cp's OEIS Frontend

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

%I A322658 #17 Dec 23 2018 15:58:55
%S A322658 36,72,105,144,195,200,255,288,315,324,345,385,392,400,450,495,525,
%T A322658 576,585,648,675,735,784,800,805,825,855,882,900,945,975,1035,1152,
%U A322658 1155,1295,1296,1305,1323,1365,1395,1425,1449,1463,1485,1547,1568,1575,1600,1645,1665,1755,1764,1785
%N A322658 Integers whose set of proper divisors, excluding 1, can be partitioned into two nonempty subsets having equal sum.
%C A322658 Called half-layered numbers in Behzadipour link.
%H A322658 Alois P. Heinz, <a href="/A322658/b322658.txt">Table of n, a(n) for n = 1..2000</a>
%H A322658 Hussein Behzadipour, <a href="https://arxiv.org/abs/1812.07233">Two-layered numbers</a>, arXiv:1812.07233 [math.NT], 2018.
%e A322658 36 is a term with {2, 3, 4, 18} and B = {6, 9, 12} having equal sums 27.
%p A322658 a:= proc(n) option remember; local k, l, t, b; b:=
%p A322658       proc(m, i) option remember; m=0 or i>0 and
%p A322658         (b(m, i-1) or l[i]<=m and b(m-l[i], i-1)) end;
%p A322658       for k from 1+`if`(n=1, 1, a(n-1)) do
%p A322658         if isprime(k) then next fi;
%p A322658         l:= sort([(numtheory[divisors](k) minus {1, k})[]]);
%p A322658         t:= add(i, i=l);
%p A322658         if t::even then forget(b);
%p A322658           if b(t/2, nops(l)) then return k fi
%p A322658         fi
%p A322658       od
%p A322658     end:
%p A322658 seq(a(n), n=1..60);  # _Alois P. Heinz_, Dec 22 2018
%t A322658 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 *)
%o A322658 (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];
%o A322658 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
%Y A322658 Cf. A083207, A246198, A322657.
%K A322658 nonn
%O A322658 1,1
%A A322658 _Michel Marcus_, Dec 22 2018