A371808 -id:A371808 - cp's OEIS frontend

A371806 Exponents k > 0 of powers of 2 such that the decimal expansion of 2^k contains more than one nonoverlapping 666 substring.

220, 222, 529, 624, 648, 702, 714, 844, 846, 850, 859, 924, 925, 929, 931, 979, 981, 983, 1062, 1088, 1133, 1135, 1219, 1230, 1241, 1259, 1310, 1343, 1349, 1384, 1394, 1467, 1472, 1495, 1503, 1524, 1550, 1589, 1627, 1631, 1642, 1652, 1656, 1663, 1679, 1744, 1751

Offset: 1

Paolo Xausa, Apr 06 2024

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

See A371808 for a variant where overlapping substrings are counted as distinct.

			220 is a term because 2^220 contains more than one nonoverlapping 666 substring 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.

Subsequence of A007356 and of A371808.

Cf. A371807.

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

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

A371809 Number of 666 substrings contained in the decimal expansion of the n-th apocalyptic number, where overlapping substrings are counted as distinct.

Original entry on oeis.org

1, 1, 1, 4, 3, 1, 1, 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, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 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, 3, 3, 1, 2, 3

Offset: 1

Paolo Xausa, Apr 06 2024

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

See A371807 for a variant counting only nonoverlapping substrings.

			a(8) = 2 because the 8th apocalyptic number (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.

Cf. A007356, A371807, A371808.

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

from itertools import islice
def agen(): # generator of terms
    pow2 = 1
    while True:
        s = str(pow2)
        c = sum(1 for i in range(len(s)-2) if s[i:i+3] == "666")
        if c > 0: yield c
        pow2 <<= 1
print(list(islice(agen(), 88))) # Michael S. Branicky, Apr 07 2024

a(n) >= A371807(n).

A372680 Integers k such that 2^k contains all powers of 2 not exceeding k as substrings.

Original entry on oeis.org

124, 192, 322, 808, 830, 957, 1757, 4067, 5489, 6616, 6724, 6794, 7065, 7727, 7728, 7736, 8253, 8938, 9438, 9989, 10194, 10195, 10271, 10350, 10389, 10397, 10445, 10475, 10611, 10835, 11107, 11500, 11606, 11758, 11835, 12089, 12304, 12398, 12501, 12548, 12645, 12694, 12695, 12734, 12820

Offset: 1

Bryle Morga, May 10 2024

It is unknown whether this sequence contains infinitely many terms.

			124 is a term; 2^124 = 21267647932558653966460912964485513216 contains 2, 4, 8, 16, 32, 64 as substrings.

Cf. A046300, A094776, A371808.

def f(m):
  a = str(2**m)
  for i in range(0, m.bit_length()):
    if str(2**i) not in a:
      return 0
  return 1
def a(n):
  m = 0
  i = 0
  while i != n:
    m += 1
    i += f(m)
  return m

cp's OEIS Frontend

A371806 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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A371809 Number of 666 substrings contained in the decimal expansion of the n-th apocalyptic number, where overlapping substrings are counted as distinct.

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A372680 Integers k such that 2^k contains all powers of 2 not exceeding k as substrings.

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Python