A322657 - OEIS

%I #16 Apr 04 2024 07:55:34

%S 36,72,144,200,288,324,392,400,450,576,648,784,800,882,900,1152,1296,

%T 1568,1600,1764,1800,1936,2178,2304,2450,2592,2704,2916,3042,3136,

%U 3200,3528,3600,3872,4050,4356,4608,4900,5000,5184,5202,5408,5832,6050,6084,6272,6400,6498

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

%C Called two-layered numbers in Behzadipour link.

%H Amiram Eldar, <a href="/A322657/b322657.txt">Table of n, a(n) for n = 1..3000</a>

%H Hussein Behzadipour, <a href="https://arxiv.org/abs/1812.07233">Two-layered numbers</a>, arXiv:1812.07233 [math.NT], 2018.

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

%p a:= proc(n) option remember; local k, l, t, b; b:=

%p       proc(m, i) option remember; m=0 or i>0 and

%p         (b(m, i-1) or l[i]<=m and b(m-l[i], i-1)) end;

%p       for k from 1+`if`(n=1, 1, a(n-1)) do

%p         l:= sort([(numtheory[divisors](k) minus {1})[]]);

%p         t:= add(i, i=l);

%p         if t::even then forget(b);

%p           if b(t/2, nops(l)) then return k fi

%p         fi

%p       od

%p     end:

%p seq(a(n), n=1..50);  # _Alois P. Heinz_, Dec 22 2018

%t 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 *)

%o (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 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

%Y Cf. A083207, A322658.

%Y Subsequence of A028982.

%K nonn

%O 1,1

%A _Michel Marcus_, Dec 22 2018