A032740 -id:A032740 - cp's OEIS frontend

A124692 Position of substring A032740(n) within the decimal expansion of 2^A032740(n).

1, 1, 3, 10, 2, 10, 5, 3, 11, 13, 2, 5, 20, 4, 3, 24, 19, 5, 17, 19, 35, 25, 53, 67, 60, 61, 33, 74, 18, 109, 17, 30, 117, 96, 66, 14, 57, 62, 97, 80, 25, 81, 38, 112, 143, 152, 2, 92, 122, 113, 129, 93, 20, 159, 138, 187, 26, 57, 35, 105, 158, 9, 50, 58, 198, 185, 182, 42, 17

Offset: 1

Cino Hilliard, Dec 25 2006, corrected Jun 26 2007

A032740 includes numbers n that are a substring of 2^n.

			A032740(1) = 6 is a substring of 64 = 2^6 starting at position 1, so a(1) = 1.
A032740(2) = 10 is a substring of 1024 = 2^10 starting at position 1, so a(2) = 1.
A032740(3) = 36 is a substring of 68719476736 = 2^36 starting at position 10, so a(3) = 10.

David W. Wilson, Table of n, a(n) for n = 1..10000

Edited by N. J. A. Sloane at the suggestion of Andrew S. Plewe, Jun 23 2007

A100129 Numbers k such that 2^k starts with k.

Original entry on oeis.org

6, 10, 1542, 77075, 113939, 1122772, 2455891300, 2830138178, 136387767490, 2111259099790, 3456955336468, 4653248164310, 10393297007134, 321249146279171, 972926121017616, 72780032758751764

Offset: 1

Mark Hudson (mrmarkhudson(AT)hotmail.com), Nov 15 2004

According to van de Lune, Erdős observed that 2^6 = 64 and 2^10 = 1024 were two examples where the decimal expansion of 2^k starts with that of k. At that time no other examples were known. Jan van de Lune computed the first 6 terms in 1978. - Juan Arias-de-Reyna, Feb 12 2016

			2^6 = 64, which begins with 6;
2^10 = 1024, which begins with 10.

J. van de Lune, A note on a problem of Erdős, Department of Pure Mathematics, ZW 87/78, Mathematisch Centrum, Amsterdam 1978, 1-3.

Cf. A064541 (2^k ending with k), A032740 (k a substring of 2^k), A131494.

f[n_] := Floor[ 10^Floor[ Log[10, n]](10^FractionalPart[n*N[ Log[10, 2], 24]])]; Do[ If[ f[n] == n, Print[n]], {n, 125000000}] (* Robert G. Wilson v, Nov 16 2004 *)

# Caveat: fails for large n due to rounding error.
from math import log10 as log
frac = lambda x: x - int(x)
is_a100129 = lambda n: 0 <= frac(n * log(2)) - frac(log(n)) < log(n + 1) - log(n) # David Radcliffe, Jun 02 2019

from itertools import count, islice
def A100129_gen(startvalue=1): # generator of terms
    a = 1<<(m:=max(startvalue,1))
    for n in count(m):
        if (s:=str(n))==str(a)[:len(s)]:
            yield n
        a <<= 1
A100129_list = list(islice(A100129_gen(),4)) # Chai Wah Wu, Apr 10 2023

The sequence contains k if and only if 0 <= {k*log_10(2)} - {log_10(k)} < log_10(k+1) - log_10(k), where {x} denotes the fractional part of x. See the van de Lune article. - David Radcliffe, Jun 02 2019

a(5) and a(6) from Robert G. Wilson v, Nov 16 2004

More terms from Robert Gerbicz, Aug 22 2006

A049301 Numbers k such that k is a substring of 3^k.

Original entry on oeis.org

7, 9, 24, 28, 57, 61, 62, 69, 71, 72, 77, 78, 80, 83, 87, 89, 95, 111, 162, 170, 174, 185, 191, 218, 222, 225, 229, 232, 249, 255, 259, 266, 267, 286, 288, 298, 314, 315, 322, 328, 329, 330, 332, 338, 351, 352, 362, 373, 376, 381, 386, 387, 414, 421, 435, 438

Offset: 1

David W. Wilson

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A000244, A032740, A049302, A049303, A049304, A049305, A049306, A049307.

ssQ[n_]:=MemberQ[Partition[IntegerDigits[3^n],IntegerLength[n],1], IntegerDigits[ n]]; Select[Range[500],ssQ] (* Harvey P. Dale, Jul 16 2013 *)

A049302 Numbers k such that k is a substring of 4^k.

