X-Men: Apocalypse (2016) |
Numbers Aplenty has to say about apocalyptic numbers:
A number of the form 2n is called apocalyptic if its digits contain "666" as a substring. The smallest apocalyptic number is 2157, which is equal to:182687704666_362864775460604089535377456991567872
while 2220 is the smallest apocalyptic number which contains two 666 groups, being equal to:16849966666_969149871666_88442938726917102321526408785780068975640576The smallest power of 2 with 3 groups is 2931.A number n such that 2n is apocalyptic is called an apocalyptic power or apocalyptic exponent.
Between 1 and 3×106 there are 3715 numbers which are non-apocalyptic exponents, the largest being 29784. In other words, it is highly probable that 2n for n≥29785 is an apocalyptic number.
Probably there are only 8 numbers, namely 2666, 3666, 5666, 6660, 6665, 6669, 11666, 26667 which contains 666 among their digits but are not apocalyptic exponents.
The first apocalyptic exponents are 157, 192, 218, 220, 222, 224, 226, 243, 245, 247, 251, 278, 285, 286, 287, 312, 355, 361, 366, 382, 384, 390, 394, 411, 434, 443, 478, 497, 499, 506, ...
Here is a link to the first 1000 apocalyptic exponents. As the numbers get larger, the frequency of numbers being non-apocalyptic exponents decreases so that, after 29874, the frequency is (probably) zero. For example, in the range from 20000 to 30000, there are only the following non-apocalyptic exponents:
20271, 20300, 20509, 20644, 20710, 21077, 21600, 21602, 22447, 22734, 23097, 23253, 24422, 24441, 25026, 25357, 25896, 26051, 26667, 29784
Why this interest in apocalyptic numbers? Well, in my diurnal age count, I've entered what might be termed an "apocalyptic phase". I'm currently 26606 days old and heading toward 26666 (the latter occurs a day before my 73rd birthday). Interestingly, as can be seen in the range of numbers above, 26667 is the second last non-apocalyptic exponent.
Numbers Aplenty also shows the smallest 3 × 3 magic square made of consecutive apocalyptic numbers. See Figure 1.
Figure 1 |
The apocalyptic powers comprise OEIS A007356:
A007356 | Apocalyptic powers: 2n contains 666. |
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