A321043 -id:A321043 - cp's OEIS frontend

A030000 a(n) is the smallest nonnegative number k such that the decimal expansion of 2^k contains the string n.

10, 0, 1, 5, 2, 8, 4, 15, 3, 12, 10, 40, 7, 17, 18, 21, 4, 27, 30, 13, 11, 18, 43, 41, 10, 8, 18, 15, 7, 32, 22, 17, 5, 25, 27, 25, 16, 30, 14, 42, 12, 22, 19, 22, 18, 28, 42, 31, 11, 32, 52, 9, 19, 16, 25, 16, 8, 20, 33, 33, 23, 58, 18, 14, 6, 16, 46, 24, 15, 34, 29, 21, 17, 30

Offset: 0

N. J. A. Sloane

a(n) is well-defined for all n, because 2^k can actually start with (not just contain) any finite sequence of digits without leading zeros. This follows from the facts that log_10(2) is irrational and that the set of fractional parts of n*x is dense in [0,1] if x is irrational. - Pontus von Brömssen, Jul 21 2021

			2^12 = 4096 is first power of 2 containing a 9, so a(9) = 12.

Giovanni Resta, Table of n, a(n) for n = 0..10000 (terms up to a(1000) from T. D. Noe, a(9634) corrected by Piotr Idzik)

Cf. A030001 (the actual powers of 2), A063565, A018856, A000079, A372044, A372045.

See also A321043.

import Data.List (isInfixOf, findIndex)
import Data.Maybe (fromJust)
a030000 n =
   fromJust $ findIndex (show n `isInfixOf`) $ map show a000079_list
-- Reinhard Zumkeller, Aug 04 2011

Table[ i=0; While[ StringPosition[ ToString[ 2^i ], ToString[ n ] ]=={}, i++ ]; i, {n, 0, 80} ]
snn[n_]:=Module[{k=0},While[SequenceCount[IntegerDigits[2^k],IntegerDigits[n]]==0,k++];k]; Array[snn,100,0] (* Harvey P. Dale, Mar 16 2025 *)

a(n) = {if (n==1, return (0)); my(k=1, sn = Str(n)); while (#strsplit(Str(2^k), sn) == 1, k++); k;} \\ Michel Marcus, Mar 06 2021

apply( A030000(n)={n=Str(n);for(k=0,oo,#strsplit(Str(2^k),n)>1&& return(k))}, [0..99]) \\ Also allows to search for digit strings with leading zeros, e.g., "00" => k=53. - M. F. Hasler, Jul 11 2021

def a(n):
  k, strn = 0, str(n)
  while strn not in str(2**k): k += 1
  return k
print([a(n) for n in range(74)]) # Michael S. Branicky, Mar 06 2021

a(n) <= A018856(n) for n >= 1. - Pontus von Brömssen, Jul 21 2021

More terms from Hans Havermann

A321128 Single-digit numbers in the order in which they first appear in the decimal expansions of prime numbers, followed by the two-digit numbers in the order in which they appear, then the three-digit numbers, and so on.

Original entry on oeis.org

2, 3, 5, 7, 1, 9, 4, 6, 8, 0, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 10, 12, 27, 39, 14, 49, 15, 51, 57, 16, 63, 18, 81, 91, 93, 99, 21, 22, 33, 24, 25, 26, 69, 77, 28, 30, 34, 35, 36, 38, 40, 42, 44, 45, 46, 48, 87

Offset: 1

Rémy Sigrist, Oct 27 2018

This sequence is a variant of A321043.
This sequence establishes a bijection from the positive integers to the nonnegative integers; see A320938 for the inverse.
Prime numbers appear in increasing order.

			The first terms, alongside the corresponding prime numbers, are:
  n   a(n)  Prime
  --  ----  -----
   1     2      2
   2     3      3
   3     5      5
   4     7      7
   5     1     11
   6     9     19
   7     4     41
   8     6     61
   9     8     83
  10     0    101
  11    11     11
  12    13     13
  ...
  30    89     89
  31    97     97
  32    10    101
  33    12    127
  34    27    127
  35    39    139
  36    14    149
  37    49    149

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, Colored scatterplot of the first 100000 terms
Rémy Sigrist, Scatterplot of the first 100000 terms of the analog for the Euler totient function (A000010)
Rémy Sigrist, PARI program for A321128

Cf. A000010, A000040, A320938 (inverse), A321043.

PARI
```
See Links section.
```

A330384 Single-digit numbers in order of appearance in decimal expansions of composite numbers, followed by two-digit numbers in order of appearance, then three-digit numbers and so forth.

Original entry on oeis.org

4, 6, 8, 9, 1, 0, 2, 5, 7, 3, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28, 30, 32, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45, 46, 48, 49, 50, 51, 52, 54, 55, 56, 57, 58, 60, 62, 63, 64, 65, 66, 68, 69, 70, 72, 74, 75, 76, 77, 78, 80, 81, 82, 84, 85, 86

Offset: 1

David Williams, Dec 12 2019

			From _Rémy Sigrist_, Dec 24 2019: (Start)
The first terms, alongside the corresponding composite numbers, are:
  n   a(n)  Composite
  --  ----  ---------
   1     4          4
   2     6          6
   3     8          8
   4     9          9
   5     1         10
   6     0         10
   7     2         12
   8     5         15
   9     7         27
  10     3         30
  11    10         10
  12    12         12
  13    14         14
  ...
  79    99         99
  80    11        110
  81    17        117
  82    19        119
  83    23        123
(End)

Rémy Sigrist, Table of n, a(n) for n = 1..11000
Rémy Sigrist, Colored scatterplot of the first 110000 terms (red pixels indicate terms that are prime)
Rémy Sigrist, PARI program for A330384

Cf. A002808, A321043, A321128.

PARI
```
\\ See Links section.
```

More terms from Rémy Sigrist, Dec 24 2019

cp's OEIS Frontend

A030000 a(n) is the smallest nonnegative number k such that the decimal expansion of 2^k contains the string n.

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A321128 Single-digit numbers in the order in which they first appear in the decimal expansions of prime numbers, followed by the two-digit numbers in the order in which they appear, then the three-digit numbers, and so on.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

A330384 Single-digit numbers in order of appearance in decimal expansions of composite numbers, followed by two-digit numbers in order of appearance, then three-digit numbers and so forth.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

PARI

Extensions