A215737 -id:A215737 - cp's OEIS frontend

A215731 a(n) is the smallest m for which the decimal representation of 11^m contains n consecutive identical digits.

0, 1, 8, 39, 156, 482, 1323, 2983, 9443, 39879, 214747, 296095, 296095, 5541239, 8621384, 30789328

Offset: 1

V. Raman, Aug 22 2012

			The decimal representation of 11^39879 contains ten consecutive 6s, and is the least such power with such a string of digits.

Cf. A215737 (the repeated digits), A045875, A215727, A215728, A215729, A215730.

mostDigits[t_] := Module[{lastDigit = t[[1]], record = 1, cnt = 1}, Do[If[t[[n]] == lastDigit, cnt++, If[cnt > record, record = cnt]; cnt = 1; lastDigit = t[[n]]], {n, 2, Length[t]}]; If[cnt > record, record = cnt] ; record]; nn = 10; t = Table[-1, {nn}]; n = -1; While[Min[t] == -1, n++; c = mostDigits[IntegerDigits[11^n]]; If[c > nn, c = nn]; While[c > 0 && t[[c]] == -1, t[[c]] = n; c--]]; t (* T. D. Noe, Apr 29 2013 *)

def A215731(n):
    l, x = [str(d)*n for d in range(10)], 1
    for m in range(10**9):
        s = str(x)
        for k in l:
            if k in s:
                return m
        x *= 11
    return 'search limit reached'
# Chai Wah Wu, Dec 17 2014

a(10) discovered by "Wick" (See http://www.mersenneforum.org/showpost.php?p=334789&postcount=89).  Definition clarified and all terms to a(10) verified by Daran Gill, Mar 24 2013
a(11) discovered by Tom Womack (See http://www.mersenneforum.org/showpost.php?p=337916&postcount=105), Rick van der Hoorn, Apr 24 2013
a(12)-a(13) from Giovanni Resta, Apr 25 2013
Corrected a(12), Rick van der Hoorn, Apr 28 2013
a(14) from Giovanni Resta, Apr 18 2016
a(15) from Bert Dobbelaere, Feb 15 2019
a(16) from Paul Geneau de Lamarlière, Oct 03 2024

A217171 a(n) is the first digit (from the left) to appear six times in succession in the decimal representation n^A217161(n).

Original entry on oeis.org

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

Offset: 2

V. Raman, Sep 27 2012

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

Cf. A217161.

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

Table[k = 1; While[d = IntegerDigits[n^k]; df = Partition[Differences[d], 5, 1]; ! MemberQ[df, {0, 0, 0, 0, 0}], k++]; d[[Position[df, {0, 0, 0, 0, 0}][[1, 1]]]], {n, 2, 100}] (* T. D. Noe, Oct 02 2012 *)

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 *)

A217168 a(n) is the first digit (from the left) to appear three times in succession in the decimal representation of n^A217158(n).

Original entry on oeis.org

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

Offset: 2

V. Raman, Sep 27 2012

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

Cf. A217158.

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

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

A217169 a(n) is the first digit (from the left) to appear four times in succession in the decimal representation of n^A217159(n).

Original entry on oeis.org

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

Offset: 2

V. Raman, Sep 27 2012

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

Cf. A217159.

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

Table[k = 1; While[d = IntegerDigits[n^k]; df = Partition[Differences[d], 3, 1]; ! MemberQ[df, {0, 0, 0}], k++]; d[[Position[df, {0, 0, 0}][[1, 1]]]], {n, 2, 100}] (* T. D. Noe, Oct 02 2012 *)

A217170 a(n) is the first digit (from the left) to appear five times in succession in the decimal representation of n^A217160(n).

Original entry on oeis.org

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

Offset: 2

V. Raman, Sep 27 2012

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

Cf. A217160.

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

Table[k = 1; While[d = IntegerDigits[n^k]; df = Partition[Differences[d], 4, 1]; ! MemberQ[df, {0, 0, 0, 0}], k++]; d[[Position[df, {0, 0, 0, 0}][[1, 1]]]], {n, 2, 100}] (* T. D. Noe, Oct 02 2012 *)

A217172 a(n) is the first digit (from the left) to appear seven times in succession in the decimal representation of n^A217162(n).

Original entry on oeis.org

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

Offset: 2

V. Raman, Sep 27 2012

V. Raman, Table of n, a(n) for n = 2..1000

Cf. A217162.

