Theorem: Abundancy of Integers in form 945 + 630n

A large number of integers of the form $945 + 630 n$, for $n \in \Z_{\ge 0}$, are abundant.
The first counterexample is for $n = 52$.
{{WIP|Add this sequence to the number pages}}

Proof:
{{begin-eqn}}
{{eqn | n = n = 0
      | l = \map {\sigma_1} {945} - 945
      | r = 1920 - 945
      | c = {{DSFLink|945}}
}}
{{eqn | r = 975
      | c = 
}}
{{eqn | n = n = 1
      | l = \map {\sigma_1} {1575} - 1575
      | r = 3224 - 1575
      | c = {{DSFLink|1575}}
}}
{{eqn | r = 1649
      | c = 
}}
{{eqn | n = n = 2
      | l = \map {\sigma_1} {2205} - 2205
      | r = 4446 - 2205
      | c = {{DSFLink|2205}}
}}
{{eqn | r = 2241
      | c = 
}}
{{eqn | n = n = 3
      | l = \map {\sigma_1} {2835} - 2835
      | r = 5808 - 2835
      | c = {{DSFLink|2835}}
}}
{{eqn | r = 2973
      | c = 
}}
{{eqn | n = n = 4
      | l = \map {\sigma_1} {3465} - 3465
      | r = 7488 - 3465
      | c = {{DSFLink|3465}}
}}
{{eqn | r = 4023
      | c = 
}}
{{eqn | n = n = 5
      | l = \map {\sigma_1} {4095} - 4095
      | r = 8736 - 4095
      | c = {{DSFLink|4095}}
}}
{{eqn | r = 4641
      | c = 
}}
{{eqn | n = n = 6
      | l = \map {\sigma_1} {4725} - 4725
      | r = 9920 - 4725
      | c = {{DSFLink|4725}}
}}
{{eqn | r = 5195
      | c = 
}}
{{eqn | n = n = 7
      | l = \map {\sigma_1} {5355} - 5355
      | r = 11 \, 232 - 5355
      | c = {{DSFLink|5355}}
}}
{{eqn | r = 5877
      | c = 
}}
{{eqn | n = n = 8
      | l = \map {\sigma_1} {5985} - 5985
      | r = 12 \, 480 - 5985
      | c = {{DSFLink|5985}}
}}
{{eqn | r = 6495
      | c = 
}}
{{eqn | n = n = 9
      | l = \map {\sigma_1} {6615} - 6615
      | r = 13 \, 680 - 6615
      | c = {{DSFLink|6615}}
}}
{{eqn | r = 7065
      | c = 
}}
{{eqn | n = n = 10
      | l = \map {\sigma_1} {7245} - 7245
      | r = 14 \, 976 - 7245
      | c = {{DSFLink|7245}}
}}
{{eqn | r = 7731
      | c = 
}}
{{eqn | n = n = 11
      | l = \map {\sigma_1} {7875} - 7875
      | r = 16 \, 224 - 7875
      | c = {{DSFLink|7875}}
}}
{{eqn | r = 8349
      | c = 
}}
{{eqn | n = n = 12
      | l = \map {\sigma_1} {8505} - 8505
      | r = 17 \, 472 - 8505
      | c = {{DSFLink|8505}}
}}
{{eqn | r = 8967
      | c = 
}}
{{eqn | n = n = 13
      | l = \map {\sigma_1} {9135} - 9135
      | r = 18 \, 720 - 9135
      | c = {{DSFLink|9135}}
}}
{{eqn | r = 9585
      | c = 
}}
{{eqn | n = n = 14
      | l = \map {\sigma_1} {9765} - 9765
      | r = 19 \, 968 - 9765
      | c = {{DSFLink|9765}}
}}
{{eqn | r = 10 \, 203
      | c = 
}}
{{eqn | n = n = 15
      | l = \map {\sigma_1} {10 \, 395} - 10 \, 395
      | r = 23 \, 040 - 10 \, 395
      | c = {{DSFLink|10,395|10 \, 395}}
}}
{{eqn | r = 12 \, 645
      | c = 
}}
{{eqn | n = n = 16
      | l = \map {\sigma_1} {11 \, 025} - 11\,025
      | r = 22 \, 971 - 11\,025
      | c = {{DSFLink|11,025|11\,025}}
}}
{{eqn | r = 11\,946
      | c = 
}}
{{eqn | n = n = 17
      | l = \map {\sigma_1} {11\,655}  - 11\,655
      | r = 23 \, 712 - 11\,655
      | c = {{DSFLink|11,655|11\,655}}
}}
{{eqn | r = 12 \, 057
      | c = 
}}
{{eqn | n = n = 18
      | l = \map {\sigma_1} {12\,285} - 12\,285
