A239066 -id:A239066 - cp's OEIS frontend

A239067 Triangle read by rows: row n lists the smallest positive ideal symmetric 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, 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.

A239068 Triangle read by rows: row n lists the smallest positive ideal non-symmetric multigrade of degree n, or 2n+2 zeros if none.

Original entry on oeis.org

1, 6, 9, 3, 3, 10, 1, 10, 12, 23, 3, 5, 16, 22, 1, 7, 17, 26, 30, 2, 5, 21, 22, 31, 1, 10, 18, 35, 37, 47, 2, 7, 25, 26, 43, 45, 1, 19, 20, 51, 57, 80, 82, 2, 12, 31, 40, 69, 71, 85, 1, 8, 24, 51, 54, 82, 83, 97, 2, 6, 27, 43, 64, 73, 89, 96

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. See A239067.
Any ideal multigrade x1,x2;y1,y2 of degree 1 is symmetric, since x1+x2 = y1+y2. Ideal non-symmetric multigrades are known only for degrees 2,3,4,5,6,7. The ones for degrees 5,6,7 are only conjecturally the smallest ones.

			1, 6, 9; 3, 3, 10
1, 10, 12, 23; 3, 5, 16, 22
1, 7, 17, 26, 30; 2, 5, 21, 22, 31
1, 10, 18, 35, 37, 47; 2, 7, 25, 26, 43, 45
1, 19, 20, 51, 57, 80, 82; 2, 12, 31, 40, 69, 71, 85
1, 8, 24, 51, 54, 82, 83, 97; 2, 6, 27, 43, 64, 73, 89, 96
1, 6, 9; 3, 3, 10 is an ideal non-symmetric multigrade of degree 2 as 1+10 != 6+3 and 1^1 + 6^1 + 9^1 = 16 = 3^1 + 3^1 + 10^1 and 1^2 + 6^2 + 9^2 = 118 = 3^2 + 3^2 + 10^2.

Cf. A239066, A239067.

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

Showing 1-5 of 5 results.

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A239068 Triangle read by rows: row n lists the smallest positive ideal non-symmetric multigrade of degree n, or 2n+2 zeros if none.

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

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

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

A237435 Analog of A237434 for 16-tuples.

Views

Author

Keywords

Comments

Examples

Crossrefs

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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

Extensions