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%I #13 Mar 11 2021 03:32:23
%S 3510,3770,5670,5810,6790,7630,7910,9590,9730,544310,740870,2070970,
%T 4017310,4095190,5368510,5569690,5762330,5838770,5855290,5856130,
%U 5887630,5902470,5985770,6006070,6039530,6075370,6083630,6181210,6259610,6471290,7038710,7065730,7285390
%N Unitary pseudoperfect numbers k such that no subset of the nontrivial unitary divisors {d|k : 1 < d < k, gcd(d, k/d) = 1} adds up to k.
%C Numbers that are the sum of a proper subset of their aliquot unitary divisors but are not the sum of any subset of their nontrivial unitary divisors.
%C The unitary perfect numbers (A002827) which are a subset of the unitary pseudoperfect numbers (A293188) are excluded from this sequence since otherwise they would all be trivial terms: if k is a unitary perfect number then the sum of the divisors {d|k : 1 < d < k, gcd(d, k/d) = 1} is k-1, so any subset of them has a sum smaller than k.
%C The unitary pseudoperfect numbers are thus a disjoint union of the unitary perfect numbers, this sequence and A342398.
%C The unitary abundant numbers (A034683) are a disjoint union of the unitary weird numbers (A064114), this sequence and A342398.
%H Amiram Eldar, <a href="/A342399/b342399.txt">Table of n, a(n) for n = 1..206</a>
%e 3510 is a term since it is a unitary pseudoperfect number, 3510 = 1 + 2 + 5 + 13 + 27 + 54 + 65 + 130 + 135 + 270 + 351 + 702 + 1755, and the set of nontrivial unitary divisors of 3510, {d|3510 : 1 < d < 3510, gcd(d, 3510/d) = 1} = {2, 5, 10, 13, 26, 27, 54, 65, 130, 135, 270, 351, 702, 1755}, has no subset that adds up to 3510.
%t q[n_] := Module[{d = Most @ Select[Divisors[n], CoprimeQ[#, n/#] &], x}, Plus @@ d > n && SeriesCoefficient[Series[Product[1 + x^d[[i]], {i, Length[d]}], {x, 0, n}], n] > 0 && SeriesCoefficient[Series[Product[1 + x^d[[i]], {i, 2, Length[d]}], {x, 0, n}], n] == 0]; Select[Range[10^4], q]
%Y The unitary version of A339343.
%Y Subsequence of A034683 and A293188.
%Y Cf. A002827, A064114, A342398.
%K nonn
%O 1,1
%A _Amiram Eldar_, Mar 10 2021