A335216 Bi-unitary Zumkeller numbers (A335215) that are not exponentially odd numbers (A268335).

48, 60, 72, 80, 90, 150, 162, 192, 240, 288, 294, 320, 336, 360, 420, 432, 448, 504, 528, 540, 560, 576, 600, 624, 630, 648, 660, 720, 726, 756, 768, 780, 792, 800, 810, 816, 832, 880, 912, 924, 936, 960, 990, 1008, 1014, 1020, 1040, 1050, 1092, 1104, 1134, 1140

Offset

1,1

Comments

Zumkeller numbers (A083207) that are exponentially odd (A268335) are also bi-unitary Zumkeller numbers (A335215), since all of their divisors are bi-unitary.

Links

Table of n, a(n) for n=1..52.

Example

48 is a term since it is not exponentially odd number (48 = 2^4 * 3 and 4 is even), and its set of bi-unitary divisors, {1, 2, 3, 6, 8, 16, 24, 48}, can be partitioned into 2 disjoint sets, whose sum is equal: 1 + 2 + 3 + 8 + 16 + 24 = 6 + 48.

Mathematica

uDivs[n_] := Select[Divisors[n], CoprimeQ[#, n/#] &]; bDivs[n_] := Select[Divisors[n], Last @ Intersection[uDivs[#], uDivs[n/#]] == 1 &]; bzQ[n_] := Module[{d = bDivs[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]]; expOddQ[n_] := AllTrue[Last /@ FactorInteger[n], OddQ]; Select[Range[1000], !expOddQ[#] && bzQ[#] &]

Crossrefs

Subsequence of A335215.

Cf. A083207, A268335, A290466.

Sequence in context: A231469 A261546 A335938 * A114821 A108098 A114505

Adjacent sequences: A335213 A335214 A335215 * A335217 A335218 A335219

Keyword

nonn

Author

Amiram Eldar, May 27 2020

Status

approved