Abundancy of Integers in form 945 + 630n
Jump to navigation
Jump to search
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 $n = 52$.
Work In Progress In particular: Add this sequence to the number pages You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by completing it. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{WIP}} from the code. |
Proof
\(\text {(n = 0)}: \quad\) | \(\ds \map {\sigma_1} {945} - 945\) | \(=\) | \(\ds 1920 - 945\) | $\sigma_1$ of $945$ | ||||||||||
\(\ds \) | \(=\) | \(\ds 975\) | ||||||||||||
\(\text {(n = 1)}: \quad\) | \(\ds \map {\sigma_1} {1575} - 1575\) | \(=\) | \(\ds 3224 - 1575\) | $\sigma_1$ of $1575$ | ||||||||||
\(\ds \) | \(=\) | \(\ds 1649\) | ||||||||||||
\(\text {(n = 2)}: \quad\) | \(\ds \map {\sigma_1} {2205} - 2205\) | \(=\) | \(\ds 4446 - 2205\) | $\sigma_1$ of $2205$ | ||||||||||
\(\ds \) | \(=\) | \(\ds 2241\) | ||||||||||||
\(\text {(n = 3)}: \quad\) | \(\ds \map {\sigma_1} {2835} - 2835\) | \(=\) | \(\ds 5808 - 2835\) | $\sigma_1$ of $2835$ | ||||||||||
\(\ds \) | \(=\) | \(\ds 2973\) | ||||||||||||
\(\text {(n = 4)}: \quad\) | \(\ds \map {\sigma_1} {3465} - 3465\) | \(=\) | \(\ds 7488 - 3465\) | $\sigma_1$ of $3465$ | ||||||||||
\(\ds \) | \(=\) | \(\ds 4023\) | ||||||||||||
\(\text {(n = 5)}: \quad\) | \(\ds \map {\sigma_1} {4095} - 4095\) | \(=\) | \(\ds 8736 - 4095\) | $\sigma_1$ of $4095$ | ||||||||||
\(\ds \) | \(=\) | \(\ds 4641\) | ||||||||||||
\(\text {(n = 6)}: \quad\) | \(\ds \map {\sigma_1} {4725} - 4725\) | \(=\) | \(\ds 9920 - 4725\) | $\sigma_1$ of $4725$ | ||||||||||
\(\ds \) | \(=\) | \(\ds 5195\) | ||||||||||||
\(\text {(n = 7)}: \quad\) | \(\ds \map {\sigma_1} {5355} - 5355\) | \(=\) | \(\ds 11 \, 232 - 5355\) | $\sigma_1$ of $5355$ | ||||||||||
\(\ds \) | \(=\) | \(\ds 5877\) | ||||||||||||
\(\text {(n = 8)}: \quad\) | \(\ds \map {\sigma_1} {5985} - 5985\) | \(=\) | \(\ds 12 \, 480 - 5985\) | $\sigma_1$ of $5985$ | ||||||||||
\(\ds \) | \(=\) | \(\ds 6495\) | ||||||||||||
\(\text {(n = 9)}: \quad\) | \(\ds \map {\sigma_1} {6615} - 6615\) | \(=\) | \(\ds 13 \, 680 - 6615\) | $\sigma_1$ of $6615$ | ||||||||||
\(\ds \) | \(=\) | \(\ds 7065\) | ||||||||||||
\(\text {(n = 10)}: \quad\) | \(\ds \map {\sigma_1} {7245} - 7245\) | \(=\) | \(\ds 14 \, 976 - 7245\) | $\sigma_1$ of $7245$ | ||||||||||
\(\ds \) | \(=\) | \(\ds 7731\) | ||||||||||||
\(\text {(n = 11)}: \quad\) | \(\ds \map {\sigma_1} {7875} - 7875\) | \(=\) | \(\ds 16 \, 224 - 7875\) | $\sigma_1$ of $7875$ | ||||||||||
\(\ds \) | \(=\) | \(\ds 8349\) | ||||||||||||
\(\text {(n = 12)}: \quad\) | \(\ds \map {\sigma_1} {8505} - 8505\) | \(=\) | \(\ds 17 \, 472 - 8505\) | $\sigma_1$ of $8505$ | ||||||||||
\(\ds \) | \(=\) | \(\ds 8967\) | ||||||||||||
