A371806 -id:A371806 - cp's OEIS frontend

A371808 Exponents k > 0 of powers of 2 such that the decimal expansion of 2^k contains more than one 666 substring (overlapping substrings are counted as distinct).

220, 222, 243, 529, 624, 648, 662, 702, 714, 838, 840, 842, 844, 846, 850, 857, 859, 867, 869, 871, 924, 925, 927, 929, 931, 975, 979, 981, 983, 1056, 1058, 1062, 1088, 1133, 1135, 1160, 1162, 1219, 1230, 1241, 1259, 1310, 1341, 1343, 1349, 1384, 1394, 1411, 1420

Offset: 1

Paolo Xausa, Apr 06 2024

A positive power of 2 containing 666 in its decimal expansion is called an apocalyptic number.

See A371806 for a variant counting only nonoverlapping substrings.

			243 is a term because 2^243 contains two (overlapping) 666 substrings in its decimal expansion:
.
                   ***
14134776518227074636666380005943348126619871175004951664972849610340958208.
                    ***

Paolo Xausa, Table of n, a(n) for n = 1..10000
Brady Haran and Tony Padilla, Apocalyptic Numbers, YouTube Numberphile video, 2024.
Eric Weisstein's World of Mathematics, Apocalyptic Number.

Subsequence of A007356.

Cf. A371806, A371809.

Select[Range[2000], StringCount[IntegerString[2^#], "666", Overlaps->True] > 1 &]

def ok(n): return (s:=str(1< 1 or "6666" in s
print([k for k in range(2000) if ok(k)]) # Michael S. Branicky, Apr 07 2024

A371807 Number of nonoverlapping 666 substrings contained in the decimal expansion of the n-th apocalyptic number.

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2

Offset: 1

Paolo Xausa, Apr 06 2024

An apocalyptic number is a positive power of 2 containing 666 in its decimal expansion.

See A371809 for a variant where overlapping substrings are counted as distinct.

			a(4) = 2 because the 4th apocalyptic number (2^220) contains two nonoverlapping 666 substrings in its decimal expansion:
2^220 = 168499(666)66969149871(666)88442938726917102321526408785780068975640576.

Paolo Xausa, Table of n, a(n) for n = 1..10000
Brady Haran and Tony Padilla, Apocalyptic Numbers, YouTube Numberphile video, 2024.
Eric Weisstein's World of Mathematics, Apocalyptic Number.

Cf. A007356, A371806, A371809.

Select[StringCount[IntegerString[2^Range[1000]], "666"], # > 0 &]

from itertools import islice
def agen(): # generator of terms
    pow2 = 1
    while True:
        s = str(pow2)
        if (c := s.count("666")) > 0: yield c
        pow2 <<= 1
print(list(islice(agen(), 88))) # Michael S. Branicky, Apr 07 2024

a(n) <= A371809(n).

cp's OEIS Frontend

A371808 Exponents k > 0 of powers of 2 such that the decimal expansion of 2^k contains more than one 666 substring (overlapping substrings are counted as distinct).

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