A112643 Odd squarefree abundant numbers.

15015, 19635, 21945, 23205, 25935, 26565, 31395, 33495, 33915, 35805, 39585, 41055, 42315, 42735, 45885, 47355, 49665, 50505, 51765, 54285, 55965, 58695, 61215, 64155, 68145, 70455, 72345, 77385, 80535, 82005, 83265, 84315, 91245

Offset: 1

Author

Labos Elemer, Sep 20 2005

Keywords

• nonn

Comments

Deviates from A046391 (does not contain 36465, 40755 for example).

The numbers of terms not exceeding 10^k, for k = 5, 6, ..., are 34, 134, 1663, 16328, 175630, 1694621, 16726454, ... . Apparently, the asymptotic density of this sequence exists and equals 0.00016... . - Amiram Eldar, Sep 02 2022

From Amiram Eldar, Jan 15 2025: (Start)

The least term that is not divisible by 5 is a(3696) = 22309287.

The least term that is not divisible by 3 is a(5607800) = 33426748355.

The least term that is coprime to 15 is 1357656019974967471687377449. (End)

Examples

			199815 = 3 * 5 * 7 * 11 * 173, with 32 divisors adding up to 400896 = 2 * 199815 + 1266.
		

Links

• Donovan Johnson, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A001221, A005101, A005231, A005408, A046391, A087248.

Programs

Maple

# see A087248 for the additional code
isA112643 := proc(n)
    isA087248(n) and type(n,'odd') ;
end proc:
for n from 1 do
    if isA112643(n) then
        print(n);
    end if;
end do: # R. J. Mathar, Nov 10 2014

Mathematica

ta = {{0}}; Do[g = n; s = DivisorSigma[1, n] - 2 * n; If[Greater[s, 0] && Equal[Abs[MoebiusMu[n]], 1] && !Equal[Mod[n, 2], 0], Print[n, PrimeFactorList[n], s]; ta = Append[ta, n]], {n, 1, 200000}];{ta = Delete[ta, 1], g}(* Elemer *)
Select[Range[1, 99999, 2], MoebiusMu[#] != 0 && DivisorSigma[1, #] > 2 # &] (* Alonso del Arte, Nov 11 2017 *)

PARI

is(n)=if(n%2==0, return(0)); my(f=factor(n)); sigma(f)>2*n && vecmax(f[,2])==1 \\ Charles R Greathouse IV, Feb 21 2017

Formula

A087248 INTERSECT A005408.

omega(a(n)) >= 5, where omega(n) = A001221(n) is the number of distinct primes dividing n. - Amiram Eldar, Jan 15 2025