A322657 Integers whose set of divisors, excluding 1, can be partitioned into two nonempty subsets having equal sum.

36, 72, 144, 200, 288, 324, 392, 400, 450, 576, 648, 784, 800, 882, 900, 1152, 1296, 1568, 1600, 1764, 1800, 1936, 2178, 2304, 2450, 2592, 2704, 2916, 3042, 3136, 3200, 3528, 3600, 3872, 4050, 4356, 4608, 4900, 5000, 5184, 5202, 5408, 5832, 6050, 6084, 6272, 6400, 6498

Offset: 1

Author

Michel Marcus, Dec 22 2018

Keywords

• nonn

Comments

Called two-layered numbers in Behzadipour link.

Examples

			36 is a term with {2, 3, 4, 36} and {6, 9, 12, 18} having equal sums 45.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..3000
• Hussein Behzadipour, Two-layered numbers, arXiv:1812.07233 [math.NT], 2018.

Crossrefs

Cf. A083207, A322658.

Subsequence of A028982.

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
        l:= sort([(numtheory[divisors](k) minus {1})[]]);
        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..50);  # Alois P. Heinz, Dec 22 2018

Mathematica

aQ[n_] := Module[{d = Rest[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, 6500], 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), d), s=sum(i=1, #dd, dd[i])); s%2==0 && part(s/2-n, dd[1..#dd-1]); \\ both after pari in A083207