A perfect number is a number for which the sum of its proper divisors is exactly equal to the number. For example, the sum of the proper divisors of $28$ would be $1 + 2 + 4 + 7 + 14 = 28$, which means that $28$ is a perfect number.
A number $n$ is called abundant if this sum exceeds $n$.
Suppose $n$ was odd. Then $\frac{n}{2}$ is not a divisor, since $\frac{n}{2}$ is not an integer.
The only way I can see it being abundant is if $\frac{n}{3}$ was an integer, and then we got proper divisors below $\frac{n}{3}$ that summed up to be greater than $\frac{2n}{3}$.
Take a look, for example, at the first 1000 abundant numbers:
12 18 20 24 30 36 40 42 48 54
56 60 66 70 72 78 80 84 88 90
96 100 102 104 108 112 114 120 126 132
138 140 144 150 156 160 162 168 174 176
180 186 192 196 198 200 204 208 210 216
220 222 224 228 234 240 246 252 258 260
264 270 272 276 280 282 288 294 300 304
306 308 312 318 320 324 330 336 340 342
348 350 352 354 360 364 366 368 372 378
380 384 390 392 396 400 402 408 414 416
420 426 432 438 440 444 448 450 456 460
462 464 468 474 476 480 486 490 492 498
500 504 510 516 520 522 528 532 534 540
544 546 550 552 558 560 564 570 572 576
580 582 588 594 600 606 608 612 616 618
620 624 630 636 640 642 644 648 650 654
660 666 672 678 680 684 690 696 700 702
704 708 714 720 726 728 732 736 738 740
744 748 750 756 760 762 768 770 774 780
784 786 792 798 800 804 810 812 816 820
822 828 832 834 836 840 846 852 858 860
864 868 870 876 880 882 888 894 896 900
906 910 912 918 920 924 928 930 936 940
942 945 948 952 954 960 966 968 972 978
980 984 990 992 996 1000
How can we get started on proving no odd number can be abundant?
There are odd abundant numbers. The first is $945$ and is in your list.
OEIS: Odd abundant numbers