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Exponents k > 0 of powers of 2 such that the decimal expansion of 2^k contains more than one nonoverlapping 666 substring.

3

`%I #28 Apr 10 2024 09:18:16
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`%S 220,222,529,624,648,702,714,844,846,850,859,924,925,929,931,979,981,
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`%T 983,1062,1088,1133,1135,1219,1230,1241,1259,1310,1343,1349,1384,1394,
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`%U 1467,1472,1495,1503,1524,1550,1589,1627,1631,1642,1652,1656,1663,1679,1744,1751
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`%N Exponents k > 0 of powers of 2 such that the decimal expansion of 2^k contains more than one nonoverlapping 666 substring.
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`%C A positive power of 2 containing 666 in its decimal expansion is called an apocalyptic number.
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`%C See A371808 for a variant where overlapping substrings are counted as distinct.
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`%H Paolo Xausa, <a href="/A371806/b371806.txt">Table of n, a(n) for n = 1..10000</a>
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`%H Brady Haran and Tony Padilla, <a href="https://www.youtube.com/watch?v=0LkBwCSMsX4">Apocalyptic Numbers</a>, YouTube Numberphile video, 2024.
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`%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/ApocalypticNumber.html">Apocalyptic Number</a>.
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`%e 220 is a term because 2^220 contains more than one nonoverlapping 666 substring in its decimal expansion:
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`%e 2^220 = 168499(666)66969149871(666)88442938726917102321526408785780068975640576.
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`%t Select[Range[2000], StringCount[IntegerString[2^#], "666"] > 1 &]
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`%o (Python)
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`%o def ok(n): return str(1<<n).count("666") > 1
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`%o print([k for k in range(2000) if ok(k)]) # _Michael S. Branicky_, Apr 07 2024
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`%Y Subsequence of A007356 and of A371808.
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`%Y Cf. A371807.
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`%K nonn,easy,base
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`%O 1,1
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`%A _Paolo Xausa_, Apr 06 2024
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