A064114 Unitary weird numbers: unitary abundant (A034683) but not unitary pseudoperfect (A293188).

70, 4030, 5390, 5830, 10430, 10570, 10990, 11410, 11690, 11830, 12110, 12530, 12670, 13370, 13510, 13790, 13930, 14770, 15610, 15890, 16030, 16310, 16730, 16870, 17010, 17570, 17990, 18410, 18830, 18970, 19390, 19670, 19810, 20230, 20510, 21490, 21770, 21910

Offset

1,1

Comments

Terms that are not (regular) weird (A006037): 5390, 11830, 17010, 20230, 25270, 37030, 51030, 58870, 67270, 93170, 95830, ... - Amiram Eldar, Dec 01 2018

Conjecture: All the terms are divisible by 10 (tested on the first 10^6 terms). - Amiram Eldar, Oct 19 2019

The numbers of terms not exceeding 10^k, for k = 1, 2, ..., are , 0, 1, 1, 4, 205, 1680, 14302, 165369, 1682383, 16326260, ... . Apparently, the asymptotic density of this sequence exists and equals 0.0016... . - Amiram Eldar, Jan 24 2023

Links

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

Example

70 is in the sequence since the sum of its proper unitary divisors, 1, 2, 5, 7, 10, 14, 35 is 74 > 70, yet no subset of these divisors has the sum 74.

Mathematica

udiv[n_] := Select[Divisors[n], GCD[#, n/#] == 1 &]; weirdQ[n_] := Module[{d = Most[udiv[n]]}, If[Total[d] < n, False, c = SeriesCoefficient[Series[Product[1 + x^d[[i]], {i, Length[d]}], {x, 0, n}], n]; c == 0]]; Select[Range[100000], weirdQ] (* Amiram Eldar, Dec 01 2018 *)

Crossrefs

Cf. A006037, A034683, A293188.

Sequence in context: A275295 A306984 A292986 * A354281 A348631 A354283

Adjacent sequences: A064111 A064112 A064113 * A064115 A064116 A064117

Keyword

nonn

Author

Naohiro Nomoto, Sep 08 2001

Extensions

a(25)-a(38) from Amiram Eldar, Dec 01 2018

Status

approved