A215236 -id:A215236 - cp's OEIS frontend

A216063 a(n) is the conjectured highest power of n which has no two identical digits in succession.

126, 133, 63, 32, 26, 27, 42, 33, 1, 16, 15, 11, 76, 15, 26, 19, 18, 8, 1, 45, 38, 19, 12, 16, 30, 22, 11, 21, 1, 16, 16, 11, 12, 11, 13, 10, 23, 10, 1, 22, 19, 6, 18, 25, 23, 11, 10, 6, 1, 6, 8, 20, 14, 17, 11, 13, 14, 13, 1, 15, 14, 17, 21, 16, 16, 9, 4, 11

Offset: 2

V. Raman, Sep 01 2012

Contribution from Charles R Greathouse IV, Sep 17 2012: (Start)
a(n) = 0 for infinitely many n; such n have positive density in this sequence. Question: are such n of density 1?
A naive heuristic suggests that there are infinitely many n such that a(n) = 6 but only finitely many a(n) such that a(n) > 6. This suggests a weaker conjecture: this sequence is bounded. (End)

			3^133 = 2865014852390475710679572105323242035759805416923029389510561523 which has no two adjacent identical digits.

T. D. Noe, Table of n, a(n) for n = 2..1000 (terms 2 to 99 from V. Raman).

Cf. A043096, A216064, A216065, A215236.

Table[mx = 0; Do[If[! MemberQ[Differences[IntegerDigits[n^k]], 0], mx = k], {k, 1000}]; mx, {n, 2, 100}] (* T. D. Noe, Sep 17 2012 *)

isA043096(n)=my(v=digits(n));for(i=2,#v,if(v[i]==v[i-1],return(0)));1
a(n)=my(best=0); if(n==14,76,for(k=1, max(9,94\sqrt(log(n))), if(isA043096(n^k), best=k)); best ) \\ (conjectural) Charles R Greathouse IV, Sep 17 2012

A217157 a(n) is the least value of k such that the decimal expansion of n^k contains two consecutive identical digits.

Original entry on oeis.org

16, 11, 8, 11, 5, 6, 6, 6, 2, 1, 2, 9, 3, 2, 4, 7, 5, 5, 2, 2, 1, 6, 4, 6, 5, 4, 8, 5, 2, 6, 5, 1, 2, 2, 3, 7, 2, 4, 2, 5, 3, 4, 1, 3, 2, 2, 3, 3, 2, 7, 4, 3, 6, 1, 4, 4, 2, 4, 2, 3, 2, 3, 3, 2, 1, 2, 3, 4, 2, 3, 7, 6, 3, 6, 2, 1, 3, 4, 2, 3, 3, 2, 5, 2, 4, 6

Offset: 2

V. Raman, Sep 27 2012

Least number m such that n^m is a term of A171901 - Chai Wah Wu, Feb 20 2019
Conjecture: 1 <= a(n) <= 16 for n > 1 and a(n) < 16 for n > 2. - Chai Wah Wu, Feb 20 2019
a(n) >= 1 for all n > 1 and is bounded: see link for proof. - Robert Israel, Feb 21 2019

V. Raman, Table of n, a(n) for n = 2..10000
Robert Israel, Proof that A217157 >= 1 and is bounded

Cf. A045875, A215236.

Cf. A215727, A215728, A215729, A215730, A215731, A171901, A306305.

f:= proc(n) local L,k;
   for k from 1 do
      L:= convert(n^k,base,10);
      if has(L[2..-1]-L[1..-2],0) then return k fi
   od
end proc:
map(f, [$2..100]); # Robert Israel, Feb 21 2019

Table[k = 1; While[! MemberQ[Differences[IntegerDigits[n^k]], 0], k++]; k, {n, 2, 100}] (* T. D. Noe, Oct 01 2012 *)

def A217157(n):
    m, k = 1, n
    while True:
        s = str(k)
        for i in range(1,len(s)):
            if s[i] == s[i-1]:
                return m
        m += 1
        k *= n # Chai Wah Wu, Feb 20 2019

a(A171901(n)) = 1. - Chai Wah Wu, Feb 20 2019

a(n) = A215236(n) + 1. - Georg Fischer, Nov 25 2020

A217167 a(n) is the first digit (from the left) to appear two times in succession in the decimal representation of n^A217157(n).

Original entry on oeis.org

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

Offset: 2

V. Raman, Sep 27 2012

Note that 109^2 = 11881. So by looking at the digits from the left, we have a(109) = 1.

V. Raman, Table of n, a(n) for n = 2..10000

Cf. A217157.
Cf. A215732, A215236.
Cf. A215733, A215734, A215735, A215736, A215737.

Table[k = 1; While[d = IntegerDigits[n^k]; ! MemberQ[df = Differences[d], 0], k++]; d[[Position[df, 0][[1, 1]]]], {n, 2, 100}] (* T. D. Noe, Oct 02 2012 *)

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A216063 a(n) is the conjectured highest power of n which has no two identical digits in succession.

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A217157 a(n) is the least value of k such that the decimal expansion of n^k contains two consecutive identical digits.

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A217167 a(n) is the first digit (from the left) to appear two times in succession in the decimal representation of n^A217157(n).

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