A343153 Factor balanced numbers. See Comments for definition.

12, 16, 35, 36, 39, 49, 292, 315, 1352, 2115, 2116, 4868, 5006, 7264, 7733, 8316, 17063, 17458, 27911, 38556, 55092, 65198, 135650, 138237, 146289, 177478, 378308, 388093, 489155, 703298, 880172, 1050673, 1254720, 1566986, 1716020, 2452849, 3143529, 3418375, 11394618, 14265355, 15479597

Offset: 1

Author

Scott R. Shannon, Apr 07 2021

Keywords

• nonn

Comments

Start by writing 1 and then the ordered prime factors of all subsequent numbers in either of two columns so that the difference of the sum of the numbers is each column is a minimum. The sequence lists the numbers such that on writing the number's final prime factor the two columns' sums are equal. There are only 59 terms less than 500 million.

See A343154 for the column sum when both column sums are equal.

See A343186 for the terms when starting from 2.

Examples

			Start by writing 1 in the left column. Then continue to write the ordered factors of each number in either the left or right column so the difference of the column sums is minimized. The resulting columns are:
.
  Number  |   Factors  |  Left Sum  |  Right Sum
    1            -           1            0
    2            2           1            2
    3            3           4            2
    4           2,2          6            4
    5            5           6            9
    6           2,3         11            9
    7            7          11           16
    8          2,2,2        17           16
    9           3,3         20           19
   10           2,5         25           21
   11           11          25           32
   12          2,2,3        32           32
.
at which point all the factors of 12 have been added and the column sums are equal, so 12 is the first Factor balanced number. Continuing,
.
   13           13          45           32
   14           2,7         45           41
   15           3,5         45           49
   16         2,2,2,2       51           51
.
so 16 is the next Factor balanced number.
See the attached file for the calculations up to the 17th term.
		

Links

• Scott R. Shannon, Calculations for the first 17 terms.

Crossrefs

Cf. A343154, A343186 (starting from 2), A027746, A000040, A001222.