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

36, 72, 144, 180, 200, 252, 288, 324, 360, 392, 396, 400, 468, 504, 576, 600, 612, 648, 684, 720, 784, 792, 800, 828, 900, 936, 1008, 1044, 1116, 1152, 1176, 1200, 1224, 1260, 1296, 1332, 1368, 1400, 1440, 1476, 1548, 1568, 1584, 1600, 1620, 1656, 1692, 1764

Offset: 1

Author

Amiram Eldar, Dec 25 2020

Keywords

• nonn

Comments

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).

The coreful perfect numbers (A307958) are a subsequence.

Examples

			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.
		

Crossrefs

A307958 is a subsequence.

Subsequence of A308053.

Cf. A007947, A057723.

Similar sequences: A083207, A290466, A335197, A335142, A335215, A335218.

Programs

Mathematica

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]

Python

from itertools import count, islice
from sympy import primefactors, divisors
def A339979_gen(startvalue=1): # generator of terms >= startvalue
    for n in count(max(startvalue,1)):
        f = primefactors(n)
        d = [x for x in divisors(n) if primefactors(x)==f]
        s = sum(d)
        if s&1^1 and n<<1<=s:
            d = d[:-1]
            s2, ld = (s>>1)-n, len(d)
            z = [[0 for  in range(s2+1)] for  in range(ld+1)]
            for i in range(1, ld+1):
                y = min(d[i-1], s2+1)
                z[i][:y] = z[i-1][:y]
                for j in range(y,s2+1):
                    z[i][j] = max(z[i-1][j],z[i-1][j-y]+y)
                if z[i][s2] == s2:
                    yield n
                    break
A339979_list = list(islice(A339979_gen(),20)) # Chai Wah Wu, Feb 14 2023