Euler's sum of powers conjecture is that T(n,k) = 0 for n > k > 1, but this conjecture is not true: T(4,3) = 422481, T(5,4) = 144, there are no known counterexamples for n >= 6.

There are no known 0s for k > 2.

Conjecture: If T(n,k) = 0, then T(r,k) = T(n,s) = T(r,s) = 0 for all r >= n, 2 <= s <= k.

Examples

			Table begins:
  n\k |  1   2       3    4   5   6     7     8
  ----+----------------------------------------
   1  |  1   2       3    4   5   6     7     8
   2  |  1   5       3    2   4   3     4     4
   3  |  1   0       6    7   4   3     5     2
   4  |  1   0  422481  353   5   3     9    13
   5  |  1   0       ?  144  72  12    23    14
   6  |  1   0       ?    ?   ?   ?  1141   251
   7  |  1   0       ?    ?   ?   ?   568   102
   8  |  1   0       ?    ?   ?   ?     ?  1409
T(2,5) = 4 because 4^2 = 1^2 + 1^2 + 1^2 + 2^2 + 3^2 and there is no smaller square that is the sum of 5 positive squares.
T(4,3) = 422481 because 422481^4 = 95800^4 + 217519^4 + 414560^4 and there is no smaller 4th power that is the sum of 3 positive 4th powers.
T(7,7) = 568 because 568^7 = 127^7 + 258^7 + 266^7 + 413^7 + 430^7 + 439^7 + 525^7 and there is no smaller 7th power that is the sum of 7 positive 7th powers.
		

Links

• Ed Pegg Jr., Power Sums, Math Games, November 13 2006.
• James Waldby, A Table of Fifth Powers equal to Sums of Five Fifth Powers, 2009.
• Eric Weisstein's World of Mathematics, Euler's sum of powers conjecture
• Eric Weisstein's World of Mathematics, Diophantine Equation--3rd Powers
• Eric Weisstein's World of Mathematics, Diophantine Equation--4th Powers
• Eric Weisstein's World of Mathematics, Diophantine Equation--5th Powers
• Eric Weisstein's World of Mathematics, Diophantine Equation--6th Powers
• Eric Weisstein's World of Mathematics, Diophantine Equation--7th Powers
• Eric Weisstein's World of Mathematics, Diophantine Equation--8th Powers
• Wikipedia, Euler's sum of powers conjecture

Crossrefs

Cf. A007666 (main diagonal), A264764 (subdiagonal for k = n-1).

Cf. A175610 and A003828 (both for a(19) = T(4,3) = 422481).

Cf. A003294 and A039664 (both for a(25) = T(4,4) = 353).

Cf. A134341 (for a(33) = T(5,4) = 144).

Cf. A063922 and A063923 (both for a(41) = T(5,5) = 72).

Cf. A130012, A130022 (these two sequences are not rows of this table, since they require DISTINCT n-th powers, but this table does not have that requirement).

Programs

PARI

/* return 0 instead of 1 for n=1, and oo loop when T(n, k)=0 */ A347773(p, n, s, m)={ /* Check whether s can be written as sum of n positive p-th powers not larger than m^p. If so, return the base a of the largest term a^p. */ s>n*m^p && return; n==1&&return(ispower(s, p, &n)*n); /* if s and m are not given, s>=n and m are arbitrary. */ !s&&for(m=round(sqrtn(n, p)), 9e9, A347773(p, n, m^p, m-1)&&return(m)); for(a=ceil(sqrtn(s\n, p)), min(sqrtn(max(0, s-n+1), p), m), A347773(p, n-1, s-a^p, a)&&return(a)); } /* after M. F. Hasler in A007666 */ /* Just enter "A347773(n, k)" to get T(n, k) */

Formula

T(n,1) = 1.

T(1,k) = k.

T(n,2) = 0 for n > 2.

T(n,n) = A007666(n).

T(n,n-1) = A264764(n).

T(3,k) <= A130012(k).

T(4,k) <= A130022(k).