Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%I #9 Jun 09 2020 03:31:26
%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 3-abundant numbers (A068403) or 3-perfect numbers (A005820). None of the 6 known 3-perfect numbers are terms of this sequence. If there is a term that is 3-perfect, 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