\(\text {(n = 13)}: \quad\) | \(\ds \map {\sigma_1} {9135} - 9135\) | \(=\) | \(\ds 18 \, 720 - 9135\) | $\sigma_1$ of $9135$ | ||||||||||
\(\ds \) | \(=\) | \(\ds 9585\) | ||||||||||||
\(\text {(n = 14)}: \quad\) | \(\ds \map {\sigma_1} {9765} - 9765\) | \(=\) | \(\ds 19 \, 968 - 9765\) | $\sigma_1$ of $9765$ | ||||||||||
\(\ds \) | \(=\) | \(\ds 10 \, 203\) | ||||||||||||
\(\text {(n = 15)}: \quad\) | \(\ds \map {\sigma_1} {10 \, 395} - 10 \, 395\) | \(=\) | \(\ds 23 \, 040 - 10 \, 395\) | $\sigma_1$ of $10 \, 395$ | ||||||||||
\(\ds \) | \(=\) | \(\ds 12 \, 645\) | ||||||||||||
\(\text {(n = 16)}: \quad\) | \(\ds \map {\sigma_1} {11 \, 025} - 11\,025\) | \(=\) | \(\ds 22 \, 971 - 11\,025\) | $\sigma_1$ of $11\,025$ | ||||||||||
\(\ds \) | \(=\) | \(\ds 11\,946\) | ||||||||||||
\(\text {(n = 17)}: \quad\) | \(\ds \map {\sigma_1} {11\,655} - 11\,655\) | \(=\) | \(\ds 23 \, 712 - 11\,655\) | $\sigma_1$ of $11\,655$ | ||||||||||
\(\ds \) | \(=\) | \(\ds 12 \, 057\) | ||||||||||||
\(\text {(n = 18)}: \quad\) | \(\ds \map {\sigma_1} {12\,285} - 12\,285\) | \(=\) | \(\ds 26\,880 - 12\,285\) | $\sigma_1$ of $12\,285$ | ||||||||||
\(\ds \) | \(=\) | \(\ds 14\,595\) | ||||||||||||
\(\text {(n = 19)}: \quad\) | \(\ds \map {\sigma_1} {12\,915} - 12\,915\) | \(=\) | \(\ds 26 \, 208 - 12\,915\) | $\sigma_1$ of $12\,915$ | ||||||||||
\(\ds \) | \(=\) | \(\ds 13 \, 293\) | ||||||||||||
\(\text {(n = 20)}: \quad\) | \(\ds \map {\sigma_1} {13\,545} - 13\,545\) | \(=\) | \(\ds 27 \, 456 - 13\,545\) | $\sigma_1$ of $13\,545$ | ||||||||||
\(\ds \) | \(=\) | \(\ds 13 \, 911\) | ||||||||||||
\(\text {(n = 21)}: \quad\) | \(\ds \map {\sigma_1} {14\,175} - 14\,175\) | \(=\) | \(\ds 30 \, 008 - 14\,175\) | $\sigma_1$ of $14\,175$ | ||||||||||
\(\ds \) | \(=\) | \(\ds 15 \, 833\) | ||||||||||||
\(\text {(n = 22)}: \quad\) | \(\ds \map {\sigma_1} {14\,805} - 14\,805\) | \(=\) | \(\ds 29 \, 952 - 14\,805\) | $\sigma_1$ of $14\,805$ | ||||||||||
\(\ds \) | \(=\) | \(\ds 15 \, 147\) | ||||||||||||
\(\text {(n = 23)}: \quad\) | \(\ds \map {\sigma_1} {15\,435} - 15\,435\) | \(=\) | \(\ds 31 \, 200 - 15\,435\) | $\sigma_1$ of $15\,435$ | ||||||||||
\(\ds \) | \(=\) | \(\ds 15 \, 765\) | ||||||||||||
\(\text {(n = 24)}: \quad\) | \(\ds \map {\sigma_1} {16\,065} - 16\,065\) | \(=\) | \(\ds 34 \, 560 - 16\,065\) | $\sigma_1$ of $16\,065$ | ||||||||||
\(\ds \) | \(=\) | \(\ds 18 \, 495\) | ||||||||||||
\(\text {(n = 25)}: \quad\) | \(\ds \map {\sigma_1} {16\,695} - 16\,695\) | \(=\) | \(\ds 33 \, 696 - 16\,695\) | $\sigma_1$ of $16\,695$ | ||||||||||
\(\ds \) | \(=\) | \(\ds 17 \, 001\) | ||||||||||||
\(\text {(n = 26)}: \quad\) | \(\ds \map {\sigma_1} {17\,325} - 17\,325\) | \(=\) | \(\ds 38 \, 688 - 17\,325\) | $\sigma_1$ of $17\,325$ | ||||||||||
