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

1,1

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.

Links

Table of n, a(n) for n=1..25.

Example

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.

Mathematica

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

Prog

(Python)
from itertools import 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

Crossrefs

Subsequence of A083207.

Cf. A000005, A000396, A005820, A334409.

Sequence in context: A007691 A348031 A260508 * A065997 A348035 A354073

Adjacent sequences: A334407 A334408 A334409 * A334411 A334412 A334413

Keyword

nonn,more

Author

Amiram Eldar, Apr 27 2020

Extensions

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

Status

approved