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

1,1

Comments

Called two-layered numbers in Behzadipour link.

Links

Amiram Eldar, Table of n, a(n) for n = 1..3000

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

Example

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

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 *)

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

Crossrefs

Cf. A083207, A322658.

Subsequence of A028982.

Sequence in context: A153642 A119843 A339979 * A364930 A339980 A347892

Adjacent sequences: A322654 A322655 A322656 * A322658 A322659 A322660

Keyword

nonn

Author

Michel Marcus, Dec 22 2018

Status

approved