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

1,1

Comments

Called half-layered numbers in Behzadipour link.

Links

Alois P. Heinz, Table of n, a(n) for n = 1..2000

Hussein Behzadipour, Two-layered numbers, arXiv:1812.07233 [math.NT], 2018.

Example

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

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 *)

Prog

(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) && (x<n)), d), s=sum(i=1, #dd, dd[i])); if (#dd, s%2==0 && part(s/2-vecmax(dd), dd[1..#dd-1])); \\ both after pari in A083207

Crossrefs

Cf. A083207, A246198, A322657.

Sequence in context: A363216 A363169 A114127 * A224830 A044102 A043370

Adjacent sequences: A322655 A322656 A322657 * A322659 A322660 A322661

Keyword

nonn

Author

Michel Marcus, Dec 22 2018

Status

approved