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A334409 Numbers m such that the sum of the first k divisors and the last k divisors of m is equal to 2*m for some k that is smaller than half of the number of divisors of m.

%I A334409 #16 Feb 19 2022 10:04:01
%S A334409 36,152812,6112576,72702928,154286848,397955025,15356519488,
%T A334409 23003680492,35755623784,93789539668,302122464256,351155553970,
%U A334409 1081806148665,1090488143872,1663167899025,2233955122576
%N A334409 Numbers m such that the sum of the first k divisors and the last k divisors of m is equal to 2*m for some k that is smaller than half of the number of divisors of m.
%C A334409 If k is allowed to be equal to half of the number of divisors of m, then the perfect numbers (A000396) will be terms.
%C A334409 a(17) > 10^13. 3021194449732665786499072 is also a term. - _Giovanni Resta_, May 09 2020
%e A334409 36 is a term since its divisors are {1, 2, 3, 4, 6, 9, 12, 18, 36} and the sum of the first 3 and last 3 divisors is (1 + 2 + 3) + (12 + 18 + 36) = 72 = 2 * 36.
%t A334409 seqQ[n_] := Module[{d = Divisors[n]}, nd = Length[d]; nd2 = Ceiling[nd/2] - 1; s = Accumulate[d[[1 ;; nd2]] + n/d[[1 ;; nd2]]]; MemberQ[s, 2*n]]; Select[Range[10^6], seqQ]
%o A334409 (Python)
%o A334409 from itertools import count, islice, accumulate
%o A334409 from sympy import divisors
%o A334409 def A334409_gen(startvalue=1): # generator of terms >= startvalue
%o A334409     for n in count(max(startvalue,1)):
%o A334409         ds = divisors(n)
%o A334409         if any(s==2*n for s in accumulate(ds[i]+ds[-1-i] for i in range((len(ds)-1)//2))):
%o A334409             yield n
%o A334409 A334409_list = list(islice(A334409_gen(),2)) # _Chai Wah Wu_, Feb 19 2022
%Y A334409 Subsequence of A005835 and A334405.
%Y A334409 A variant of A194472 and A318168.
%Y A334409 Cf. A000396, A334410.
%K A334409 nonn,more
%O A334409 1,1
%A A334409 _Amiram Eldar_, Apr 27 2020
%E A334409 a(8)-a(16) from _Giovanni Resta_, May 06 2020