      | r = 26\,880 - 12\,285
      | c = {{DSFLink|12,285|12\,285}}
}}
{{eqn | r = 14\,595
      | c = 
}}
{{eqn | n = n = 19
      | l = \map {\sigma_1} {12\,915} - 12\,915
      | r = 26 \, 208 - 12\,915
      | c = {{DSFLink|12,915|12\,915}}
}}
{{eqn | r = 13 \, 293
      | c = 
}}
{{eqn | n = n = 20
      | l = \map {\sigma_1} {13\,545} - 13\,545
      | r = 27 \, 456 - 13\,545
      | c = {{DSFLink|13,545|13\,545}}
}}
{{eqn | r = 13 \, 911
      | c = 
}}
{{eqn | n = n = 21
      | l = \map {\sigma_1} {14\,175} - 14\,175
      | r = 30 \, 008 - 14\,175
      | c = {{DSFLink|14,175|14\,175}}
}}
{{eqn | r = 15 \, 833
      | c = 
}}
{{eqn | n = n = 22
      | l = \map {\sigma_1} {14\,805} - 14\,805
      | r = 29 \, 952 - 14\,805
      | c = {{DSFLink|14,805|14\,805}}
}}
{{eqn | r = 15 \, 147
      | c = 
}}
{{eqn | n = n = 23
      | l = \map {\sigma_1} {15\,435} - 15\,435
      | r = 31 \, 200 - 15\,435
      | c = {{DSFLink|15,435|15\,435}}
}}
{{eqn | r = 15 \, 765
      | c = 
}}
{{eqn | n = n = 24
      | l = \map {\sigma_1} {16\,065} - 16\,065
      | r = 34 \, 560 - 16\,065
      | c = {{DSFLink|16,065|16\,065}}
}}
{{eqn | r = 18 \, 495
      | c = 
}}
{{eqn | n = n = 25
      | l = \map {\sigma_1} {16\,695} - 16\,695
      | r = 33 \, 696 - 16\,695
      | c = {{DSFLink|16,695|16\,695}}
}}
{{eqn | r = 17 \, 001
      | c = 
}}
{{eqn | n = n = 26
      | l = \map {\sigma_1} {17\,325} - 17\,325
      | r = 38 \, 688 - 17\,325
      | c = {{DSFLink|17,325|17\,325}}
}}
{{eqn | r = 21 \, 363
      | c = 
}}
{{eqn | n = n = 27
      | l = \map {\sigma_1} {17\,955} - 17\,955
      | r = 38 \, 400 - 17\,955
      | c = {{DSFLink|17,955|17\,955}}
}}
{{eqn | r = 20 \, 445
      | c = 
}}
{{eqn | n = n = 28
      | l = \map {\sigma_1} {18\,585} - 18\,585
      | r = 37 \, 440 - 18\,585
      | c = {{DSFLink|18,585|18\,585}}
}}
{{eqn | r = 18 \, 855
      | c = 
}}
{{eqn | n = n = 29
      | l = \map {\sigma_1} {19\,215} - 19\,215
      | r = 38 \, 688 - 19\,215
      | c = {{DSFLink|19,215|19\,215}}
}}
{{eqn | r = 19 \, 473
      | c = 
}}
{{eqn | n = n = 30
      | l = \map {\sigma_1} {19\,845} - 19\,845
      | r = 41 \, 382 - 19\,845
      | c = {{DSFLink|19,845|19\,845}}
}}
{{eqn | r = 21 \, 537
      | c = 
}}
{{eqn | n = n = 31
      | l = \map {\sigma_1} {20\,475} - 20\,475
      | r = 45 \, 136 - 20\,475
      | c = {{DSFLink|20,475|20\,475}}
}}
{{eqn | r = 24 \, 661
      | c = 
}}
{{eqn | n = n = 32
      | l = \map {\sigma_1} {21\,105} - 21\,105
      | r = 42 \, 432 - 21\,105
      | c = {{DSFLink|21,105|21\,105}}
}}
{{eqn | r = 21 \, 327
      | c = 
}}
{{eqn | n = n = 33
      | l = \map {\sigma_1} {21\,735} - 21\,735
      | r = 46 \, 080 - 21\,735
      | c = {{DSFLink|21,735|21\,735}}
}}
{{eqn | r = 24 \, 705
      | c = 
}}
{{eqn | n = n = 34
      | l = \map {\sigma_1} {22\,365} - 22\,365
      | r = 44 \, 928 - 22\,365
      | c = {{DSFLink|22,365|22\,365}}
}}
{{eqn | r = 22 \, 563
      | c = 
}}
{{eqn | n = n = 35
      | l = \map {\sigma_1} {22\,995} - 22\,995
      | r = 46 \, 176 - 22\,995
      | c = {{DSFLink|22,995|22\,995}}
}}
