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 A225819 #29 Jan 05 2025 19:51:40 %S A225819 1,2,3,4,6,7,9,10,12,14,16,17,19,21,23,25,27,29,31,33,35,37,39,42,44, %T A225819 46,48,51,53,55,58,60,62,65,67,70,72,75,77,80,82,85,88,90,93,96,98, %U A225819 101,104,106,109,112,115,117,120,123,126,129,132,134,137,140,143,146,149,152,155 %N A225819 Consider the set of n-tuples such that the sum of cubes of the elements is equal to square of their sum; sequence gives largest element in all such tuples. %C A225819 Conjecture [Sen]: lim inf log_n a(n) >= 5/4. %H A225819 Balarka Sen, <a href="/A225819/b225819.txt">Table of n, a(n) for n = 1..500</a> %H A225819 John Mason, <a href="http://www.jstor.org/stable/3620469">Generalising 'sums of cubes equal to squares of sums'</a>, The Mathematical Gazette 85:502 (2001), pp. 50-58. %H A225819 W. R. Utz, The Diophantine Equation (x_1 + x_2 + ... + x_n)^2 = x_1^3 + x_2^3 + ... + x_n^3, Fibonacci Quarterly 15:1 (1977), pp. 14, 16. <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/15-1/utz-a.pdf">Part 1</a>, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/15-1/utz-b.pdf">part 2</a>. %F A225819 n <= a(n) <= n^(4/3), see A158649. %e A225819 Call an n-multiset with the sum of cubes of the elements equal to square of their sum an n-SCESS. %e A225819 a(6) = 7 since the only 6-SCESS with the largest element >= 7 are (2, 4, 4, 5, 5, 7), (3, 3, 3, 3, 5, 7), (3, 4, 5, 5, 6, 7), (3, 5, 5, 5, 6, 7) and (4, 5, 5, 6, 6, 7) and none have an element larger than 7. %e A225819 a(7) = 9 since the only 7-SCESS with the largest element >= 9 are (4, 4, 4, 5, 5, 5, 9), (4, 5, 5, 5, 6, 6, 9) and (6, 6, 6, 6, 6, 6, 9) and none have an element larger than 9. %e A225819 a(8) = 10 since the only 8-SCESS with the largest element >= 10 are (2, 5, 5, 5, 5, 5, 6, 10), (2, 6, 6, 6, 6, 6, 6, 10), (3, 4, 5, 5, 5, 6, 7, 10), (3, 4, 5, 5, 6, 6, 7, 10), (3, 5, 5, 5, 6, 7, 7, 10), (3, 6, 6, 6, 7, 7, 7, 10), (4, 4, 4, 4, 4, 4, 6, 10), (4, 4, 4, 4, 5, 5, 7, 10), (4, 5, 5, 6, 6, 7, 8, 10), (5, 5, 5, 7, 7, 7, 8, 10) and (6, 6, 6, 6, 6, 6, 9, 10) and none have an element larger than 10. %o A225819 (PARI) a(n)=my(v=vector(n, i, 1), N=n^(4/3), m=n); while(v[#v]<N, v[1]++; if(v[1]>N, for(i=2, N, if(v[i]<N, v[i]++; for(j=2, i-1, v[j]=v[i]); v[1]=max(v[i],m); break))); if(sum(i=1, n, v[i]^3)==sum(i=1, n, v[i])^2, m=max(m,v[1])));m %Y A225819 Cf. A158649, A225808. %K A225819 nonn %O A225819 1,2 %A A225819 _Charles R Greathouse IV_, _Jimmy Zotos_, and _Balarka Sen_, Jul 30 2013