Abundancy of Integers in form 945 + 630n

Theorem

A large number of integers of the form $945 + 630 n$, for $n \in \Z_{\ge 0}$, are abundant.

The first counterexample is for $n = 52$.

Proof

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

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