A239068 -id:A239068 - cp's OEIS frontend

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

Jonathan Sondow, Mar 09 2014

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.

			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.

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.

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

Cf. A237434, A237435, A239067, A239068.

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

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

A239067 Triangle read by rows: row n lists the smallest positive ideal symmetric multigrade of degree n, or 2n+2 zeros if none.

Original entry on oeis.org

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, 28, 59, 65, 90, 102, 2, 14, 39, 45, 76, 85, 103, 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

Jonathan Sondow, Mar 10 2014

The main entry for this topic is A239066.
A multigrade x1<=x2<=...<=xs; y1<=y2<=...<=ys is "symmetric" if x1+ys = x2+y(s-1) = ... = xs+y1 when s is odd, or x1+xs = x2+x(s-1) = ... = x(s/2)+x((s/2)+1) = y1+ys = y2+y(s-1) = ... = y(s/2)+y((s/2)+1) when s is even. For non-symmetric ones, see A239068.
The ideal symmetric multigrades of degrees 5,6,7,8,9,10 are only conjecturally the smallest ones.

			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, 28, 59, 65, 90, 102; 2, 14, 39, 45, 76, 85, 103
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 symmetric multigrade of degree 2 as 1+5 = 4+2 = 4+2 and 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.
1, 4, 5, 8; 2, 2, 7, 7 is an ideal symmetric multigrade of degree 3 as 1+8 = 4+5 = 2+7 = 2+7 and 1^1 + 4^1 + 5^1 + 8^1 = 18 = 2^1 + 2^1 + 7^1 + 7^1 and 1^2 + 4^2 + 5^2 + 8^2 = 106 = 2^2 + 2^2 + 7^2 + 7^2 and 1^3 + 4^3 + 5^3 + 8^3 = 702 = 2^3 + 2^3 + 7^3 + 7^3.

C. Starr, Notes on Listener Crossword 4595 by Elap, The Mathematical Gazette (July 2021), Vol. 105, Issue 563, 291-298.

Cf. A239066, A239068.

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

A237434 Primitive, symmetric octuples of distinct numbers a,b,c,d,x,y,z,w with 0

Original entry on oeis.org

1, 5, 8, 12, 2, 3, 10, 11, 1, 8, 10, 17, 2, 5, 13, 16, 1, 10, 12, 23, 3, 5, 16, 22

Offset: 1

Jonathan Sondow, Feb 07 2014

If a,b,c,d,x,y,z,w satisfies the (in)equalities in the definition, then so does the translate a-t,b-t,c-t,d-t,x-t,y-t,z-t,w-t, for t

Bennett, Minculete, and Tetiva show that there do not exist distinct numbers a,b,c,x,y,z with 0

In this 6-term multigrade problem, if the restriction a<=x

			1 + 5 + 8 + 12 = 26 = 2 + 3 + 10 + 11.
1^2 + 5^2 + 8^2 + 12^2 = 234 = 2^2 + 3^2 + 10^2 + 11^2.
1^3 + 5^3 + 8^3 + 12^3 = 2366 = 2^3 + 3^3 + 10^3 + 11^3.
1 + 12 = 5 + 8 = 2 + 11 = 3 + 10 = 13.

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.

Grahame Bennett, Nicuşor Minculete, and Marian Tetiva, Problem 11635: When Sums of Powers Determine Their Terms, Amer. Math. Monthly, 121 (2014), p. 89.
Tito Piezas III, Equal Sums of Like Powers and the Prouhet-Tarry-Escott (PTE) Problem
Tito Piezas III and Eric W. Weisstein, Multigrade Equation, MathWorld
Carlos Rivera, Puzzle 65.- Multigrade Relations, The Prime Puzzles & Problems Connection
Chen Shuwen, Equal Sums of Like Powers

Cf. A237435, A239066, A239067, A239068.

A237435 Analog of A237434 for 16-tuples.

Original entry on oeis.org

1, 5, 10, 24, 28, 42, 47, 51, 2, 3, 12, 21, 31, 40, 49, 50

Offset: 1

Jonathan Sondow, Feb 08 2014

See A237434 for comments, references, and links.

