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A354283 Weird numbers k such that k-1 and k+1 are both sums of subsets of the aliquot divisors of k.

%I A354283 #8 May 24 2022 06:14:32
%S A354283 70,4030,5830,17272,243892,4199030,11339816,11547352,13885970,
%T A354283 24450010,31699430,32284330,34041370,34169630,42251930,50761810,
%U A354283 67727110,67820390,85389368,89283592,141659096,146764264,162079768,173482552,259858324,410832532,411643576,486224072
%N A354283 Weird numbers k such that k-1 and k+1 are both sums of subsets of the aliquot divisors of k.
%C A354283 There are 17270452 weird numbers below 10^10 and only 42 are in this sequence.
%H A354283 Amiram Eldar, <a href="/A354283/b354283.txt">Table of n, a(n) for n = 1..42</a>
%e A354283 70 is a term since it is a weird number, its aliquot divisors are {1, 2, 5, 7, 10, 14, 35}, 69 = 1 + 2 + 7 + 10 + 14 + 35, and 71 = 5 + 7 + 10 + 14 + 35.
%t A354283 q[n_] := Module[{d = Most @ Divisors[n], x, s, c}, If[Plus @@ d <= n, False, s = Series[Product[1 + x^d[[i]], {i, Length[d]}], {x, 0, n + 1}]; c = SeriesCoefficient[s, #] & /@ (n + {-1, 0, 1}); c[[1]] > 0 && c[[2]] == 0 && c[[3]] > 0]]; Select[Range[10000], q]
%o A354283 (PARI) is(n, d=divisors(n)[^-1], s=vecsum(d))={s>n && !is_A005835(n, d, s) && is_A005835(n-1, d, s) && is_A005835(n+1, d, s)}; \\ using is_A005835() by _M. F. Hasler_ at A005835
%Y A354283 Intersection of A354281 and A354282.
%Y A354283 Subsequence of A006037.
%Y A354283 Cf. A005835.
%K A354283 nonn
%O A354283 1,1
%A A354283 _Amiram Eldar_, May 22 2022