\(\ds \) | \(=\) | \(\ds 21 \, 363\) | ||||||||||||
\(\text {(n = 27)}: \quad\) | \(\ds \map {\sigma_1} {17\,955} - 17\,955\) | \(=\) | \(\ds 38 \, 400 - 17\,955\) | $\sigma_1$ of $17\,955$ | ||||||||||
\(\ds \) | \(=\) | \(\ds 20 \, 445\) | ||||||||||||
\(\text {(n = 28)}: \quad\) | \(\ds \map {\sigma_1} {18\,585} - 18\,585\) | \(=\) | \(\ds 37 \, 440 - 18\,585\) | $\sigma_1$ of $18\,585$ | ||||||||||
\(\ds \) | \(=\) | \(\ds 18 \, 855\) | ||||||||||||
\(\text {(n = 29)}: \quad\) | \(\ds \map {\sigma_1} {19\,215} - 19\,215\) | \(=\) | \(\ds 38 \, 688 - 19\,215\) | $\sigma_1$ of $19\,215$ | ||||||||||
\(\ds \) | \(=\) | \(\ds 19 \, 473\) | ||||||||||||
\(\text {(n = 30)}: \quad\) | \(\ds \map {\sigma_1} {19\,845} - 19\,845\) | \(=\) | \(\ds 41 \, 382 - 19\,845\) | $\sigma_1$ of $19\,845$ | ||||||||||
\(\ds \) | \(=\) | \(\ds 21 \, 537\) | ||||||||||||
\(\text {(n = 31)}: \quad\) | \(\ds \map {\sigma_1} {20\,475} - 20\,475\) | \(=\) | \(\ds 45 \, 136 - 20\,475\) | $\sigma_1$ of $20\,475$ | ||||||||||
\(\ds \) | \(=\) | \(\ds 24 \, 661\) | ||||||||||||
\(\text {(n = 32)}: \quad\) | \(\ds \map {\sigma_1} {21\,105} - 21\,105\) | \(=\) | \(\ds 42 \, 432 - 21\,105\) | $\sigma_1$ of $21\,105$ | ||||||||||
\(\ds \) | \(=\) | \(\ds 21 \, 327\) | ||||||||||||
\(\text {(n = 33)}: \quad\) | \(\ds \map {\sigma_1} {21\,735} - 21\,735\) | \(=\) | \(\ds 46 \, 080 - 21\,735\) | $\sigma_1$ of $21\,735$ | ||||||||||
\(\ds \) | \(=\) | \(\ds 24 \, 705\) | ||||||||||||
\(\text {(n = 34)}: \quad\) | \(\ds \map {\sigma_1} {22\,365} - 22\,365\) | \(=\) | \(\ds 44 \, 928 - 22\,365\) | $\sigma_1$ of $22\,365$ | ||||||||||
\(\ds \) | \(=\) | \(\ds 22 \, 563\) | ||||||||||||
\(\text {(n = 35)}: \quad\) | \(\ds \map {\sigma_1} {22\,995} - 22\,995\) | \(=\) | \(\ds 46 \, 176 - 22\,995\) | $\sigma_1$ of $22\,995$ | ||||||||||
\(\ds \) | \(=\) | \(\ds 23 \, 181\) | ||||||||||||
\(\text {(n = 36)}: \quad\) | \(\ds \map {\sigma_1} {23\,625} - 23\,625\) | \(=\) | \(\ds 49 \, 920 - 23\,625\) | $\sigma_1$ of $23\,625$ | ||||||||||
\(\ds \) | \(=\) | \(\ds 26 \, 295\) | ||||||||||||
\(\text {(n = 37)}: \quad\) | \(\ds \map {\sigma_1} {24\,255} - 24\,255\) | \(=\) | \(\ds 53 \, 352 - 24\,255\) | $\sigma_1$ of $24\,255$ | ||||||||||
\(\ds \) | \(=\) | \(\ds 29 \, 097\) | ||||||||||||
\(\text {(n = 38)}: \quad\) | \(\ds \map {\sigma_1} {24\,885} - 24\,885\) | \(=\) | \(\ds 49 \, 920 - 24\,885\) | $\sigma_1$ of $24\,885$ | ||||||||||
\(\ds \) | \(=\) | \(\ds 25 \, 035\) | ||||||||||||
\(\text {(n = 39)}: \quad\) | \(\ds \map {\sigma_1} {25\,515} - 25\,515\) | \(=\) | \(\ds 52 \, 464 - 25\,515\) | $\sigma_1$ of $25\,515$ | ||||||||||
\(\ds \) | \(=\) | \(\ds 26 \, 949\) | ||||||||||||
