Definition:Pluperfect Digital Invariant/Examples
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Examples of Pluperfect Digital Invariants
$3$ Digit Pluperfect Digital Invariants
| \(\ds 153\) | \(=\) | \(\ds 1 + 125 + 27\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 1^3 + 5^3 + 3^3\) |
| \(\ds 370\) | \(=\) | \(\ds 27 + 343 + 0\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 3^3 + 7^3 + 0^3\) |
| \(\ds 371\) | \(=\) | \(\ds 27 + 343 + 1\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 3^3 + 7^3 + 1^3\) |
| \(\ds 407\) | \(=\) | \(\ds 64 + 0 + 343\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 4^3 + 0^3 + 7^3\) |
$4$ Digit Pluperfect Digital Invariants
| \(\ds 1634\) | \(=\) | \(\ds 1 + 1296 + 81 + 256\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 1^4 + 6^4 + 3^4 + 4^4\) |
| \(\ds 8208\) | \(=\) | \(\ds 4096 + 16 + 0 + 4096\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 8^4 + 2^4 + 0^4 + 8^4\) |
| \(\ds 9474\) | \(=\) | \(\ds 6561 + 256 + 2401 + 256\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 9^4 + 4^4 + 7^4 + 4^4\) |
$5$ Digit Pluperfect Digital Invariants
| \(\ds 54 \, 748\) | \(=\) | \(\ds 3125 + 1024 + 16 \, 807 + 1024 + 32 \, 768\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 5^5 + 4^5 + 7^5 + 4^5 + 8^5\) |
| \(\ds 92 \, 727\) | \(=\) | \(\ds 59 \, 049 + 32 + 16 \, 807 + 32 + 16 \, 807\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 9^5 + 2^5 + 7^5 + 2^5 + 7^5\) |
| \(\ds 93 \, 084\) | \(=\) | \(\ds 59 \, 049 + 243 + 0 + 32 \, 768 + 1024\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 9^5 + 3^5 + 0^5 + 8^5 + 4^5\) |
$6$ Digit Pluperfect Digital Invariants
| \(\ds 548 \, 834\) | \(=\) | \(\ds 15 \, 625 + 4096 + 262 \, 144 + 262 \, 144 + 729 + 4096\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 5^6 + 4^6 + 8^6 + 8^6 + 3^6 + 4^6\) |
$7$ Digit Pluperfect Digital Invariants
| \(\ds 1 \, 741 \, 725\) | \(=\) | \(\ds 1 + 823 \, 543 + 16 \, 384 + 1 + 823 \, 543 + 128 + 78 \, 125\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 1^7 + 7^7 + 4^7 + 1^7 + 7^7 + 2^7 + 5^7\) |
| \(\ds 4 \, 210 \, 818\) | \(=\) | \(\ds 16 \, 384 + 128 + 1 + 0 + 2 \, 097 \, 152 + 1 + 2 \, 097 \, 152\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 4^7 + 2^7 + 1^7 + 0^7 + 8^7 + 1^7 + 8^7\) |
| \(\ds 9 \, 800 \, 817\) | \(=\) | \(\ds 4 \, 782 \, 969 + 2 \, 097 \, 152 + 0 + 0 + 2 \, 097 \, 152 + 1 + 823 \, 543\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 9^7 + 8^7 + 0^7 + 0^7 + 8^7 + 1^7 + 7^7\) |
| \(\ds 9 \, 926 \, 315\) | \(=\) | \(\ds 4 \, 782 \, 969 + 4 \, 782 \, 969 + 128 + 279 \, 936 + 2187 + 1 + 78 \, 125\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 9^7 + 9^7 + 2^7 + 6^7 + 3^7 + 1^7 + 5^7\) |
$8$ Digit Pluperfect Digital Invariants
| \(\ds 24 \, 678 \, 050\) | \(=\) | \(\ds 256 + 65 \, 536 + 1 \, 679 \, 616 + 5 \, 764 \, 801 + 16 \, 777 \, 216 + 0 + 390 \, 625 + 0\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 2^8 + 4^8 + 6^8 + 7^8 + 8^8 + 0^8 + 5^8 + 0^8\) |
| \(\ds 24 \, 678 \, 051\) | \(=\) | \(\ds 256 + 65 \, 536 + 1 \, 679 \, 616 + 5 \, 764 \, 801 + 16 \, 777 \, 216 + 0 + 390 \, 625 + 1\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 2^8 + 4^8 + 6^8 + 7^8 + 8^8 + 0^8 + 5^8 + 1^8\) |
| \(\ds 88 \, 593 \, 477\) | \(=\) | \(\ds 16 \, 777 \, 216 + 16 \, 777 \, 216 + 390 \, 625 + 43 \, 046 \, 721 + 6561 + 65 \, 536 + 5 \, 764 \, 801 + 5 \, 764 \, 801\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 8^8 + 8^8 + 5^8 + 9^8 + 3^8 + 4^8 + 7^8 + 7^8\) |
$9$ Digit Pluperfect Digital Invariants
| \(\ds 146 \, 511 \, 208\) | \(=\) | \(\ds 1 + 262 \, 144 + 10 \, 077 \, 696 + 1 \, 953 \, 125 + 1 + 1 + 512 + 0 + 134 \, 217 \, 728\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 1^9 + 4^9 + 6^9 + 5^9 + 1^9 + 1^9 + 2^9 + 0^9 + 8^9\) |
| \(\ds 472 \, 335 \, 975\) | \(=\) | \(\ds 262 \, 144 + 40 \, 353 \, 607 + 512 + 19 \, 683 + 19 \, 683 + 1 \, 953 \, 125 + 387 \, 420 \, 489 + 40 \, 353 \, 607 + 1 \, 953 \, 125\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 4^9 + 7^9 + 2^9 + 3^9 + 3^9 + 5^9 + 9^9 + 7^9 + 5^9\) |
| \(\ds 534 \, 494 \, 836\) | \(=\) | \(\ds 1 \, 953 \, 125 + 19 \, 683 + 262 \, 144 + 262 \, 144 + 387 \, 420 \, 489 + 262 \, 144 + 134 \, 217 \, 728 + 19 \, 683 + 10 \, 077 \, 696\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 5^9 + 3^9 + 4^9 + 4^9 + 9^9 + 4^9 + 8^9 + 3^9 + 6^9\) |
| \(\ds 912 \, 985 \, 153\) | \(=\) | \(\ds 387 \, 420 \, 489 + 1 + 512 + 387 \, 420 \, 489 + 134 \, 217 \, 728 + 1 \, 953 \, 125 + 1 + 1 \, 953 \, 125 + 19 \, 683\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 9^9 + 1^9 + 2^9 + 9^9 + 8^9 + 5^9 + 1^9 + 5^9 + 3^9\) |
$10$ Digit Pluperfect Digital Invariants
| \(\ds 4 \, 679 \, 307 \, 774\) | \(=\) | \(\ds 1 \, 048 \, 576 + 60 \, 466 \, 176 + 282 \, 475 \, 249 + 3 \, 486 \, 784 \, 401 + 59 \, 049 + 0 + 282 \, 475 \, 249 + 282 \, 475 \, 249 + 282 \, 475 \, 249 + 1 \, 048 \, 576\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 4^{10} + 6^{10} + 7^{10} + 9^{10} + 3^{10} + 0^{10} + 7^{10} + 7^{10} + 7^{10} + 4^{10}\) |