Abundancy of Integers in form 945 + 630n

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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

$\blacksquare$


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