{{eqn | r = 23 \, 181
      | c = 
}}
{{eqn | n = n = 36
      | l = \map {\sigma_1} {23\,625} - 23\,625
      | r = 49 \, 920 - 23\,625
      | c = {{DSFLink|23,625|23\,625}}
}}
{{eqn | r = 26 \, 295
      | c = 
}}
{{eqn | n = n = 37
      | l = \map {\sigma_1} {24\,255} - 24\,255
      | r = 53 \, 352 - 24\,255
      | c = {{DSFLink|24,255|24\,255}}
}}
{{eqn | r = 29 \, 097
      | c = 
}}
{{eqn | n = n = 38
      | l = \map {\sigma_1} {24\,885} - 24\,885
      | r = 49 \, 920 - 24\,885
      | c = {{DSFLink|24,885|24\,885}}
}}
{{eqn | r = 25 \, 035
      | c = 
}}
{{eqn | n = n = 39
      | l = \map {\sigma_1} {25\,515} - 25\,515
      | r = 52 \, 464 - 25\,515
      | c = {{DSFLink|25,515|25\,515}}
}}
{{eqn | r = 26 \, 949
      | c = 
}}
{{eqn | n = n = 40
      | l = \map {\sigma_1} {26\,145} - 26\,145
      | r = 52 \, 416 - 26\,145
      | c = {{DSFLink|26,145|26\,145}}
}}
{{eqn | r = 26 \, 271
      | c = 
}}
{{eqn | n = n = 41
      | l = \map {\sigma_1} {26\,775} - 26\,775
      | r = 58 \, 032 - 26\,775
      | c = {{DSFLink|26,775|26\,775}}
}}
{{eqn | r = 31 \, 257
      | c = 
}}
{{eqn | n = n = 42
      | l = \map {\sigma_1} {27\,405} - 27\,405
      | r = 57 \, 600 - 27\,405
      | c = {{DSFLink|27,405|27\,405}}
}}
{{eqn | r = 30 \, 195
      | c = 
}}
{{eqn | n = n = 43
      | l = \map {\sigma_1} {28\,035} - 28\,035
      | r = 56 \, 160 - 28\,035
      | c = {{DSFLink|28,035|28\,035}}
}}
{{eqn | r = 28 \, 125
      | c = 
}}
{{eqn | n = n = 44
      | l = \map {\sigma_1} {28\,665} - 28\,665
      | r = 62 \, 244 - 28\,665
      | c = {{DSFLink|28,665|28\,665}}
}}
{{eqn | r = 33 \, 579
      | c = 
}}
{{eqn | n = n = 45
      | l = \map {\sigma_1} {29\,295} - 29\,295
      | r = 61 \, 440 - 29\,295
      | c = {{DSFLink|29,295|29\,295}}
}}
{{eqn | r = 32 \, 145
      | c = 
}}
{{eqn | n = n = 46
      | l = \map {\sigma_1} {29\,925} - 29\,925
      | r = 64 \, 480 - 29\,925
      | c = {{DSFLink|29,925|29\,925}}
}}
{{eqn | r = 34, 555
      | c = 
}}
{{eqn | n = n = 47
      | l = \map {\sigma_1} {30\,555} - 30\,555
      | r = 61 \, 152 - 30\,555
      | c = {{DSFLink|30,555|30\,555}}
}}
{{eqn | r = 30 \, 597
      | c = 
}}
{{eqn | n = n = 48
      | l = \map {\sigma_1} {31\,185} - 31\,185
      | r = 69 \, 696 - 31\,185
      | c = {{DSFLink|31,185|31\,185}}
}}
{{eqn | r = 38 \, 511
      | c = 
}}
{{eqn | n = n = 49
      | l = \map {\sigma_1} {31\,815} - 31\,815
      | r = 63 \, 648 - 31\,815
      | c = {{DSFLink|31,815|31\,815}}
}}
{{eqn | r = 31 \, 833
      | c = 
}}
{{eqn | n = n = 50
      | l = \map {\sigma_1} {32\,445} - 32\,445
      | r = 64 \, 896 - 32\,445
      | c = {{DSFLink|32,445|32\,445}}
}}
{{eqn | r = 32 \, 451
      | c = 
}}
{{eqn | n = n = 51
      | l = \map {\sigma_1} {33\,075} - 33\,075
      | r = 70 \, 680 - 33\,075
      | c = {{DSFLink|33,075|33\,075}}
}}
{{eqn | r = 37 \, 605
      | c = 
}}
{{eqn | n = n = 52
      | l = \map {\sigma_1} {33\,705} - 33\,705
      | r = 67 \, 392 - 33\,705
      | c = {{DSFLink|33,705|33\,705}}
}}
{{eqn | r = 33 \, 687
      | c = and so $33\,705$ is not abundant
}}
{{end-eqn}}
{{qed}}