A030000 -id:A030000 - cp's OEIS frontend

A062523 6^a(n) is smallest nonnegative power of 6 containing the string 'n'.

9, 0, 3, 2, 6, 6, 1, 5, 12, 4, 9, 16, 4, 13, 28, 18, 3, 10, 15, 21, 26, 3, 22, 12, 27, 26, 17, 7, 16, 4, 13, 22, 24, 12, 27, 19, 2, 21, 22, 30, 13, 14, 22, 25, 17, 15, 6, 15, 28, 15, 21, 31, 46, 23, 28, 18, 6, 15, 20, 17, 10, 8, 11, 33, 14, 6, 6, 8, 18, 9, 11, 22, 26, 17, 16, 33

Offset: 0

Robert G. Wilson v, Jun 24 2001

Cf. A030000. Essentially the same as A063569.

Table[k = 0; While[ StringPosition[ ToString[6^k], ToString[n] ] == {}, k++ ]; k, {n, 0, 75} ]
Module[{nn=100,p},p=Table[{n,6^n},{n,0,nn}];Table[SelectFirst[p,SequenceCount[ IntegerDigits[ #[[2]]],IntegerDigits[k]]>0&],{k,0,nn}]][[;;,1]] (* Harvey P. Dale, Jun 02 2023 *)

A062526 9^a(n) is smallest power of 9 containing the string 'n'.

Original entry on oeis.org

5, 0, 3, 6, 5, 4, 4, 3, 2, 1, 11, 15, 18, 11, 6, 18, 17, 19, 13, 23, 9, 8, 14, 21, 12, 13, 26, 28, 12, 3, 8, 6, 13, 24, 10, 18, 12, 27, 9, 27, 10, 6, 9, 8, 6, 14, 8, 7, 9, 5, 18, 16, 18, 6, 13, 35, 4, 20, 13, 5, 11, 4, 23, 18, 12, 4, 17, 8, 24, 7, 22, 17, 3, 26, 9, 30

Offset: 0

Robert G. Wilson v, Jun 24 2001

Table[k = 0; While[ StringPosition[ ToString[9^k], ToString[n] ] == {}, k++ ]; k, {n, 0, 75} ]

Cf. A030000.

A062520 3^a(n) is smallest nonnegative power of 3 containing n.

Original entry on oeis.org

10, 0, 3, 1, 5, 8, 8, 3, 4, 2, 21, 19, 17, 22, 11, 13, 17, 11, 7, 9, 18, 7, 19, 13, 5, 26, 19, 3, 24, 6, 16, 12, 13, 31, 15, 21, 24, 29, 18, 31, 17, 12, 18, 5, 12, 28, 16, 11, 15, 10, 35, 32, 33, 12, 26, 27, 8, 40, 26, 10, 21, 8, 19, 17, 24, 8, 33, 16, 9, 14, 35, 11, 6, 29, 18, 47

Offset: 0

Robert G. Wilson v, Jun 24 2001

			a(1) = 0 since 3^0 = 1. a(2) = a(7) = a(27) = 3 because 3^3 = 27.

Robert Israel, Table of n, a(n) for n = 0..9999

Cf. A030000. Essentially the same as A063566. Cf. A000244.

N:= 99:
count:= 1: A["0"]:= 10:
for n from 0 while count <= N do
  S:= convert(3^n,string); nS:= length(S);
  for m from 1 to 2 while count <= N do
    for i from 1 to nS+1-m while count <= N do
      if S[i] <> "0" and not assigned(A[S[i..i+m-1]]) then
         count:= count+1; A[S[i..i+m-1]]:= n;
      fi
    od
  od
od:
seq(A[convert(n,string)],n=0..N); # Robert Israel, Jun 14 2018

Table[k = 0; While[ StringPosition[ ToString[3^k], ToString[n] ] == {}, k++ ]; k, {n, 0, 75} ]

def a(n):
    s, k = str(n), 0
    while s not in str(3**k): k += 1
    return k
print([a(n) for n in range(76)]) # Michael S. Branicky, Oct 04 2021

A062521 4^a(n) is smallest nonnegative power of 4 containing the string 'n'.

Original entry on oeis.org

5, 0, 4, 7, 1, 4, 2, 10, 7, 6, 5, 20, 25, 35, 9, 29, 2, 17, 15, 11, 28, 9, 36, 29, 5, 4, 9, 19, 24, 16, 11, 37, 38, 43, 35, 14, 8, 15, 7, 21, 6, 11, 16, 11, 9, 14, 21, 18, 10, 16, 26, 20, 30, 8, 14, 8, 4, 10, 25, 22, 22, 29, 9, 7, 3, 8, 23, 12, 14, 17, 23, 13, 12, 15, 15, 22

Offset: 0

Robert G. Wilson v, Jun 24 2001

Cf. A030000. Essentially the same as A063567.

