A334410 Numbers m such that the sum of the first k divisors of m, for some k, is equal to the sum of its other divisors.

6, 28, 120, 496, 672, 8128, 35640, 199584, 523776, 2142720, 12999168, 33550336, 459818240, 1476304896, 2836487808, 6039429120, 6399679104, 8589869056, 36639203328, 51001180160, 137438691328, 266653296000, 658470384000, 2732372020224, 6164773235712

Offset: 1

Author

Amiram Eldar, Apr 27 2020

Keywords

• nonn
• more

Comments

Includes all the perfect numbers (A000396), for them k = d(m) - 1 and the even 3-perfect numbers (A005820), for them k = d(m) - 2 (where d(m) = A000005(m) is the number of divisors of m).

36639203328 is also a term.

Examples

			6 is a term since its set of divisors, {1, 2, 3, 6}, can be partitioned into two disjoint sets with equal sum, {1, 2, 3} and {6}, such that the first 3 divisors are in the first set.
		

Links

• Index entries for sequences where odd perfect numbers must occur, if they exist at all

Crossrefs

Subsequence of A083207.

Cf. A000005, A000396, A005820, A334409.

Programs

Mathematica

Select[Range[200000], MemberQ[Accumulate[(d = Divisors[#])], (Plus @@ d)/2] &]

Python

from itertools import accumulate, count, islice
from sympy import divisors
def A334410_gen(startvalue=1): # generator of terms >= startvalue
    for n in count(max(startvalue,1)):
        ds = divisors(n)
        s = sum(ds)
        if s % 2 == 0 and any(2*a==s for a in accumulate(ds)):
            yield n
A334410_list = list(islice(A334410_gen(),7)) # Chai Wah Wu, Feb 19 2022

Extensions

a(19)-a(25) from Giovanni Resta, May 08 2020