This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A347773 #13 Sep 16 2021 02:36:03
%S A347773 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
%N 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.
%C A347773 a(26) = T(5,3) is conjectured to be 0, but this has not been proved.
%C A347773 By Fermat's last theorem, T(n,2) = 0 for n > 2.
%C A347773 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.
%C A347773 There are no known 0s for k > 2.
%C A347773 Conjecture: If T(n,k) = 0, then T(r,k) = T(n,s) = T(r,s) = 0 for all r >= n, 2 <= s <= k.
%H A347773 Ed Pegg Jr., <a href="https://web.archive.org/web/20080410224256/http://www.maa.org/editorial/mathgames/mathgames_11_13_06.html">Power Sums</a>, Math Games, November 13 2006.
%H A347773 James Waldby, <a href="https://pat7.com/jp/s515-10007-t">A Table of Fifth Powers equal to Sums of Five Fifth Powers</a>, 2009.
%H A347773 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/EulersSumofPowersConjecture.html">Euler's sum of powers conjecture</a>
%H A347773 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/DiophantineEquation3rdPowers.html">Diophantine Equation--3rd Powers</a>
%H A347773 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/DiophantineEquation4thPowers.html">Diophantine Equation--4th Powers</a>
%H A347773 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/DiophantineEquation5thPowers.html">Diophantine Equation--5th Powers</a>
%H A347773 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/DiophantineEquation6thPowers.html">Diophantine Equation--6th Powers</a>
%H A347773 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/DiophantineEquation7thPowers.html">Diophantine Equation--7th Powers</a>
%H A347773 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/DiophantineEquation8thPowers.html">Diophantine Equation--8th Powers</a>
%H A347773 Wikipedia, <a href="https://en.wikipedia.org/wiki/Euler%27s_sum_of_powers_conjecture">Euler's sum of powers conjecture</a>
%F A347773 T(n,1) = 1.
%F A347773 T(1,k) = k.
%F A347773 T(n,2) = 0 for n > 2.
%F A347773 T(n,n) = A007666(n).
%F A347773 T(n,n-1) = A264764(n).
%F A347773 T(3,k) <= A130012(k).
%F A347773 T(4,k) <= A130022(k).
%e A347773 Table begins:
%e A347773 n\k | 1 2 3 4 5 6 7 8
%e A347773 ----+----------------------------------------
%e A347773 1 | 1 2 3 4 5 6 7 8
%e A347773 2 | 1 5 3 2 4 3 4 4
%e A347773 3 | 1 0 6 7 4 3 5 2
%e A347773 4 | 1 0 422481 353 5 3 9 13
%e A347773 5 | 1 0 ? 144 72 12 23 14
%e A347773 6 | 1 0 ? ? ? ? 1141 251
%e A347773 7 | 1 0 ? ? ? ? 568 102
%e A347773 8 | 1 0 ? ? ? ? ? 1409
%e A347773 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.
%e A347773 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.
%e A347773 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.
%o A347773 (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) */
%Y A347773 Cf. A007666 (main diagonal), A264764 (subdiagonal for k = n-1).
%Y A347773 Cf. A175610 and A003828 (both for a(19) = T(4,3) = 422481).
%Y A347773 Cf. A003294 and A039664 (both for a(25) = T(4,4) = 353).
%Y A347773 Cf. A134341 (for a(33) = T(5,4) = 144).
%Y A347773 Cf. A063922 and A063923 (both for a(41) = T(5,5) = 72).
%Y A347773 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).
%K A347773 nonn,tabl,more,hard
%O A347773 1,2
%A A347773 _Eric Chen_, Sep 15 2021