\(\text {(n = 40)}: \quad\) | \(\ds \map {\sigma_1} {26\,145} - 26\,145\) | \(=\) | \(\ds 52 \, 416 - 26\,145\) | $\sigma_1$ of $26\,145$ | ||||||||||
\(\ds \) | \(=\) | \(\ds 26 \, 271\) | ||||||||||||
\(\text {(n = 41)}: \quad\) | \(\ds \map {\sigma_1} {26\,775} - 26\,775\) | \(=\) | \(\ds 58 \, 032 - 26\,775\) | $\sigma_1$ of $26\,775$ | ||||||||||
\(\ds \) | \(=\) | \(\ds 31 \, 257\) | ||||||||||||
\(\text {(n = 42)}: \quad\) | \(\ds \map {\sigma_1} {27\,405} - 27\,405\) | \(=\) | \(\ds 57 \, 600 - 27\,405\) | $\sigma_1$ of $27\,405$ | ||||||||||
\(\ds \) | \(=\) | \(\ds 30 \, 195\) | ||||||||||||
\(\text {(n = 43)}: \quad\) | \(\ds \map {\sigma_1} {28\,035} - 28\,035\) | \(=\) | \(\ds 56 \, 160 - 28\,035\) | $\sigma_1$ of $28\,035$ | ||||||||||
\(\ds \) | \(=\) | \(\ds 28 \, 125\) | ||||||||||||
\(\text {(n = 44)}: \quad\) | \(\ds \map {\sigma_1} {28\,665} - 28\,665\) | \(=\) | \(\ds 62 \, 244 - 28\,665\) | $\sigma_1$ of $28\,665$ | ||||||||||
\(\ds \) | \(=\) | \(\ds 33 \, 579\) | ||||||||||||
\(\text {(n = 45)}: \quad\) | \(\ds \map {\sigma_1} {29\,295} - 29\,295\) | \(=\) | \(\ds 61 \, 440 - 29\,295\) | $\sigma_1$ of $29\,295$ | ||||||||||
\(\ds \) | \(=\) | \(\ds 32 \, 145\) | ||||||||||||
\(\text {(n = 46)}: \quad\) | \(\ds \map {\sigma_1} {29\,925} - 29\,925\) | \(=\) | \(\ds 64 \, 480 - 29\,925\) | $\sigma_1$ of $29\,925$ | ||||||||||
\(\ds \) | \(=\) | \(\ds 34, 555\) | ||||||||||||
\(\text {(n = 47)}: \quad\) | \(\ds \map {\sigma_1} {30\,555} - 30\,555\) | \(=\) | \(\ds 61 \, 152 - 30\,555\) | $\sigma_1$ of $30\,555$ | ||||||||||
\(\ds \) | \(=\) | \(\ds 30 \, 597\) | ||||||||||||
\(\text {(n = 48)}: \quad\) | \(\ds \map {\sigma_1} {31\,185} - 31\,185\) | \(=\) | \(\ds 69 \, 696 - 31\,185\) | $\sigma_1$ of $31\,185$ | ||||||||||
\(\ds \) | \(=\) | \(\ds 38 \, 511\) | ||||||||||||
\(\text {(n = 49)}: \quad\) | \(\ds \map {\sigma_1} {31\,815} - 31\,815\) | \(=\) | \(\ds 63 \, 648 - 31\,815\) | $\sigma_1$ of $31\,815$ | ||||||||||
\(\ds \) | \(=\) | \(\ds 31 \, 833\) | ||||||||||||
\(\text {(n = 50)}: \quad\) | \(\ds \map {\sigma_1} {32\,445} - 32\,445\) | \(=\) | \(\ds 64 \, 896 - 32\,445\) | $\sigma_1$ of $32\,445$ | ||||||||||
\(\ds \) | \(=\) | \(\ds 32 \, 451\) | ||||||||||||
\(\text {(n = 51)}: \quad\) | \(\ds \map {\sigma_1} {33\,075} - 33\,075\) | \(=\) | \(\ds 70 \, 680 - 33\,075\) | $\sigma_1$ of $33\,075$ | ||||||||||
\(\ds \) | \(=\) | \(\ds 37 \, 605\) | ||||||||||||
\(\text {(n = 52)}: \quad\) | \(\ds \map {\sigma_1} {33\,705} - 33\,705\) | \(=\) | \(\ds 67 \, 392 - 33\,705\) | $\sigma_1$ of $33\,705$ | ||||||||||
\(\ds \) | \(=\) | \(\ds 33 \, 687\) | and so $33\,705$ is not abundant |
$\blacksquare$
Sources
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $33,705$