%I
%S 840,2940,7260,9240,10140,10920,13860,14280,15960,16380,17160,18480,
%T 19320,20580,21420,21840,22440,23100,23940,24024,24360,25080,26040,
%U 26520,27300,28560,29640,30360,30870,31080,31920,32340,34440,34650,35700,35880,36120,36960
%N Numbers that are both unitary pseudoperfect (A293188) and nonunitary pseudoperfect (A327945).
%C All the terms are nonsquarefree (since squarefree numbers do not have nonunitary divisors).
%C All the terms are either 3abundant numbers (A068403) or 3perfect numbers (A005820). None of the 6 known 3perfect numbers are terms of this sequence. If there is a term that is 3perfect, it is also a unitary perfect (A002827) and a nonunitary perfect (A064591).
%H Amiram Eldar, <a href="/A335141/b335141.txt">Table of n, a(n) for n = 1..400</a>
%e 840 is a term since its aliquot unitary divisors are {1, 3, 5, 7, 8, 15, 21, 24, 35, 40, 56, 105, 120, 168, 280} and 1 + 5 + 7 + 8 + 15 + 35 + 40 + 56 + 105 + 120 + 168 + 280 = 840, and its nonunitary divisors are {2, 4, 6, 10, 12, 14, 20, 28, 30, 42, 60, 70, 84, 140, 210, 420} and 70 + 140 + 210 + 420 = 840.
%t pspQ[n_] := Module[{d = Divisors[n], ud, nd, x}, ud = Select[d, CoprimeQ[#, n/#] &]; nd = Complement[d, ud]; ud = Most[ud]; Plus @@ ud >= n && Plus @@ nd >= n && SeriesCoefficient[Series[Product[1 + x^ud[[i]], {i, Length[ud]}], {x, 0, n}], n] > 0 && SeriesCoefficient[Series[Product[1 + x^nd[[i]], {i, Length[nd]}], {x, 0, n}], n] > 0]; Select[Range[10^4], pspQ]
%Y Intersection of A293188 and A327945.
%Y Subsequence of A335140.
%Y Cf. A002827, A005820, A064591, A068403.
%K nonn
%O 1,1
%A _Amiram Eldar_, May 25 2020
