Miss This

1,729,404 is seven digits long. If you drop one digit at a time, you can create seven more numbers from it, each six digits long. If you add these numbers, something special happens:

1,729,404 → 729404 (missing 1) + 129404 (missing 7) + 179404 (missing 2) + 172404 + 172904 + 172944 + 172940 = 1,729,404

So 1,729,404 is narcissistic, or equal to some manipulation of its own digits. Searching for numbers like this might seem like a big task, but you can cut the search-time considerably by noting that the final two digits determine whether a number is a suitable candidate for testing. For example, what if a seven-digit number ends in …38? Then the final digit of the missing-digit sum will equal (3 x 1 + 8 x 6) modulo 10 = (3 + 48) mod 10 = 51 mod 10 = 1. This means that you don’t need to check any seven-digit number ending in …38.

But what about seven-digit numbers ending in …57? Now the final digit of the sum will equal (5 x 1 + 7 x 6) modulo 10 = (5 + 42) mod 10 = 47 mod 10 = 7. So seven-digit numbers ending in …57 are possible missing-digit narcissistic sums. Then you can test numbers ending …157, …257, …357 and so on, to determine the last-but-one digit of the sum. Using this method, one quickly finds the only two seven-digit numbers of this form in base-10:

1,729,404 → 729404 + 129404 + 179404 + 172404 + 172904 + 172944 + 172940 = 1,729,404

1,800,000 → 800000 + 100000 + 180000 + 180000 + 180000 + 180000 + 180000 = 1,800,000

What about eight-digit numbers? Only those ending in these two digits need to be checked: …00, …23, …28, …41, …46, …64, …69, …82, …87. Here are the results:

• 13,758,846 → 3758846 + 1758846 + 1358846 + 1378846 + 1375846 + 1375846 + 1375886 + 1375884 = 13,758,846
• 13,800,000 → 3800000 + 1800000 + 1300000 + 1380000 + 1380000 + 1380000 + 1380000 + 1380000 = 13,800,000
• 14,358,846 → 4358846 + 1358846 + 1458846 + 1438846 + 1435846 + 1435846 + 1435886 + 1435884 = 14,358,846
• 14,400,000 → 4400000 + 1400000 + 1400000 + 1440000 + 1440000 + 1440000 + 1440000 + 1440000 = 14,400,000
• 15,000,000 → 5000000 + 1000000 + 1500000 + 1500000 + 1500000 + 1500000 + 1500000 + 1500000 = 15,000,000
• 28,758,846 → 8758846 + 2758846 + 2858846 + 2878846 + 2875846 + 2875846 + 2875886 + 2875884 = 28,758,846
• 28,800,000 → 8800000 + 2800000 + 2800000 + 2880000 + 2880000 + 2880000 + 2880000 + 2880000 = 28,800,000
• 29,358,846 → 9358846 + 2358846 + 2958846 + 2938846 + 2935846 + 2935846 + 2935886 + 2935884 = 29,358,846
• 29,400,000 → 9400000 + 2400000 + 2900000 + 2940000 + 2940000 + 2940000 + 2940000 + 2940000 = 29,400,000

But there are no nine-digit sumbers, or nine-digit numbers that supply missing-digit narcissistic sums. What about ten-digit sumbers? There are twenty-one:

1,107,488,889; 1,107,489,042; 1,111,088,889; 1,111,089,042; 3,277,800,000; 3,281,400,000; 4,388,888,889; 4,388,889,042; 4,392,488,889; 4,392,489,042; 4,500,000,000; 5,607,488,889; 5,607,489,042; 5,611,088,889; 5,611,089,042; 7,777,800,000; 7,781,400,000; 8,888,888,889; 8,888,889,042; 8,892,488,889; 8,892,489,042 (21 numbers)

Finally, the nine eleven-digit sumbers all take this form:

30,000,000,000 → 0000000000 + 3000000000 + 3000000000 + 3000000000 + 3000000000 + 3000000000 + 3000000000 + 3000000000 + 3000000000 + 3000000000 + 3000000000 = 30,000,000,000

