A323395 -id:A323395 - cp's OEIS frontend

A323610 List of 5-powerful numbers (for the definition of k-powerful see A323395).

48, 64, 72, 80, 88, 96, 104, 112, 120, 128, 136, 144, 152, 160, 168, 176, 184, 192, 200, 208, 216, 224, 232, 240, 248, 256, 264, 272, 280, 288, 296, 304, 312, 320, 328, 336, 344, 352, 360, 368, 376, 384, 392, 400, 408, 416, 424, 432, 440, 448, 456, 464, 472, 480, 488, 496, 504

Offset: 1

Stan Wagon, Jan 19 2019

nonn

The sequence consists of the multiples of 8 that are greater than or equal to 64, together with 48. The result is due to D. Boyd, Berend, and Golan. It had been conjectured that the sequence began with 64, but Boyd discovered the set
{1,2,7,10,11,12,13,14,16,17,21,22,27,28,32,33,35,36,37,38,39,42,47,48},
which shows that 48 is 5-powerful.

D. Berend and S. Golan, Littlewood polynomials with high order zeros, Math. Comp. 75 (2006) 1541-1552.
D. Boyd, On a problem of Byrnes concerning polynomials with restricted coefficients, Math. Comp. 66 (1997) 1697-1703.
Stan Wagon, Overview Table

Cf. A323395.

A323614 List of 7-powerful numbers (for the definition of k-powerful see A323395).

Original entry on oeis.org

144, 192, 208, 224, 240, 256, 272, 288, 304, 320, 336, 352, 368, 384, 400, 416, 432, 448, 464, 480, 496, 512, 528, 544, 560, 576, 592, 608, 624, 640, 656, 672, 688, 704, 720, 736, 752, 768, 784, 800, 816, 832, 848, 864, 880, 896, 912, 928, 944, 960

Offset: 1

Stan Wagon, Jan 20 2019

nonn

The sequence consists of the multiples of 16 that are greater than or equal to 192, together with 144. The result is due to S. Golan, R. Pratt, and S. Wagon, who used integer-linear programming to find powerful sets.

S. Golan, R. Pratt, S. Wagon, Equipowerful numbers, to appear

Stan Wagon, Overview table

Join[{144},Range[192,1000,16]] (* Harvey P. Dale, Nov 27 2021 *)

A323629 List of 6-powerful numbers (for the definition of k-powerful see A323395).

Original entry on oeis.org

96, 128, 144, 160, 176, 192, 200, 208, 216, 224, 232, 240, 248, 256, 264, 272, 280, 288, 296, 304, 312, 320, 328, 336, 344, 352, 360, 368, 376, 384, 392, 400, 408, 416, 424, 432, 440, 448, 456, 464, 472, 480, 488, 496, 504, 512, 520, 528, 536

Offset: 1

Stan Wagon, Jan 20 2019

The set consists of 96, 128, 144, 160, 176, and all multiples of 8 that are greater than or equal to 192. The values 200, 216, 232, 248, 264, 280 are by Golan, Pratt, and Wagon; these are sufficient to give all further entries that are 8 (mod 16). Freiman and Litsyn proved that there is some M so that the list beyond M consists of all multiples of 8.

The linked file gives sets proving that all the given values are 6-powerful.

			a(1) = 96 because {1, 2, 7, 10, 11, 12, 13, 14, 16, 17, 21, 22, 27, 28, 32, 33, 35, 36, 37, 38, 39, 42, 47, 48, 51, 52, 53, 54, 56, 57, 63, 66, 67, 68, 71, 72, 73, 74, 77, 78, 79, 82, 88, 89, 91, 92, 93, 94} has the property that the sum of the i-th powers of this set equals the same for its complement in {1, 2, ..., 96}, for each i = 0, 1, 2, 3, 4, 5, 6.

S. Golan, R. Pratt, S. Wagon, Equipowerful numbers, to appear.

Harvey P. Dale, Table of n, a(n) for n = 1..1000
G. Freiman and S. Litsyn, Asymptotically exact bounds on the size of high-order spectral-null codes, IEE Trans. Inform. Theory 45:6 (1999) 1798-1807.
Stan Wagon, Witnessing sets for the 6-powerful numbers
Stan Wagon, Overview table
Index entries for linear recurrences with constant coefficients, signature (2, -1).

Cf. A323614, A323610, A323395.

LinearRecurrence[{2,-1},{96,128,144,160,176,192,200},50] (* Harvey P. Dale, Aug 27 2025 *)

G.f.: -8*x*(x^6+2*x^2+8*x-12)/(x-1)^2. - Alois P. Heinz, Jan 25 2019

More terms added by Stan Wagon, Jan 25 2019

A306248 Smallest m for which 2n is not m-powerful (for the definition of k-powerful see A323395).

Original entry on oeis.org

1, 2, 1, 3, 1, 3, 1, 4, 1, 3, 1, 4, 1, 3, 1, 5, 1, 3, 1, 5, 1, 3, 1, 6, 1, 3, 1, 5, 1, 3, 1, 6, 1, 3, 1, 6, 1, 3, 1, 6, 1, 3, 1, 6, 1, 3, 1, 7, 1, 3, 1, 6, 1, 3, 1, 7, 1, 3, 1, 6, 1, 3, 1, 7, 1, 3, 1, 6, 1, 3, 1, 8, 1, 3, 1, 6, 1, 3, 1, 7, 1, 3, 1, 6, 1, 3, 1, 7, 1, 3, 1, 6, 1, 3, 1

Offset: 1

Stan Wagon, Jan 31 2019

This function is known as m*(2n). For odd n all values of m*(n) are 0.

			The bipartition {1,4}, {2,3} of {1,2,3,4} has equal first power-sums. But there is no such bipartition with equal power-sums for exponents 0, 1, and 2. Therefore a(2) = 2.

S. Golan, Equal moments division of a set, Math. Comp. 77 (2008) 1695-1712.
Stan Wagon, Data for n up to 128, updated Sep 29 2019.

Cf. A323395, A323610, A323629, A323614.

a(56) corrected by Stan Wagon, May 06 2019

a(72) corrected by Stan Wagon, May 24 2019

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

cp's OEIS Frontend

A323610 List of 5-powerful numbers (for the definition of k-powerful see A323395).

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A323614 List of 7-powerful numbers (for the definition of k-powerful see A323395).

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A323629 List of 6-powerful numbers (for the definition of k-powerful see A323395).

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A306248 Smallest m for which 2n is not m-powerful (for the definition of k-powerful see A323395).

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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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