Table[k = 0; While[ StringPosition[ ToString[4^k], ToString[n] ] == {}, k++ ]; k, {n, 0, 75} ]

A062522 5^a(n) is smallest nonnegative power of 5 containing the string 'n'.

Original entry on oeis.org

8, 0, 2, 5, 11, 1, 4, 7, 7, 8, 14, 23, 3, 30, 12, 6, 20, 15, 22, 9, 13, 33, 13, 22, 12, 2, 18, 37, 11, 17, 15, 5, 19, 35, 19, 14, 20, 21, 18, 8, 12, 12, 37, 20, 12, 17, 18, 21, 11, 26, 23, 14, 16, 9, 30, 23, 6, 15, 16, 24, 24, 14, 4, 19, 20, 10, 31, 20, 21, 18, 13, 21, 18, 19, 20

Offset: 0

Robert G. Wilson v, Jun 24 2001

Table[k = 0; While[ StringPosition[ ToString[5^k], ToString[n]] == {}, k++ ]; k, {n, 0, 75} ]

Cf. A030000. Essentially the same as A063568.

A062524 7^a(n) is smallest nonnegative power of 7 containing the string 'n'.

Original entry on oeis.org

4, 0, 4, 3, 2, 7, 5, 1, 5, 2, 13, 6, 12, 12, 19, 15, 5, 6, 19, 11, 12, 22, 14, 7, 4, 30, 11, 23, 10, 16, 14, 19, 11, 16, 3, 7, 9, 19, 12, 17, 4, 12, 27, 3, 18, 21, 32, 10, 8, 2, 15, 17, 10, 9, 7, 21, 15, 8, 21, 18, 9, 15, 18, 17, 6, 27, 20, 11, 5, 16, 33, 27, 12, 11, 11, 10

Offset: 0

Robert G. Wilson v, Jun 24 2001

See A176767 for the actual powers 7^a(n). - M. F. Hasler, Oct 03 2014

Cf. A030000.

Essentially the same as A063570.

Table[k = 0; While[ StringPosition[ ToString[7^k], ToString[n] ] == {}, k++ ]; k, {n, 0, 75} ]

A331619 a(n) is the smallest positive number k such that the decimal expansion of n*2^k contains the string n.

Original entry on oeis.org

1, 4, 4, 7, 4, 9, 4, 8, 4, 10, 10, 10, 10, 10, 10, 10, 10, 7, 10, 10, 4, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 4, 10, 28, 19, 13, 15, 16, 40, 20, 30, 53, 27, 20, 27, 35, 30, 20, 53, 31, 20, 4, 8, 18, 35, 20, 10, 30, 23, 20

Offset: 0

Rémy Sigrist, Jan 22 2020

			  n   a(n)  n*2^a(n)
  --  ----  --------
   0     1         0
   1     4        16
   2     4        32
   3     7       384
   4     4        64
   5     9      2560
   6     4        96
   7     8      1792
   8     4       128
   9    10      9216
  10    10     10240

Rémy Sigrist, PARI program for A331619

Cf. A030000, A331440.

A346120 a(n) is the smallest nonnegative number k such that the decimal expansion of k! contains the string n.

Original entry on oeis.org

5, 0, 2, 8, 4, 7, 3, 6, 9, 11, 19, 22, 5, 15, 26, 25, 11, 14, 27, 29, 5, 19, 13, 18, 4, 23, 26, 13, 9, 14, 15, 32, 8, 24, 28, 17, 9, 18, 23, 11, 7, 27, 17, 15, 21, 19, 26, 12, 24, 23, 7, 19, 23, 32, 29, 17, 17, 18, 23, 25, 12, 26, 9, 26, 18, 30, 20, 15, 11, 27, 13

Offset: 0

Ilya Gutkovskiy, Jul 05 2021

			a(3) = 8 since 3 occurs in 8! = 40320, but not in 0!, 1!, 2!, ..., 7!.

Amiram Eldar, Table of n, a(n) for n = 0..10000
Richard V. Andree, Experiments with Natural Numbers, in: LeRoy C. Dalton and Henry D. Snyder (eds.), Topics for Mathematics Clubs, National Council of Teachers of Mathematics, 1973, Excursion 4, p. 64.

