A322657 -id:A322657 - cp's OEIS frontend

A322658 Integers whose set of proper divisors, excluding 1, can be partitioned into two nonempty subsets having equal sum.

36, 72, 105, 144, 195, 200, 255, 288, 315, 324, 345, 385, 392, 400, 450, 495, 525, 576, 585, 648, 675, 735, 784, 800, 805, 825, 855, 882, 900, 945, 975, 1035, 1152, 1155, 1295, 1296, 1305, 1323, 1365, 1395, 1425, 1449, 1463, 1485, 1547, 1568, 1575, 1600, 1645, 1665, 1755, 1764, 1785

Offset: 1

Michel Marcus, Dec 22 2018

nonn

Called half-layered numbers in Behzadipour link.

			36 is a term with {2, 3, 4, 18} and B = {6, 9, 12} having equal sums 27.

Alois P. Heinz, Table of n, a(n) for n = 1..2000
Hussein Behzadipour, Two-layered numbers, arXiv:1812.07233 [math.NT], 2018.

Cf. A083207, A246198, A322657.

a:= proc(n) option remember; local k, l, t, b; b:=
      proc(m, i) option remember; m=0 or i>0 and
        (b(m, i-1) or l[i]<=m and b(m-l[i], i-1)) end;
      for k from 1+`if`(n=1, 1, a(n-1)) do
        if isprime(k) then next fi;
        l:= sort([(numtheory[divisors](k) minus {1, k})[]]);
        t:= add(i, i=l);
        if t::even then forget(b);
          if b(t/2, nops(l)) then return k fi
        fi
      od
    end:
seq(a(n), n=1..60);  # Alois P. Heinz, Dec 22 2018

aQ[n_] := CompositeQ[n] && Module[{d = Rest[Most[Divisors[n]]], t, ds, x}, ds = Plus @@ d; If[Mod[ds, 2] > 0, False, t = CoefficientList[Product[1 + x^i, {i, d}], x]; t[[1 + ds/2]] > 0]]; Select[Range[2, 1785], aQ]  (* Amiram Eldar, Dec 22 2018 after T. D. Noe at A083207 *)

part(n, v)=if(n<1, return(n==0)); forstep(i=#v, 2, -1, if(part(n-v[i], v[1..i-1]), return(1))); n==v[1];
is(n)=my(d=divisors(n), dd = select(x->((x>1) && (xA083207

A382953 Numbers with at least one factorization for which the factors can be partitioned into 2 or more distinct subsets with equal sums.

Original entry on oeis.org

16, 30, 48, 54, 64, 70, 72, 84, 96, 120, 126, 128, 144, 160, 162, 180, 192, 198, 210, 216, 240, 243, 250, 252, 256, 264, 270, 280, 286, 288, 300, 308, 320, 324, 330, 336, 360, 378, 384, 390, 396, 400, 420, 432, 440, 448, 462, 468, 480, 486, 495, 504, 510, 512

Offset: 1

Charles L. Hohn, Apr 09 2025

nonn

Here "distinct" means that no partition contains the same subset of factors, e.g. 4 is not a term because {2} == {2}.
Because 2 + 2 = 2 * 2 = 4, many terms have multiple instances that differ only by factors {2, 2} vs. {4}, except in some cases where such substitutions would create indistinct subsets, e.g. while 16 is a term for partition set {{2, 2}, {4}}, {{2, 2}, {2, 2}} and {{4}, {4}} do not count as additional instances.
For primes p and integers x >= 0, p^(p+2+2x) and p^(2p+3+x) are terms.
For integers x and y >= 0, (4x+4)^(y+2) and (4x+6)^(y+3) are terms.
First few terms with record counts of instances: 16 (1 instance), 48 (2), 120 (3), 240 (6), 576 (8), 720 (9), 768 (12).
If k is a term, then 4k is also a term. - Ivan N. Ianakiev, Apr 10 2025

			a(1) = 16: 2 * 2 * 4 = 16 and 2 + 2 = 4.
a(2) = 30: 2 * 3 * 5 = 30 and 2 + 3 = 5.
a(3) = 48: 2 * 2 * 2 * 6 = 48 and 2 + 2 + 2 = 6, and also 2 * 4 * 6 = 48 and 2 + 4 = 6.
a(5) = 64: 2 * 2 * 2 * 2 * 4 = 64 and 2 + 2 + 2 = 2 + 4.
a(39) = 384: 2 * 2 * 2 * 2 * 4 * 6 = 384 and 2 + 2 + 2 = 2 + 4 = 6 (plus 4 other instances).

Cf. A083207, A322657, A255265 (subsequence).

ok[n_]:=Catch@ Block[{t, d=Divisors@n,f}, f[y_]:=Block[{L={}, r}, r[x_,m_,c_]:= If[x==1, AppendTo[L,c], r[x/#, #, Append[c,#]]& /@ Select[ Divisors@x, #>=m&];]; f[y,2,{}]; L]; Do[t=Plus@@@ s[d[[i]]]; If[d[[i]]^2!=n, Intersection[t, Plus@@@ s[n/d[[i]]]] != {} && Throw@True, Sort@t != Union@t && Throw@True],{i, 2, Ceiling[ Length@d/2]}]; False]; Select[Range@ 512,ok] (* Giovanni Resta, Apr 10 2025 *)

a382953_count(x, f=List())={my(r=x/if(#f, vecprod(Vec(f)), 1)); if(#f && r==1, my(c=0, s=vecsum(Vec(f)), d=divisors(s)); for(i=2, #d, my(z=s/d[i]); if(z1, next); listput(f, d); c+=a382953_count(x, f); listpop(f)); return(c)}
a382953_part(f, z, rvs=0, v=List())={my(c=0); if(#v==#f[2], if(sum(i=1, #v, f[1][i]*v[i])

cp's OEIS Frontend

A322658 Integers whose set of proper divisors, excluding 1, can be partitioned into two nonempty subsets having equal sum.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI