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

1,1

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.

Links

Table of n, a(n) for n=1..48.

Example

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.

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]

Prog

(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

Crossrefs

A307958 is a subsequence.

Subsequence of A308053.

Cf. A007947, A057723.

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

Sequence in context: A031479 A153642 A119843 * A322657 A364930 A339980

Adjacent sequences: A339976 A339977 A339978 * A339980 A339981 A339982

Keyword

nonn

Author

Amiram Eldar, Dec 25 2020

Status

approved