A007356 -id:A007356 - 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

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

Original entry on oeis.org

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

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).

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).

A248504 Least number k > 0 such that n^k contains 666 in its decimal representation, or 0 if no such k exists.

Original entry on oeis.org

0, 157, 34, 96, 102, 18, 70, 64, 17, 0, 42, 41, 25, 44, 30, 48, 16, 97, 30, 157, 50, 33, 15, 35, 51, 12, 35, 10, 34, 34, 34, 44, 44, 30, 47, 9, 20, 46, 23, 96, 33, 13, 42, 32, 39, 17, 8, 27, 35, 102, 22, 42, 80, 55, 28, 55, 38, 19, 48, 18, 74, 15, 31, 32, 37

Offset: 1

Talha Ali, Dec 01 2014

a(n) <= a(2) = 157 for all n <= 10^5. Is there any n for which a(n) > 157? - Robert Israel, Dec 01 2014

			a(2)=157 because 2^157=182687704666362864775460604089535377456991567872 contains '666' (see A007356).
a(3)=34 because 3^34=16677181699666568 contains '666' and belongs to A051003.

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A007356, A051003.

f:= proc(n)
local k;
if n = 10^ilog10(n) then return 0 fi;
for k from 1 do
  if StringTools[Search]("666",sprintf("%d",n^k)) <> 0 then return k fi
od
end proc;
seq(f(n), n=1..1000); # Robert Israel, Dec 01 2014

A248504[n_] := If[IntegerQ[Log10[n]], 0, Block[{k = 0}, While[StringFreeQ[IntegerString[n^++k], "666"]]; k]];
Array[A248504, 100] (* Paolo Xausa, Apr 08 2024 *)

isok(n) = {d = digits(n); for (i=1, #d-3, if ((d[i] == 6) && (d[i+1]==6) && (d[i+2]==6), return(1));); return (0);}
a(n) = {if ((n==1) || (n==10) || (ispower(n,,&p) && (p==10)), return(0)); k = 1; while (! isok(n^k), k++); k;} \\ Michel Marcus, Dec 01 2014

More terms from Alois P. Heinz, Dec 01 2014

A328375 Numbers k such that the decimal expansion of 2^k contains the substring 777.

Original entry on oeis.org

24, 40, 75, 152, 166, 179, 181, 191, 194, 199, 214, 230, 235, 260, 282, 296, 304, 311, 317, 323, 326, 332, 342, 345, 363, 370, 374, 390, 417, 424, 426, 443, 455, 468, 471, 474, 475, 483, 489, 490, 505, 512, 523, 524, 536, 540, 559, 567, 581, 584, 585, 588, 593

Offset: 1

Eder Vanzei, Oct 14 2019

The decimal expansion of 2^k ends in 7776 iff k == 40 (mod 500), so the sequence is infinite. - Jon E. Schoenfield, Oct 14 2019

Conjecture: if n > 30536, then a(n) = n + 3623. - Chai Wah Wu, Oct 26 2019

			16777216 = 2^24.

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Cf. A007356 (contains 666), A030000 (contains n).

q:= n-> searchtext("777", cat(2^n))>0:
select(q, [$1..600])[];  # Alois P. Heinz, Oct 26 2019

aQ[n_] := SequenceCount[IntegerDigits[2^n], {7, 7, 7}] > 0; Select[Range[660], aQ] (* Amiram Eldar, Oct 26 2019 *)

A328375_list = [k for k in range(1000) if '777' in str(2**k)] # Chai Wah Wu, Oct 26 2019

A371838 Decimal expansion of 666^666, the Legion's number of the first kind.

Original entry on oeis.org

2, 7, 1, 5, 4, 1, 7, 5, 9, 2, 8, 8, 7, 1, 2, 8, 5, 5, 8, 2, 6, 0, 8, 7, 4, 5, 5, 1, 7, 0, 0, 2, 1, 7, 8, 6, 0, 2, 7, 8, 3, 8, 5, 2, 1, 0, 6, 5, 0, 1, 6, 9, 8, 7, 1, 7, 8, 2, 2, 3, 0, 0, 4, 6, 9, 6, 5, 7, 8, 3, 6, 7, 5, 3, 4, 7, 8, 4, 6, 0, 3, 6, 8, 8, 0, 1, 3

Offset: 1881

Paolo Xausa, Apr 08 2024

This number has 1881 digits and contains two nonoverlapping 666 substrings.

			2715417592887128558260874551700217860278385210650169871782230046965783675...

Paolo Xausa, Table of n, a(n) for n = 1881..3761
Brady Haran and Tony Padilla, Goliath & Leviathan Numbers, YouTube Numberphile video, 2024.
Eric Weisstein's World of Mathematics, Legion's Numbers.

Cf. A007356, A051003, A115983.

Mathematica
```
IntegerDigits[666^666][[;;100]]
```

digits(666^666) \\ Michel Marcus, Apr 08 2024

afull = list(map(int, str(666**666))) # Michael S. Branicky, Apr 08 2024

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

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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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A248504 Least number k > 0 such that n^k contains 666 in its decimal representation, or 0 if no such k exists.

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A328375 Numbers k such that the decimal expansion of 2^k contains the substring 777.

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A371838 Decimal expansion of 666^666, the Legion's number of the first kind.

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