So that’s forty-one narcissistic sumbers in base-10. Not all of them are listed in Sequence A131639 at the Encyclopedia of Integer Sequences, but I think I’ve got my program working right. Other bases show similar patterns. Here are some missing-digit narcissistic sumbers in base-5:

• 1,243 → 243 + 143 + 123 + 124 = 1,243 (b=5) = 198 (b=10)
• 1,324 → 324 + 124 + 134 + 132 = 1,324 (b=5) = 214 (b=10)
• 1,331 → 331 + 131 + 131 + 133 = 1,331 (b=5) = 216 (b=10)
• 1,412 → 412 + 112 + 142 + 141 = 1,412 (b=5) = 232 (b=10)

• 100,000 → 00000 + 10000 + 10000 + 10000 + 10000 + 10000 = 100,000 (b=5) = 3,125 (b=10)
• 200,000 → 00000 + 20000 + 20000 + 20000 + 20000 + 20000 = 200,000 (b=5) = 6,250 (b=10)
• 300,000 → 00000 + 30000 + 30000 + 30000 + 30000 + 30000 = 300,000 (b=5) = 9,375 (b=10)
• 400,000 → 00000 + 40000 + 40000 + 40000 + 40000 + 40000 = 400,000 (b=5) = 12,500 (b=10)

And here are some sumbers in base-16:

5,4CD,111,0EE,EF0,542 = 4CD1110EEEF0542 + 5CD1110EEEF0542 + 54D1110EEEF0542 + 54C1110EEEF0542 + 54CD110EEEF0542 + 54CD110EEEF0542 + 54CD110EEEF0542 + 54CD111EEEF0542 + 54CD1110EEF0542 + 54CD1110EEF0542 + 54CD1110EEF0542 + 54CD1110EEE0542 + 54CD1110EEEF542 + 54CD1110EEEF042 + 54CD1110EEEF052 + 54CD1110EEEF054 (b=16) = 6,110,559,033,837,421,890 (b=10)

6,5DD,E13,CEE,EF0,542 = 5DDE13CEEEF0542 + 6DDE13CEEEF0542 + 65DE13CEEEF0542 + 65DE13CEEEF0542 + 65DD13CEEEF0542 + 65DDE3CEEEF0542 + 65DDE1CEEEF0542 + 65DDE13EEEF0542 + 65DDE13CEEF0542 + 65DDE13CEEF0542 + 65DDE13CEEF0542 + 65DDE13CEEE0542 + 65DDE13CEEEF542 + 65DDE13CEEEF042 + 65DDE13CEEEF052 + 65DDE13CEEEF054 (b=16) = 7,340,270,619,506,705,730 (b=10)

10,000,000,000,000,000 → 0000000000000000 + 1000000000000000 + 1000000000000000 + 1000000000000000 + 1000000000000000 + 1000000000000000 + 1000000000000000 + 1000000000000000 + 1000000000000000 + 1000000000000000 + 1000000000000000 + 1000000000000000 + 1000000000000000 + 1000000000000000 + 1000000000000000 + 1000000000000000 + 1000000000000000 = 10,000,000,000,000,000 (b=16) = 18,446,744,073,709,551,616 (b=10)

F0,000,000,000,000,000 → 0000000000000000 + F000000000000000 + F000000000000000 + F000000000000000 + F000000000000000 + F000000000000000 + F000000000000000 + F000000000000000 + F000000000000000 + F000000000000000 + F000000000000000 + F000000000000000 + F000000000000000 + F000000000000000 + F000000000000000 + F000000000000000 + F000000000000000 = F0,000,000,000,000,000 (b=16) = 276,701,161,105,643,274,240 (b=10)

Next I’d like to investigate sumbers created by missing two, three and more digits at a time. Here’s a taster:

1,043,101 → 43101 (missing 1 and 0) + 03101 (missing 1 and 4) + 04101 (missing 1 and 3) + 04301 + 04311 + 04310 + 13101 + 14101 + 14301 + 14311 + 14310 + 10101 + 10301 + 10311 + 10310 + 10401 + 10411 + 10410 + 10431 + 10430 + 10431 = 1,043,101 (b=5) = 18,526 (b=10)

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