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

Author

Naohiro Nomoto, Sep 08 2001

Keywords

• nonn

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

Examples

			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.
		

Links

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

Crossrefs

Cf. A006037, A034683, A293188.

Programs

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 *)

Extensions

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