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

Author

Michel Marcus, Dec 22 2018

Keywords

• nonn

Comments

Called half-layered numbers in Behzadipour link.

Examples

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

Links

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

Crossrefs

Cf. A083207, A246198, A322657.

Programs

Maple

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

Mathematica

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 *)

PARI

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