

A334972


Biunitary admirable numbers: numbers k such that there is a proper biunitary divisor d of k such that bsigma(k)  2*d = 2*k, where bsigma is the sum of biunitary divisors function (A188999).


5



24, 30, 40, 42, 48, 54, 56, 66, 70, 78, 80, 88, 102, 104, 114, 120, 138, 150, 162, 174, 186, 222, 224, 246, 258, 270, 282, 294, 318, 354, 360, 366, 402, 420, 426, 438, 448, 474, 498, 534, 540, 582, 606, 618, 630, 642, 654, 660, 672, 678, 720, 726, 762, 780, 786
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OFFSET

1,1


COMMENTS

Equivalently, numbers that are equal to the sum of their proper biunitary divisors, with one of them taken with a minus sign.
Admirable numbers (A111592) that are exponentially odd (A268335) are also biunitary admirable numbers since all of their divisors are biunitary. Terms that are not exponentially odd are 48, 80, 150, 162, 294, 360, 420, 448, 540, 630, 660, 720, 726, 780, 832, 990, ...


LINKS

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


EXAMPLE

48 is in the sequence since 48 = 1 + 2 + 3  6 + 8 + 16 + 24 is the sum of its proper biunitary divisors with one of them, 6, taken with a minus sign.


MATHEMATICA

fun[p_, e_] := If[OddQ[e], (p^(e + 1)  1)/(p  1), (p^(e + 1)  1)/(p  1)  p^(e/2)]; bsigma[1] = 1; bsigma[n_] := Times @@ (fun @@@ FactorInteger[n]); buDivQ[n_, 1] = True; buDivQ[n_, div_] := If[Mod[#2, #1] == 0, Last@Apply[Intersection, Map[Select[Divisors[#], Function[d, CoprimeQ[d, #/d]]] &, {#1, #2/#1}]] == 1, False] & @@ {div, n}; buAdmQ[n_] := (ab = bsigma[n]  2 n) > 0 && EvenQ[ab] && ab/2 < n && Divisible[n, ab/2] && buDivQ[n, ab/2]; Select[Range[1000], buAdmQ]


CROSSREFS

The biunitary version of A111592.
Subsequence of A292982.
Cf. A188999, A222266, A268335, A328328, A334974.
Sequence in context: A068544 A284174 A292982 * A109797 A129656 A334974
Adjacent sequences: A334969 A334970 A334971 * A334973 A334974 A334975


KEYWORD

nonn


AUTHOR

Amiram Eldar, May 18 2020


STATUS

approved



