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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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

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).

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

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Last modified August 14 07:04 EDT 2020. Contains 336477 sequences. (Running on oeis4.)