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

1,1

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)

Links

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

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

Example

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

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 *)

Prog

(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

Crossrefs

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

Sequence in context: A104875 A046391 A339938 * A129485 A360526 A379031

Adjacent sequences: A112640 A112641 A112642 * A112644 A112645 A112646

Keyword

nonn,changed

Author

Labos Elemer, Sep 20 2005

Status

approved