Original entry on oeis.org

6, 10, 17, 25, 36, 42, 50, 59, 60, 61, 72, 73, 78, 79, 81, 84, 86, 87, 89, 92, 93, 95, 96, 160, 200, 212, 222, 225, 227, 239, 260, 261, 269, 290, 291, 296, 300, 301, 304, 311, 313, 315, 324, 326, 327, 330, 336, 344, 345, 348, 350, 355, 362, 372, 378, 379, 381

Offset: 1

David W. Wilson

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000302, A032740, A049301, A049303, A049304, A049305, A049306, A049307.

Includes A288845.

A049303 Numbers k such that k is a substring of 5^k.

Original entry on oeis.org

2, 5, 6, 7, 9, 19, 25, 32, 34, 36, 41, 54, 55, 56, 59, 62, 64, 67, 69, 70, 71, 75, 80, 81, 82, 84, 86, 87, 89, 92, 93, 95, 96, 111, 115, 125, 128, 140, 163, 166, 177, 178, 189, 192, 205, 212, 219, 221, 226, 233, 236, 242, 258, 259, 267, 294, 303, 309, 323, 327, 329

Offset: 1

David W. Wilson

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A000351, A032740, A049301, A049302, A049304, A049305, A049306, A049307.

ss5nQ[n_]:=Module[{len=IntegerLength[n]},MemberQ[Partition[ IntegerDigits[ 5^n], len,1],IntegerDigits[n]]]; Select[Range[400],ss5nQ] (* Harvey P. Dale, Jan 06 2013 *)
Select[Range[350],SequenceCount[IntegerDigits[5^#],IntegerDigits[#]]>0&] (* Harvey P. Dale, Dec 27 2024 *)

A049304 Numbers k such that k is a substring of 6^k.

Original entry on oeis.org

6, 7, 9, 13, 21, 22, 23, 29, 39, 40, 42, 44, 45, 48, 53, 55, 56, 60, 63, 64, 65, 67, 68, 69, 70, 73, 74, 75, 76, 77, 79, 82, 83, 87, 89, 92, 93, 94, 98, 105, 107, 127, 129, 131, 134, 137, 143, 147, 152, 163, 165, 167, 174, 179, 184, 189, 197, 224, 226, 227, 234, 240

Offset: 1

David W. Wilson

			9 is in the sequence because 6^9 = 10077696 contains 9 as a substring. - _David A. Corneth_, Aug 13 2021

David A. Corneth, Table of n, a(n) for n = 1..10000

Cf. A000400, A032740, A049301, A049302, A049303, A049305, A049306, A049307.

Select[Range[250],SequenceCount[IntegerDigits[6^#],IntegerDigits[#]]>0&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Aug 03 2018 *)

is(n) = { my(digs6n, digsn, streak, i, j); digs6n = digits(6^n); digsn = digits(n); for(i = 1, #digs6n + 1 - #digsn, streak = 0; for(j = 1, #digsn, if(digs6n[i + j - 1] == digsn[j], streak++ , next(2) ) ); if(streak == #digsn, return(1) ) ); 0 } \\ David A. Corneth, Aug 13 2021

def ok(n): return str(n) in str(6**n)
print(list(filter(ok, range(241)))) # Michael S. Branicky, Aug 13 2021

A049306 Numbers k such that k is a substring of 8^k.

Original entry on oeis.org

4, 6, 7, 10, 13, 17, 18, 28, 31, 33, 36, 38, 42, 44, 47, 48, 49, 52, 54, 56, 58, 60, 63, 64, 67, 68, 69, 76, 77, 79, 81, 82, 83, 85, 86, 89, 90, 91, 94, 97, 112, 115, 124, 130, 135, 165, 173, 176, 178, 189, 193, 195, 206, 208, 215, 221, 225, 249, 251, 252, 253, 256

Offset: 1

David W. Wilson

David A. Corneth, Table of n, a(n) for n = 1..10000

Cf. A001018, A032740, A049301, A049302, A049303, A049304, A049305, A049307.

