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A322658
Integers whose set of proper divisors, excluding 1, can be partitioned into two nonempty subsets having equal sum.
2
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
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
KEYWORD
nonn
AUTHOR
Michel Marcus, Dec 22 2018
STATUS
approved