A335141 Numbers that are both unitary pseudoperfect (A293188) and nonunitary pseudoperfect (A327945).

840, 2940, 7260, 9240, 10140, 10920, 13860, 14280, 15960, 16380, 17160, 18480, 19320, 20580, 21420, 21840, 22440, 23100, 23940, 24024, 24360, 25080, 26040, 26520, 27300, 28560, 29640, 30360, 30870, 31080, 31920, 32340, 34440, 34650, 35700, 35880, 36120, 36960

Offset: 1

Author

Amiram Eldar, May 25 2020

Keywords

• nonn

Comments

All the terms are nonsquarefree (since squarefree numbers do not have nonunitary divisors).

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).

Examples

			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.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..400

Crossrefs

Intersection of A293188 and A327945.

Subsequence of A335140.

Cf. A002827, A005820, A064591, A068403.

Programs

Mathematica

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]