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A239066 Triangle read by rows: row n lists the smallest positive ideal multigrade of degree n, or 2n+2 zeros if none.

%I A239066 #46 Feb 16 2025 08:33:21
%S A239066 1,3,2,2,1,4,4,2,2,5,1,4,5,8,2,2,7,7,1,5,9,17,18,2,3,11,15,19,1,4,6,
%T A239066 12,14,17,2,2,9,9,16,16,1,19,20,51,57,80,82,2,12,31,40,69,71,85,1,5,
%U A239066 10,24,28,42,47,51,2,3,12,21,31,40,49,50,1,25,31,84,87,134,158,182,198,2,18,42,66,113,116,169,175,199,1,13,126,214,215,413,414,502,615,627,6,7,134,183,243,385,445,494,621,622
%N A239066 Triangle read by rows: row n lists the smallest positive ideal multigrade of degree n, or 2n+2 zeros if none.
%C A239066 Each row begins with 1 or 0. The n-th row has 2n+2 terms.
%C A239066 A "positive multigrade of degree n" and size s is a pair of distinct multisets of positive integers x1 <= x2 <= ... <= xs; y1 <= y2 <= ... <= ys such that x1^k + x2^k + ... + xs^k = y1^k + y2^k + ... + ys^k for k=1,2,...,n. A multigrade is "ideal" if s=n+1 (the smallest possible size for a multigrade of degree n).
%C A239066 Ideal multigrades are known only for degrees < 11 and degree 12. The ideal multigrades of degrees 5,6,7,8,9,10 are only conjecturally the smallest ones.
%C A239066 A multigrade is a solution of the Prouhet-Tarry-Escott problem.
%C A239066 For symmetric and non-symmetric multigrades, see A239067 and A239068.
%D A239066 A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, pp. 162-165.
%D A239066 L. E. Dickson, History of the theory of numbers, vol. II: Diophantine Analysis, reprint, Chelsea, New York, 1966, pp. 705-716.
%D A239066 R. K. Guy, Unsolved Problems in Number Theory, D1.
%D A239066 G. H. Hardy and E. M. Wright, "The Four-Square Theorem" and "The Problem of Prouhet and Tarry: The Number P(k,j)." §20.5 and 21.9 in An Introduction to the Theory of Numbers, 5th ed. Oxford, England: Clarendon Press, pp. 302-306 and 328-329, 1979.
%H A239066 P. Borwein, <a href="https://books.google.com/books?id=A_ITwN13J6YC&amp;q=guises#v=snippet&amp;q=guises&amp;f=false">Computational Excursions in Analysis and Number Theory</a>, Springer, 2002, pp. 85-95.
%H A239066 Peter Borwein, Petr Lisonek and Colin Percival, <a href="https://doi.org/10.1090/S0025-5718-02-01504-1">Computational investigations of the Prouhet-Tarry-Escott Problem</a>,
%H A239066 Math. Comp., 72 (2003), 2063-2070.
%H A239066 T. Piezas III, <a href="https://sites.google.com/view/tpiezas/019-equal-sums-of-like-powers-and-the-prouhet-tarry-escott-pte-problem">Equal Sums of Like Powers and the Prouhet-Tarry-Escott (PTE) Problem</a>
%H A239066 T. Piezas III and E. W. Weisstein, <a href="https://mathworld.wolfram.com/MultigradeEquation.html">Multigrade Equation</a>, MathWorld
%H A239066 C. Rivera, <a href="http://www.primepuzzles.net/puzzles/puzz_065.htm">Puzzle 65.- Multigrade Relations</a>, The Prime Puzzles & Problems Connection
%H A239066 Chen Shuwen, <a href="http://euler.free.fr/eslp/eslp.htm">Equal Sums of Like Powers</a>
%H A239066 Chen Shuwen, <a href="http://euler.free.fr/eslp/TarryPrb.htm">The Prouhet-Tarry-Escott Problem</a>
%H A239066 C. Starr, <a href="https://doi.org/10.1017/mag.2021.61">Notes on Listener Crossword 4595 by Elap</a>, The Mathematical Gazette (July 2021), Vol. 105, Issue 563, 291-298.
%H A239066 Wikipedia, <a href="http://en.wikipedia.org/wiki/Prouhet-Tarry-Escott_problem">Prouhet-Tarry-Escott problem</a>
%H A239066 Wikipedia (French), <a href="http://fr.wikipedia.org/wiki/Problème_de_Prouhet-Tarry-Escott">Problème de Prouhet-Tarry-Escott</a>
%F A239066 a(n^2 + n - 1) = 1 or 0.
%e A239066 1, 3; 2, 2
%e A239066 1, 4, 4; 2, 2, 5
%e A239066 1, 4, 5, 8; 2, 2, 7, 7
%e A239066 1, 5, 9, 17, 18; 2, 3, 11, 15, 19
%e A239066 1, 4, 6, 12, 14, 17; 2, 2, 9, 9, 16, 16
%e A239066 1, 19, 20, 51, 57, 80, 82; 2, 12, 31, 40, 69, 71, 85
%e A239066 1, 5, 10, 24, 28, 42, 47, 51; 2, 3, 12, 21, 31, 40, 49, 50
%e A239066 1, 25, 31, 84, 87, 134, 158, 182, 198; 2, 18, 42, 66, 113, 116, 169, 175, 199
%e A239066 1, 13, 126, 214, 215, 413, 414, 502, 615, 627; 6, 7, 134, 183, 243, 385, 445, 494, 621, 622
%e A239066 1, 4, 4; 2, 2, 5 is an ideal multigrade of degree 2 as 1^1 + 4^1 + 4^1 = 9 = 2^1 + 2^1 + 5^1 and 1^2 + 4^2 + 4^2 = 33 = 2^2 + 2^2 + 5^2.
%Y A239066 Cf. A237434, A237435, A239067, A239068.
%Y A239066 Cf. A362039 (for a related problem with sets of primes instead of multisets of positive integers).
%K A239066 hard,nonn,tabf
%O A239066 1,2
%A A239066 _Jonathan Sondow_, Mar 09 2014