A215732 -id:A215732 - cp's OEIS frontend

A045875 a(n) is the smallest m for which the decimal representation of 2^m contains n consecutive identical digits.

0, 16, 24, 41, 220, 971, 972, 8554, 42485, 42486, 271979, 1039315, 1727602, 6855865, 63416789, 106892452, 356677212

Offset: 1

a(9) > 20000. - David Wasserman, Feb 16 2002
a(11) > 250000. - Robert G. Wilson v, Oct 21 2010
a(15) > 10297974. - T. D. Noe, Sep 08 2012
a(17) > 107000000. - Paul Geneau de Lamarlière, Feb 04 2024

Cf. A215732 (the digits).

a[n_] := Block[{k = 0}, While[ !MemberQ[ Length /@ Split@ IntegerDigits[2^k], n], k++ ]; k]; Table[a[n], {n, 6}] (* Robert G. Wilson v, Oct 21 2010 *)

def A045875(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 *= 2
    return 'search limit reached'
# Chai Wah Wu, Dec 17 2014

More terms from David Wasserman, Feb 16 2002
a(9) and a(10) from Robert G. Wilson v, Oct 21 2010
a(11)-a(13) added by T. D. Noe, Sep 04 2012
a(14) added by T. D. Noe, Sep 06 2012
Definition clarified by Daran Gill, Mar 24 2013
a(15) from Bert Dobbelaere, Feb 25 2019
a(16) from Paul Geneau de Lamarlière, Feb 04 2024
a(17) from Paul Geneau de Lamarlière, Sep 24 2024

A215733 a(n) is the first digit to appear n times in succession in a power of 3.

Original entry on oeis.org

1, 7, 8, 5, 5, 8, 2, 1, 2, 2, 2, 4, 5, 8, 2, 4, 4

Offset: 1

V. Raman, Aug 22 2012

Cf. A215727, A045875, A215732.

def A215733(n):
    l, x = [str(d)*n for d in range(10)], 1
    for m in range(10**9):
        s = str(x)
        for k in range(10):
            if l[k] in s:
                return k
        x *= 3
    return 'search limit reached'
# Chai Wah Wu, Dec 17 2014

a(12) from Chai Wah Wu, Dec 17 2014
a(13)-a(14) from Giovanni Resta, Apr 20 2016
a(15) from Bert Dobbelaere, Mar 04 2019
a(16)-a(17) from Bert Dobbelaere, Mar 20 2019

A215734 a(n) is the first digit to appear n times in succession in a power of 5.

Original entry on oeis.org

1, 8, 8, 7, 4, 2, 8, 5, 5, 7, 5, 7, 4, 1, 5, 7

Offset: 1

V. Raman, Aug 22 2012

Cf. A215728, A045875, A215732.

a(13)-a(14) from Giovanni Resta, Apr 19 2016

a(15)-a(16) from Bert Dobbelaere, Feb 13 2019

A215735 a(n) is the first digit to appear n times in succession in a power of 6.

Original entry on oeis.org

1, 7, 7, 0, 2, 1, 2, 7, 1, 1, 0, 5, 7, 6, 3, 7

Offset: 1

V. Raman, Aug 22 2012

See A215729 for the powers.

Cf. A045875, A215732.

a(11) added by V. Raman, Nov 23 2013
a(12)-a(13) from Giovanni Resta, Apr 19 2016
a(14) from Bert Dobbelaere, Feb 15 2019
a(15) from Paul Geneau de Lamarlière, Jun 26 2024
a(16) from Paul Geneau de Lamarlière, Jul 12 2024

A215737 a(n) is the first digit to appear n times in succession in a power of 11.

Original entry on oeis.org

1, 1, 8, 7, 6, 0, 6, 0, 9, 6, 6, 2, 2, 7, 6, 9

Offset: 1

V. Raman, Aug 22 2012

Cf. A001020 (powers of 11), A045875, A215731, A215732.

n = 1; x = 1; lst = {};
For[i = 1, i <= 10000, i++,
z = Split[IntegerDigits[x]]; a = Length /@ z; b = Max[a];
For[j = n, j <= b, j++,
  AppendTo[lst, First[First[Part[z, First[Position[a, b]]]]]]; n++
]; x = 11 x ]; lst  (* Robert Price, Mar 16 2019 *)

def A215737(n):
    a, s = 1, tuple(str(i)*n for i in range(10))
    while True:
        a *= 11
        t = str(a)
        for i, x in enumerate(s):
            if x in t:
                return i # Chai Wah Wu, Mar 30 2021

a(10)-a(13) added by V. Raman, Apr 30 2012, in correspondence with A215731.
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

A215736 a(n) is the first digit to appear n times in succession in a power of 7.

Original entry on oeis.org

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

Offset: 1

V. Raman, Aug 22 2012

Cf. A215730, A045875, A215732.

n = 1; x = 1; lst = {};
For[i = 1, i <= 10000, i++,
 z = Split[IntegerDigits[x]]; a = Length /@ z; b = Max[a];
 For[j = n, j <= b, j++,
  AppendTo[lst, First[First[Part[z, First[Position[a, b]]]]]]; n++
]; x = 7 x ]; lst (* Robert Price, Mar 16 2019 *)

a(10) added by V. Raman, Nov 23 2013
a(11)-a(13) from Giovanni Resta, Apr 19 2016
a(14)-a(15) from Bert Dobbelaere, Feb 15 2019
a(16) from Paul Geneau de Lamarlière, Jul 16 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 *)

cp's OEIS Frontend

A045875 a(n) is the smallest m for which the decimal representation of 2^m contains n consecutive identical digits.

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A215733 a(n) is the first digit to appear n times in succession in a power of 3.

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A215734 a(n) is the first digit to appear n times in succession in a power of 5.

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A215735 a(n) is the first digit to appear n times in succession in a power of 6.

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A215737 a(n) is the first digit to appear n times in succession in a power of 11.

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A215736 a(n) is the first digit to appear n times in succession in a power of 7.

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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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