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A339979 Coreful Zumkeller numbers: numbers whose set of coreful divisors can be partitioned into two disjoint sets of equal sum.

%I A339979 #11 Feb 16 2023 05:13:38
%S A339979 36,72,144,180,200,252,288,324,360,392,396,400,468,504,576,600,612,
%T A339979 648,684,720,784,792,800,828,900,936,1008,1044,1116,1152,1176,1200,
%U A339979 1224,1260,1296,1332,1368,1400,1440,1476,1548,1568,1584,1600,1620,1656,1692,1764
%N A339979 Coreful Zumkeller numbers: numbers whose set of coreful divisors can be partitioned into two disjoint sets of equal sum.
%C A339979 A coreful divisor d of a number k is a divisor with the same set of distinct prime factors as k, or rad(d) = rad(k), where rad(k) is the largest squarefree divisor of k (A007947).
%C A339979 The coreful perfect numbers (A307958) are a subsequence.
%e A339979 36 is a term since its set of coreful divisors, {6, 12, 18, 36}, can be partitioned into the two disjoint sets, {6, 12, 18} and {36}, whose sums are equal: 6 + 12 + 18 = 36.
%t A339979 corZumQ[n_] := Module[{r = Times @@ FactorInteger[n][[;; , 1]], d, sum, x}, d = r * Divisors[n/r]; (sum = Plus @@ d) >= 2*n && EvenQ[sum] && CoefficientList[Product[1 + x^i, {i, d}], x][[1 + sum/2]] > 0]; Select[Range[1800], corZumQ]
%o A339979 (Python)
%o A339979 from itertools import count, islice
%o A339979 from sympy import primefactors, divisors
%o A339979 def A339979_gen(startvalue=1): # generator of terms >= startvalue
%o A339979     for n in count(max(startvalue,1)):
%o A339979         f = primefactors(n)
%o A339979         d = [x for x in divisors(n) if primefactors(x)==f]
%o A339979         s = sum(d)
%o A339979         if s&1^1 and n<<1<=s:
%o A339979             d = d[:-1]
%o A339979             s2, ld = (s>>1)-n, len(d)
%o A339979             z = [[0 for _ in range(s2+1)] for _ in range(ld+1)]
%o A339979             for i in range(1, ld+1):
%o A339979                 y = min(d[i-1], s2+1)
%o A339979                 z[i][:y] = z[i-1][:y]
%o A339979                 for j in range(y,s2+1):
%o A339979                     z[i][j] = max(z[i-1][j],z[i-1][j-y]+y)
%o A339979                 if z[i][s2] == s2:
%o A339979                     yield n
%o A339979                     break
%o A339979 A339979_list = list(islice(A339979_gen(),20)) # _Chai Wah Wu_, Feb 14 2023
%Y A339979 A307958 is a subsequence.
%Y A339979 Subsequence of A308053.
%Y A339979 Cf. A007947, A057723.
%Y A339979 Similar sequences: A083207, A290466, A335197, A335142, A335215, A335218.
%K A339979 nonn
%O A339979 1,1
%A A339979 _Amiram Eldar_, Dec 25 2020