cp's OEIS Frontend

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.

%I A334410 #27 Jun 18 2025 10:10:04
%S A334410 6,28,120,496,672,8128,35640,199584,523776,2142720,12999168,33550336,
%T A334410 459818240,1476304896,2836487808,6039429120,6399679104,8589869056,
%U A334410 36639203328,51001180160,137438691328,266653296000,658470384000,2732372020224,6164773235712
%N 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.
%C A334410 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).
%C A334410 36639203328 is also a term.
%H A334410 <a href="/index/O#opnseqs">Index entries for sequences where odd perfect numbers must occur, if they exist at all</a>
%e A334410 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.
%t A334410 Select[Range[200000], MemberQ[Accumulate[(d = Divisors[#])], (Plus @@ d)/2] &]
%o A334410 (Python)
%o A334410 from itertools import accumulate, count, islice
%o A334410 from sympy import divisors
%o A334410 def A334410_gen(startvalue=1): # generator of terms >= startvalue
%o A334410     for n in count(max(startvalue,1)):
%o A334410         ds = divisors(n)
%o A334410         s = sum(ds)
%o A334410         if s % 2 == 0 and any(2*a==s for a in accumulate(ds)):
%o A334410             yield n
%o A334410 A334410_list = list(islice(A334410_gen(),7)) # _Chai Wah Wu_, Feb 19 2022
%Y A334410 Subsequence of A083207.
%Y A334410 Cf. A000005, A000396, A005820, A334409.
%K A334410 nonn,more
%O A334410 1,1
%A A334410 _Amiram Eldar_, Apr 27 2020
%E A334410 a(19)-a(25) from _Giovanni Resta_, May 08 2020