			1^k + 5^k + 10^k + 24^k + 28^k + 42^k + 47^k + 51^k =
2^k + 3^k + 12^k + 21^k + 31^k + 40^k + 49^k + 50^k =
208, 8060, 347464, 15713636, 730337608, 34538543780, 1652662609624,
for k = 1,2,3,4,5,6,7, respectively.
1+51 = 5+47 = 10+42 = 24+28 = 2+50 = 3+49 = 12+40 = 21+31 = 52.

Cf. A237434, A239066, A239067, A239068.

A362039 Least number s such that there are 2 different sets of primes { a1, a2, ..., an } and { b1, b2, ..., bn } with the integers in each set having the same sum s, the same sum of squares, etc., up to and including the same sum of (n-1)-st powers.

Original entry on oeis.org

16, 55, 120, 433, 378

Offset: 2

Jean-Marc Rebert, Apr 15 2023

We are to find the least number s such that there is a solution in primes to the system of equations:
a1^k + a2^k + ... + an^k = b1^k + b2^k + ... + bn^k, (k = 1, 2, ..., n-1) and {a1, ..., an} != {b1, ..., bn}.
a(7), a(8) are respectively <= 2399, 348592.

			a(2) = 16, because 3 + 13 = 16 = 5 + 11 and no lesser sum of 2 distinct primes has this property.
a(3) = 55, because 7 + 19 + 29 = 55 = 11 + 13 + 31 and 7^2 + 19^2 + 29^2 = 1251 = 11^2 + 13^2 + 31^2, and no lesser sum of 3 distinct primes has this property.
a(4) = 120, because with u = [13, 29, 31, 47] and v = [17, 19, 41, 43], Sum_{i=1..4} u(i) = 120  = Sum_{i=1..4} v(i) and Sum_{i=1..4} u(i)^2 = 4100 = Sum_{i=1..4} v(i)^2 and Sum_{i=1..4} u(i)^3 = 1602000 = Sum_{i=1..4} v(i)^3 and no lesser sum of 4 distinct primes has this property.
From _Andrew Howroyd_, Apr 18 2023: (Start)
a(5) = 433 with {13, 59, 67, 131, 163} and {23, 31, 103, 109, 167}.
a(6) = 378 with {17, 37, 43, 83, 89, 109} and {19, 29, 53, 73, 97, 107}.
(End)

Carlos Rivera, Problem 67. Multigrade Prime Solutions, The Prime Puzzles & Problems Connection.
Chen Shuwen, The Prouhet-Tarry-Escott Problem, Equal Sums of Like Powers.

Cf. A087747, A117929, A239066, A239067, A239068, A323395.

\\ Call with pr=1 to also print solution sets.
a(n, pr=0)={
  forstep(s=3*n, oo, 2, my(P=vector(s,i,primepi(i)), X=primes(P[s]));
    local(found=0, M=Map(), V=vector(n));
    my(onSet()=my(key=vector(n-2, j, sum(i=1, n, V[i]^(j+1))), z);
      if(mapisdefined(M,key,&z), found++; if(pr, print(V, z)), mapput(M,key,V)));
    my(recurse(r,m,k)=if(k==0, onSet(), for(m=max(k,P[(r-1)\k])+1, min(m, P[r-3*(k-1)]), V[k]=X[m]; self()(r-X[m], m-1, k-1)) ));
    recurse(s, #X, n);
    if(found, return(s));
  )
} \\ Andrew Howroyd, Apr 18 2023

a(2) = min({k >= 1 : A117929(k) >= 2}) = Min_{m >= 2} A087747(m) = A087747(2). - Peter Munn, May 01 2023

a(5)-a(6) from Andrew Howroyd, Apr 18 2023

Edited by Peter Munn, May 01 2023

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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.

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A239067 Triangle read by rows: row n lists the smallest positive ideal symmetric multigrade of degree n, or 2n+2 zeros if none.

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A237434 Primitive, symmetric octuples of distinct numbers a,b,c,d,x,y,z,w with 0

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A237435 Analog of A237434 for 16-tuples.

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A362039 Least number s such that there are 2 different sets of primes { a1, a2, ..., an } and { b1, b2, ..., bn } with the integers in each set having the same sum s, the same sum of squares, etc., up to and including the same sum of (n-1)-st powers.

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