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)


            NORMAL MODE IS INTEGER
 
            INTERNAL FUNCTION(ND)
            ENTRY TO ODDSUM.
            SUM = 1
            SQN = SQRT.(ND)
            THROUGH CHECK, FOR CN=3, 2, CN.G.SQN
            TM = ND/CN
            WHENEVER TM*CN.E.ND
                SUM = SUM + CN
                WHENEVER TM.NE.CN, SUM = SUM + TM
CHECK       END OF CONDITIONAL
            FUNCTION RETURN SUM
            END OF FUNCTION
 
            SEEN = 0
            NUM = 1
 
            THROUGH SHOW, FOR NUM=1, 2, SEEN.G.1000
            WHENEVER NUM.L.ODDSUM.(NUM)
                SEEN = SEEN + 1
                WHENEVER SEEN.LE.25 .OR. SEEN.E.1000, 
          0         PRINT FORMAT OUTFMT,SEEN,NUM,ODDSUM.(NUM)
SHOW        END OF CONDITIONAL
 
BILION      THROUGH BILION, FOR NUM=NUM, 2, 
          0     NUM.G.1000000000 .AND. NUM.L.ODDSUM.(NUM)
 
            PRINT FORMAT HUGENO,NUM,ODDSUM.(NUM)
 
            VECTOR VALUES OUTFMT = 
          0     $4HNO. ,I4,S1,3HIS ,I6,S1,7HDIVSUM ,I6*$
            VECTOR VALUES HUGENO = 
          0     $25HFIRST ABOVE 1 BILLION IS ,I10,S1,7HDIVSUM ,I10*$
            END OF PROGRAM