Cf. A000142, A030000, A038690, A082058.

a[n_] := (k = 0; While[! MatchQ[IntegerDigits[k!], {_, Sequence @@ IntegerDigits[n], _}], k++]; k); Table[a[n], {n, 0, 70}]

a(n) = my(k=0, s=Str(n)); while (#strsplit(Str(k!), s) < 2, k++); k; \\ Michel Marcus, Jul 05 2021

def A346120(n):
    s, k, f = str(n), 0, 1
    while s not in str(f):
        k += 1
        f *= k
    return k # Chai Wah Wu, Jul 05 2021

A371887 a(1) = 1; for n > 1, a(n) is the smallest positive integer k such that the digits of 2^k contain 2^a(n-1) as a proper substring.

Original entry on oeis.org

1, 5, 15, 507

Offset: 1

Adam Vulic, Apr 11 2024

From David A. Corneth, Apr 11 2024: (Start)
This sequence is well defined as A030000 is well defined; every finite string of digits is contained in some power of 2.
An upper bound for a(n), n > 1, can be found by solving 2^k == 2^a(n-1) (mod 10^m) where m is the number of digits of 2^a(n-1) (cf. A034887). This gives a(n) <= k = a(n-1) + 4*5^(m-1) (cf. A005054). So a(5) <= 507 + 4*5^152, which is about 7*10^106. (End)

			a(2) is the smallest k > 0 such that the digits of 2^k contain 2^a(1) = 2^1 = 2 as a proper substring, so a(2) = 5. (2^5 = 32.)
a(3) is the smallest k > 0 such that the digits of 2^k contain 2^a(2) = 32 as a proper substring, so a(3) = 15. (2^15 = 32768.)

Brady Haran, Apocalyptic Numbers, Numberphile video, 2024.

Cf. A018856, A005054, A030000, A034887.

k = 0; Rest@ NestList[(While[SequenceCount[IntegerDigits[2^k], IntegerDigits[2^#]] == 0, k++]; k++; k - 1) &, 1, 4] (* Michael De Vlieger, Apr 19 2024 *)

A175389 Smallest nonnegative number k such that 2^k contains n, 2n and 3n as substrings of its decimal expansion.

Original entry on oeis.org

10, 17, 18, 29, 50, 87, 86, 31, 70, 62, 101, 147, 86, 124, 93, 144, 82, 81, 157, 113, 100, 110, 146, 110, 88, 96, 141, 158, 94, 69, 79, 75, 123, 244, 192, 297, 181, 168, 128, 255, 101, 140, 197, 182, 147, 228, 111, 189, 224, 303, 288, 510, 321, 289, 232, 432, 342

Offset: 0

Zak Seidov, Apr 27 2010

			2^10 = 1024 is the smallest power of 2 containing a 0, so a(0) = 10.
2^101 = 2535301200456458802993406410752 is the smallest power of 2 containing 10, 20, and 30 as substrings, so a(10) = 101.

Robert Israel, Table of n, a(n) for n = 0..1000

Cf. A030000 (Susanna's sequence: smallest nonnegative number k such that 2^k contains n).

N:= 100: # to get a(0) to a(N)
R:= 'R': count:= 0:
for k from 0 while count < N+1 do
   t:= 2^k;
   d:= ilog10(t);
   V:= select(`<=`,{seq(seq(floor(t/10^i) mod 10^j, j=1..d+1-i),
      i=0..d)},3*N);
   V3:= select(t -> t <= N and has(V,2*t) and has(V,3*t), V);
   for v in V3 do
     if not assigned(R[v]) then
       count:= count+1;
       R[v]:= k;
     fi
   od;
od:
seq(R[n],n=0..N); # Robert Israel, Jul 19 2016

Table[SelectFirst[Range[0, 10^3], Function[k, Length@ DeleteCases[
Map[SequencePosition[IntegerDigits[2^k], IntegerDigits@ #] &, n Range@ 3] /. {} -> 0, m_ /; m == 0] == 3]], {n, 0, 56}] (* Michael De Vlieger, Jul 19 2016, Version 10.1 *)

cp's OEIS Frontend

A062523 6^a(n) is smallest nonnegative power of 6 containing the string 'n'.

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A062526 9^a(n) is smallest power of 9 containing the string 'n'.

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A062520 3^a(n) is smallest nonnegative power of 3 containing n.

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A062521 4^a(n) is smallest nonnegative power of 4 containing the string 'n'.

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A062522 5^a(n) is smallest nonnegative power of 5 containing the string 'n'.

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A062524 7^a(n) is smallest nonnegative power of 7 containing the string 'n'.

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A331619 a(n) is the smallest positive number k such that the decimal expansion of n*2^k contains the string n.

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A346120 a(n) is the smallest nonnegative number k such that the decimal expansion of k! contains the string n.

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A371887 a(1) = 1; for n > 1, a(n) is the smallest positive integer k such that the digits of 2^k contain 2^a(n-1) as a proper substring.

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A175389 Smallest nonnegative number k such that 2^k contains n, 2n and 3n as substrings of its decimal expansion.

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