A259081 -id:A259081 - cp's OEIS frontend

A094776 a(n) = largest k such that the decimal representation of 2^k does not contain the digit n.

86, 91, 168, 153, 107, 71, 93, 71, 78, 108

Offset: 0

Michael Taktikos, Jun 09 2004

These values are only conjectural.

The sequence could be extended to any nonnegative integer index n defining a(n) to be the largest k such that n does not appear as substring in the decimal expansion of 2^k. I conjecture that for n = 10, 11, 12, ... it continues (2000, 3020, 1942, 1465, 1859, 2507, 1950, 1849, 1850, ...). For example, curiously enough, the largest power of 2 in which the string "10" does not appear seems to be 2^2000. - M. F. Hasler, Feb 10 2023

			a(0) = 86 because 2^86 = 77371252455336267181195264 is conjectured to be the highest power of 2 that doesn't contain the digit 0.

J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 71, p. 25, Ellipses, Paris 2008.

Tanya Khovanova, 86 Conjecture, T. K.'s Math Blog, Feb. 15, 2011.
Popular Computing (Calabasas, CA), Two Tables, Vol. 1, (No. 9, Dec 1973), page PC9-16.

Cf. A259081 - A259083.
Cf. A027870 and A065712 - A065744 (number of '0's, ..., '9's in 2^n).
Cf. A034293 (numbers k such that 2^k has no '2').

f[n_] := Block[{a = {}, k = 1}, While[k < 10000, If[ Position[ Union[ IntegerDigits[ 2^k, 10]], n] == {}, AppendTo[a, k]]; k++ ]; a]; Table[ f[n][[ -1]], {n, 0, 9}] (* Robert G. Wilson v, Jun 12 2004 *)

A094776(n,L=10*20^#Str(n))={forstep(k=L, 0, -1, foreach(digits(1<M. F. Hasler, Feb 13 2023

def A094776(n, L=0):
   n = str(n)
   for k in range(L if L else 10*20**len(n), 0, -1):
      if n not in str(2**k): return k # M. F. Hasler, Feb 13 2023

A259086 a(n) = largest k such that the decimal representation of prime(n)^k does not contain the digit 2.

Original entry on oeis.org

168, 59, 1, 28, 38, 25, 16, 22, 28, 20, 22, 7, 19, 20, 4, 10, 27, 11, 8, 13, 13, 12, 5, 23, 23, 18, 42, 7, 31, 4, 10, 13, 11, 5, 11, 11, 8, 12, 12, 9, 7, 10, 10, 4, 7, 10, 3, 18, 7, 0, 8, 3, 6, 2, 8, 4, 12, 5, 4, 5, 6, 4, 8, 16, 2, 3, 5, 2, 7, 11, 12, 1, 5, 9

Offset: 1

N. J. A. Sloane, Jun 18 2015

These values are only conjectural.

Hiroaki Yamanouchi, Table of n, a(n) for n = 1..500
Popular Computing (Calabasas, CA), Two Tables, Vol. 1, (No. 9, Dec 1973), page PC9-16.

Cf. A094776, A259081-A259085.

a(14)-a(74) from Hiroaki Yamanouchi, Jun 19 2015

A259083 a(n) = largest k such that the decimal representation of 7^k does not contain the digit n.

Original entry on oeis.org

35, 30, 28, 20, 29, 25, 33, 39, 33, 61

Offset: 0

N. J. A. Sloane, Jun 18 2015

These values are only conjectural.

Popular Computing (Calabasas, CA), Two Tables, Vol. 1, (No. 9, Dec 1973), page PC9-16.

Cf. A094776, A259081, A259082.

A259085 a(n) = largest k such that the decimal representation of prime(n)^k does not contain the digit 1, or -1 if no such k exists.

Original entry on oeis.org

91, 43, 42, 30, -1, 14, 13, 23, 7, 3, -1, 6, -1, 3, 5, 2, 19, -1, 9, -1, 17, 5, 6, 9, 6, -1, -1, -1, 11, -1, 13, -1, 7, -1, 5, -1, 5, 3, 13, 6, 7, -1, -1, 7, 10, 15, -1, 2, 5, 7, 2, 9, -1, -1, 2, 6, 1, -1, 2, -1, 21, 2, 3, -1, 3, 11, -1, 5, 11, 3, 3, 3, 5, 1

Offset: 1

N. J. A. Sloane, Jun 18 2015

These values are only conjectural.

Hiroaki Yamanouchi, Table of n, a(n) for n = 1..500
Popular Computing (Calabasas, CA), Two Tables, Vol. 1, (No. 9, Dec 1973), page PC9-16.

Cf. A094776, A259081-A259086.

a(14)-a(74) from Hiroaki Yamanouchi, Jun 19 2015

A259084 a(n) = largest k such that the decimal representation of prime(n)^k does not contain the digit 0.

Original entry on oeis.org

86, 68, 58, 35, 41, 14, 27, 44, 10, 14, 16, 16, 9, 10, 8, 7, 14, 16, 14, 8, 6, 9, 4, 23, 8, 0, 14, 10, 12, 10, 6, 14, 5, 8, 5, 13, 7, 16, 7, 17, 6, 3, 9, 9, 16, 7, 12, 11, 4, 13, 7, 16, 8, 9, 3, 10, 4, 9, 6, 4, 5, 13, 3, 12, 7, 9, 6, 8, 4, 39, 13, 12, 10, 4

Offset: 1

N. J. A. Sloane, Jun 18 2015

These values are only conjectural.

a(n) = 0 if prime(n) is in A062800. - Robert Israel, Jun 19 2015

			a(1)=86 because 2^86 = 77371252455336267181195264 is conjectured to be the highest power of 2 that doesn't contain the digit 0.

Hiroaki Yamanouchi, Table of n, a(n) for n = 1..500
Popular Computing (Calabasas, CA), Two Tables, Vol. 1, (No. 9, Dec 1973), page PC9-16.

Cf. A062800, A094776, A259081-A259086.

N:= 100: K:= 100:  # to get a(1) to a(N), searching up to k = K
for n from 1 to N do
  p:= ithprime(n);
  A[n]:= 0;
  for k from 1 to K do
    if not has(convert(p^k,base,10),0) then
       A[n]:= k
    fi
  od
od:
seq(A[n],n=1..N); # Robert Israel, Jun 19 2015

a(14)-a(57) from Hiroaki Yamanouchi, Jun 19 2015

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A094776 a(n) = largest k such that the decimal representation of 2^k does not contain the digit n.

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A259086 a(n) = largest k such that the decimal representation of prime(n)^k does not contain the digit 2.

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A259083 a(n) = largest k such that the decimal representation of 7^k does not contain the digit n.

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A259085 a(n) = largest k such that the decimal representation of prime(n)^k does not contain the digit 1, or -1 if no such k exists.

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A259084 a(n) = largest k such that the decimal representation of prime(n)^k does not contain the digit 0.

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