Select[Range[300],SequenceCount[IntegerDigits[8^#],IntegerDigits[#]]>0&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Dec 16 2018 *)

def ok(n): return str(n) in str(8**n)
print(list(filter(ok, range(257)))) # Michael S. Branicky, Aug 13 2021

A049305 Numbers k such that k is a substring of 7^k.

Original entry on oeis.org

3, 4, 6, 8, 12, 15, 20, 40, 42, 43, 50, 53, 55, 59, 60, 61, 62, 69, 72, 73, 74, 75, 78, 79, 80, 81, 83, 86, 87, 88, 89, 93, 94, 95, 96, 97, 99, 100, 103, 111, 113, 114, 118, 164, 165, 185, 193, 200, 207, 210, 215, 220, 230, 232, 238, 241, 243, 250, 253, 254, 255

Offset: 1

David W. Wilson

Cf. A000420, A032740, A049301, A049302, A049303, A049304, A049306, A049307.

def ok(n): return str(n) in str(7**n)
print(list(filter(ok, range(256)))) # Michael S. Branicky, Aug 13 2021

A049307 Numbers k such that k is a substring of 9^k.

Original entry on oeis.org

5, 7, 9, 25, 26, 31, 37, 43, 46, 47, 48, 53, 59, 60, 61, 63, 68, 69, 70, 72, 74, 76, 80, 85, 87, 88, 89, 91, 94, 97, 101, 104, 107, 124, 132, 135, 140, 148, 158, 166, 170, 180, 187, 190, 199, 209, 211, 215, 231, 243, 244, 256, 266, 270, 271, 279, 283, 288, 289, 291

Offset: 1

David W. Wilson

Cf. A032740, A049301, A049302, A049303, A049304, A049305, A049306.

A178246 Numbers m such that all digits of m, including repetitions, occur among the digits of 2^m.

Original entry on oeis.org

6, 10, 14, 17, 21, 25, 27, 28, 29, 30, 31, 35, 36, 37, 39, 44, 45, 46, 47, 48, 49, 50, 51, 52, 54, 56, 57, 58, 60, 61, 62, 63, 64, 66, 67, 68, 69, 70, 72, 73, 74, 75, 76, 77, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 92, 93, 94, 95, 96, 97, 98, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 113, 114, 115, 116, 117

Offset: 1

Michel Lagneau, Dec 20 2010

The sequence shows subsets of consecutive numbers.

153 is assumed to be the largest integer missing in this sequence. - Alois P. Heinz, Jan 28 2022

			17 is a term because the digits 1 and 7 are included in the number 2^17 = 131072;
3 is not a term because the digit 3 is not in the number 2^3 = 8.
33 is not a term because 2^33 = 8589934592 does not have 2 digits 3.
153 is not in the sequence because the digit 3 is not in the number 2^153 = 11417981541647679048466287755595961091061972992.

Michel Marcus, Table of n, a(n) for n = 1..962

Cf. A000079, A064827, A032740.

Reap[Do[a = DigitCount[2^n]; b = DigitCount[n]; If[Min[a-b] >= 0, Sow[n]], {n, 10^3}]][[2, 1]]

isok(m) = {my(d=digits(m), dd=digits(2^m)); for (i=0, 9, if (#select(x->(x==i), d) > #select(x->(x==i), dd), return (0));); return(1);} \\ Michel Marcus, Jan 28 2022

Name clarified by Michel Marcus, Jan 30 2022

cp's OEIS Frontend

A124692 Position of substring A032740(n) within the decimal expansion of 2^A032740(n).

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A100129 Numbers k such that 2^k starts with k.

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A049301 Numbers k such that k is a substring of 3^k.

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A049302 Numbers k such that k is a substring of 4^k.

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A049303 Numbers k such that k is a substring of 5^k.

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A049304 Numbers k such that k is a substring of 6^k.

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A049306 Numbers k such that k is a substring of 8^k.

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A049305 Numbers k such that k is a substring of 7^k.

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Python

A049307 Numbers k such that k is a substring of 9^k.

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A178246 Numbers m such that all digits of m, including repetitions, occur among the digits of 2^m.

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