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

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, 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, 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

Offset: 1

Author

Jonathan Sondow, Mar 09 2014

Keywords

• hard
• nonn
• tabf

Comments

Each row begins with 1 or 0. The n-th row has 2n+2 terms.

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).

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.

A multigrade is a solution of the Prouhet-Tarry-Escott problem.

For symmetric and non-symmetric multigrades, see A239067 and A239068.

Examples

			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, 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, 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
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.
		

References

• A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, pp. 162-165.
• L. E. Dickson, History of the theory of numbers, vol. II: Diophantine Analysis, reprint, Chelsea, New York, 1966, pp. 705-716.
• R. K. Guy, Unsolved Problems in Number Theory, D1.
• 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.

Links

• P. Borwein, Computational Excursions in Analysis and Number Theory, Springer, 2002, pp. 85-95.
• Peter Borwein, Petr Lisonek and Colin Percival, Computational investigations of the Prouhet-Tarry-Escott Problem,
• Math. Comp., 72 (2003), 2063-2070.
• T. Piezas III, Equal Sums of Like Powers and the Prouhet-Tarry-Escott (PTE) Problem
• T. Piezas III and E. W. Weisstein, Multigrade Equation, MathWorld
• C. Rivera, Puzzle 65.- Multigrade Relations, The Prime Puzzles & Problems Connection
• Chen Shuwen, Equal Sums of Like Powers
• Chen Shuwen, The Prouhet-Tarry-Escott Problem
• C. Starr, Notes on Listener Crossword 4595 by Elap, The Mathematical Gazette (July 2021), Vol. 105, Issue 563, 291-298.
• Wikipedia, Prouhet-Tarry-Escott problem
• Wikipedia (French), Problème de Prouhet-Tarry-Escott

Crossrefs

Cf. A237434, A237435, A239067, A239068.

Cf. A362039 (for a related problem with sets of primes instead of multisets of positive integers).

Formula

a(n^2 + n - 1) = 1 or 0.