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.

16, 55, 120, 433, 378

Offset

2,1

Comments

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.

Links

Table of n, a(n) for n=2..6.

Carlos Rivera, Problem 67. Multigrade Prime Solutions, The Prime Puzzles & Problems Connection.

Chen Shuwen, The Prouhet-Tarry-Escott Problem, Equal Sums of Like Powers.

Formula

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

Example

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)

Prog

(PARI) \\ 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

Crossrefs

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

Sequence in context: A122658 A224040 A244805 * A188838 A256049 A152096

Adjacent sequences: A362036 A362037 A362038 * A362040 A362041 A362042

Keyword

nonn,more

Author

Jean-Marc Rebert, Apr 15 2023

Extensions

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

Edited by Peter Munn, May 01 2023

Status

approved