

A335141


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


1



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
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OFFSET

1,1


COMMENTS

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


LINKS

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


EXAMPLE

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.


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]


CROSSREFS

Intersection of A293188 and A327945.
Subsequence of A335140.
Cf. A002827, A005820, A064591, A068403.
Sequence in context: A232099 A005952 A260473 * A177021 A276161 A135038
Adjacent sequences: A335137 A335138 A335140 * A335142 A335143 A335144


KEYWORD

nonn


AUTHOR

Amiram Eldar, May 25 2020


STATUS

approved



