A087747 -id:A087747 - cp's OEIS frontend

A352305 a(n) is the (conjectured) largest even number that can be expressed as the sum of two distinct primes in exactly n ways.

6, 38, 68, 128, 158, 188, 398, 362, 458, 542, 632, 692, 602, 992, 808, 908, 1112, 1238, 1412, 1418, 1718, 1544, 1574, 1622, 1682, 2048, 2252, 2018, 2672, 2042, 2558, 2936, 2504, 2978, 2966, 3092, 3218, 3242, 3272, 3506, 3632, 3754, 4022, 4058, 4052, 4412, 4448, 4478

Offset: 0

Ilya Gutkovskiy, Mar 11 2022

nonn

			a(5) = 188 because 188 = 7 + 181 = 31 + 157 = 37 + 151 = 61 + 127 = 79 + 109 and it is conjectured that 188 is the last term of A080854.

Cf. A000954, A038609, A061357, A077914, A077969, A078299, A080854, A080862, A087747, A117929, A212613.

More terms from Hugo Pfoertner, Dec 18 2024

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

A365272 a(n) is the least positive integer that can be expressed as the sum of two distinct prime powers (A000961) in exactly n ways.

Original entry on oeis.org

1, 3, 5, 9, 12, 20, 30, 36, 48, 66, 72, 84, 90, 120, 144, 132, 150, 192, 180, 246, 264, 210, 252, 270, 294, 300, 330, 486, 360, 516, 522, 468, 390, 462, 420, 480, 540, 510, 570, 600, 714, 756, 936, 750, 690, 660, 630, 810, 780, 924, 870, 1296, 930, 1122, 1404, 840

Offset: 0

Ilya Gutkovskiy, Sep 07 2023

nonn

			For n = 3: 9 = 1 + 8 = 2 + 7 = 4 + 5.

Cf. A000961, A087747, A225099, A341132.

cp's OEIS Frontend

A352305 a(n) is the (conjectured) largest even number that can be expressed as the sum of two distinct primes in exactly n ways.

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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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A365272 a(n) is the least positive integer that can be expressed as the sum of two distinct prime powers (A000961) in exactly n ways.

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