An Abundant number is a number n for which the   sum of divisors   σ(n) > 2n,

or,   equivalently,   the   sum of proper divisors   (or aliquot sum)       s(n) > n.



E.G.

12   is abundant, it has the proper divisors     1,2,3,4 & 6     which sum to   16   ( > 12 or n);

       or alternately,   has the sigma sum of   1,2,3,4,6 & 12   which sum to   28   ( > 24 or 2n).



Abundant numbers are common, though even abundant numbers seem to be much more common than odd abundant numbers.

To make things more interesting, this task is specifically about finding   odd abundant numbers.



Task
Find and display here: at least the first 25 abundant odd numbers and either their proper divisor sum or sigma sum.
Find and display here: the one thousandth abundant odd number and either its proper divisor sum or sigma sum.
Find and display here: the first abundant odd number greater than one billion (109) and either its proper divisor sum or sigma sum.


References

   OEIS:A005231: Odd abundant numbers (odd numbers n whose sum of divisors exceeds 2n)
   American Journal of Mathematics, Vol. 35, No. 4 (Oct., 1913), pp. 413-422 - Finiteness of the Odd Perfect and Primitive Abundant Numbers with n Distinct Prime Factors (LE Dickson)


genit(brk1,brk2,brk3)={tcnt=0;
print("First 25 abundant odd numbers:");
forstep(n=1,999999999999999999,2,
if(tcnt==brk2&&n<brk3,next);
if(sigma(n)<=2*n,next); 
tcnt+=1; 
if(tcnt>brk1&&tcnt<brk2,next);
if(n>=brk3 && sigma(n)>2*n,print("The first odd abundant number greater than 1000000000 is ",n," with sigma = ",sigma(n) );break);
if(tcnt==brk2,print("The 1000th odd abundant number is ",n," with sigma = ",sigma(n) );next);
print(n," with sigma = ",sigma(n)));}

Output:

(11:14) gp > genit(25,1000,1000000000 )
First 25 abundant odd numbers:
945 with sigma = 1920
1575 with sigma = 3224
2205 with sigma = 4446
2835 with sigma = 5808
3465 with sigma = 7488
4095 with sigma = 8736
4725 with sigma = 9920
5355 with sigma = 11232
5775 with sigma = 11904
5985 with sigma = 12480
6435 with sigma = 13104
6615 with sigma = 13680
6825 with sigma = 13888
7245 with sigma = 14976
7425 with sigma = 14880
7875 with sigma = 16224
8085 with sigma = 16416
8415 with sigma = 16848
8505 with sigma = 17472
8925 with sigma = 17856
9135 with sigma = 18720
9555 with sigma = 19152
9765 with sigma = 19968
10395 with sigma = 23040
11025 with sigma = 22971
The 1000th odd abundant number is 492975 with sigma = 1012336
The first odd abundant number greater than 1000000000 is 1000000575 with sigma = 2083561584
(11:24) gp >
