A335140 -id:A335140 - cp's OEIS frontend

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

Amiram Eldar, May 25 2020

nonn

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).

			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.

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

Intersection of A293188 and A327945.
Subsequence of A335140.
Cf. A002827, A005820, A064591, A068403.

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]

A335201 Unitary Zumkeller numbers (A290466) that are not squarefree.

Original entry on oeis.org

60, 90, 150, 294, 420, 630, 660, 726, 750, 780, 840, 924, 990, 1014, 1020, 1050, 1092, 1140, 1170, 1380, 1386, 1428, 1470, 1530, 1596, 1638, 1650, 1710, 1734, 1740, 1860, 1890, 1950, 2058, 2070, 2142, 2166, 2220, 2460, 2550, 2580, 2610, 2790, 2820, 2850, 2940

Offset: 1

Amiram Eldar, May 26 2020

nonn

Zumkeller numbers (A083207) that are squarefree (A005117) are also unitary Zumkeller numbers (A290466), since all of their divisors are unitary.

First differs from A335140 at n = 39.

			60 is a term since it is nonsquarefree, and its unitary divisors, {1, 3, 4, 5, 12, 15, 20, 60}, can be partitioned into 2 disjoint sets whose sum is equal: 1 + 3 + 4 + 5 + 12 + 15 + 20 = 60.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Intersection of A013929 and A290466.

Cf. A005117, A083207.

uzQ[n_] :=  Module[{d = Select[Divisors[n], CoprimeQ[#, n/#] &], sum, x}, sum = Plus @@ d; If[sum < 2*n || OddQ[sum], False, CoefficientList[Product[1 + x^i, {i, d}], x][[1 + sum/2]] > 0]]; Select[Range[3000], !SquareFreeQ[#] && uzQ[#] &]

cp's OEIS Frontend

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

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