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

1,2

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.

Links

Table of n, a(n) for n=1..25.

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

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).

Example

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.

Prog

(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) */

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).

Sequence in context: A062105 A210563 A236311 * A368563 A093412 A119355

Adjacent sequences: A347770 A347771 A347772 * A347774 A347775 A347776

Keyword

nonn,tabl,more,hard

Author

Eric Chen, Sep 15 2021

Status

approved