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

Table[k = 1; While[d = IntegerDigits[n^k]; df = Partition[Differences[d], 6, 1]; ! MemberQ[df, {0, 0, 0, 0, 0, 0}],  k++]; d[[Position[df, {0, 0, 0, 0, 0, 0}][[1, 1]]]], {n, 2, 100}] (* T. D. Noe, Oct 02 2012 *)

A217175 a(n) is the first digit (from the left) to appear n times in succession in the decimal representation of the Fibonacci(A217165(n)).

Original entry on oeis.org

0, 5, 7, 7, 1, 5, 7, 7, 3, 2, 4, 3, 4, 2, 4

Offset: 1

V. Raman, Sep 27 2012

Cf. A217165.

Cf. A000045, A215732, A215733, A215734, A215735, A215736, A215737.

k = 0; Join[{0}, Table[While[d = IntegerDigits[Fibonacci[k]]; prt = Partition[Differences[d], n - 1, 1]; ! MemberQ[prt, Table[0, {n - 1}]], k++]; d[[Position[prt, Table[0, {n - 1}]][[1, 1]]]], {n, 2, 8}]] (* T. D. Noe, Oct 03 2012 *)

def A217175(n):
    if n == 1:
        return 0
    else:
        l, y, x = [str(d)*n for d in range(10)], 0, 1
        for m in range(1, 10**9):
            s = str(x)
            for k in range(10):
                if l[k] in s:
                    return k
            y, x = x, y+x
        return 'search limit reached'
# Chai Wah Wu, Dec 17 2014

a(10)-a(11) from Chai Wah Wu, Dec 17 2014

a(12)-a(15) from Nick Hobson, Feb 14 2024

A217176 a(n) is the first digit (from the left) to appear n times in succession in the decimal representation of the Lucas(A217166(n)).

Original entry on oeis.org

2, 1, 3, 0, 2, 2, 9, 7, 2, 1, 1, 5, 5, 7, 7, 9

Offset: 1

V. Raman, Sep 27 2012

Cf. A217166.

Cf. A000032, A215732, A215733, A215734, A215735, A215736, A215737.

k = 0; Join[{2}, Table[While[d = IntegerDigits[LucasL[k]]; prt = Partition[Differences[d], n - 1, 1]; ! MemberQ[prt, Table[0, {n - 1}]], k++]; d[[Position[prt, Table[0, {n - 1}]][[1, 1]]]], {n, 2, 8}]] (* T. D. Noe, Oct 03 2012 *)

def A217176(n):
    if n == 1:
        return 2
    else:
        l, y, x = [str(d)*n for d in range(10)], 2, 1
        for m in range(1, 10**9):
            s = str(x)
            for k in range(10):
                if l[k] in s:
                    return k
            y, x = x, y+x
        return 'search limit reached'
# Chai Wah Wu, Dec 17 2014

a(11) from Chai Wah Wu, Dec 17 2014

a(12)-a(16) from Nick Hobson, Feb 03 2024

A217173 a(n) is the first digit (from the left) to appear eight times in succession in the decimal representation of n^A217163(n).

Original entry on oeis.org

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

Offset: 2

V. Raman, Sep 27 2012

Cf. A217163.

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

Table[k = 1; While[d = IntegerDigits[n^k]; df = Partition[Differences[d], 7, 1]; ! MemberQ[df, {0, 0, 0, 0, 0, 0, 0}],  k++]; d[[Position[df, {0, 0, 0, 0, 0, 0, 0}][[1, 1]]]], {n, 2, 10}] (* T. D. Noe, Oct 02 2012 *)

cp's OEIS Frontend

A215731 a(n) is the smallest m for which the decimal representation of 11^m contains n consecutive identical digits.

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A217171 a(n) is the first digit (from the left) to appear six times in succession in the decimal representation n^A217161(n).

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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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A217168 a(n) is the first digit (from the left) to appear three times in succession in the decimal representation of n^A217158(n).

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A217169 a(n) is the first digit (from the left) to appear four times in succession in the decimal representation of n^A217159(n).

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A217170 a(n) is the first digit (from the left) to appear five times in succession in the decimal representation of n^A217160(n).

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A217172 a(n) is the first digit (from the left) to appear seven times in succession in the decimal representation of n^A217162(n).

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A217175 a(n) is the first digit (from the left) to appear n times in succession in the decimal representation of the Fibonacci(A217165(n)).

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A217176 a(n) is the first digit (from the left) to appear n times in succession in the decimal representation of the Lucas(A217166(n)).

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A217173 a(n) is the first digit (from the left) to appear eight times in succession in the decimal representation of n^A217163(n).

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