A094776 -id:A094776 - cp's OEIS frontend

A259091 Smallest k such that 2^k contains two adjacent copies of n in its decimal expansion.

53, 40, 43, 25, 18, 16, 46, 24, 19, 33, 378, 313, 170, 374, 361, 359, 64, 34, 507, 151, 348, 246, 314, 284, 349, 314, 261, 151, 385, 166, 156, 364, 65, 219, 371, 359, 503, 148, 155, 352, 349, 308, 247, 255, 192, 387, 165, 149, 171, 150, 210, 155, 209, 101, 505

Offset: 0

N. J. A. Sloane, Jun 18 2015

The multi-digit generalization of A171132. - R. J. Mathar, Jul 06 2015

			2^53 = 9007199254740992 contains two adjacent 0's.

Chai Wah Wu, Table of n, a(n) for n = 0..1000
Popular Computing (Calabasas, CA), Two Tables, Vol. 1, (No. 9, Dec 1973), page PC9-16.

Cf. A006889, A131535, A131536, A259089, A063565, A259092.

Table[k = 0; While[! SequenceCount[IntegerDigits[2^k], Flatten[ConstantArray[IntegerDigits[n], 2]]] > 0, k++]; k, {n, 0, 100}] (* Robert Price, May 17 2019 *)

def A259091(n):
    s, k, k2 = str(n)*2, 0, 1
    while True:
        if s in str(k2):
            return k
        k += 1
        k2 *= 2 # Chai Wah Wu, Jun 18 2015

More terms from Chai Wah Wu, Jun 18 2015

A259092 Smallest k such that 2^k contains three adjacent copies of n in its decimal expansion.

Original entry on oeis.org

242, 42, 43, 83, 44, 41, 157, 24, 39, 50, 949, 1841, 3661, 1798, 1701, 1161, 1806, 391, 1890, 2053, 950, 1164, 2354, 1807, 3816, 1800, 1799, 818, 1702, 2115, 904, 1798, 1807, 2270, 392, 1699, 3022, 394, 2054, 1758, 1804, 2300, 2720, 2403, 3396, 1133, 1808, 3820

Offset: 0

N. J. A. Sloane, Jun 18 2015

The multi-digit generalization of A171242. - R. J. Mathar, Jul 06 2015

			2^242 = 7067388259113537318333190002971674063309935587502475832486424805170479104 contains three adjacent 0's.

Chai Wah Wu, Table of n, a(n) for n = 0..1000
Popular Computing (Calabasas, CA), Two Tables, Vol. 1, (No. 9, Dec 1973), page PC9-16.

Cf. A006889, A131535, A131536, A259089, A063565, A259091.

Table[k = 0; While[! SequenceCount[IntegerDigits[2^k], Flatten[ConstantArray[IntegerDigits[n], 3]]] > 0, k++]; k, {n, 0, 50}] (* Robert Price, May 17 2019 *)

def A259092(n):
    s, k, k2 = str(n)*3, 0, 1
    while True:
        if s in str(k2):
            return k
        k += 1
        k2 *= 2 # Chai Wah Wu, Jun 18 2015

More terms from Chai Wah Wu, Jun 18 2015

A131543 Exponent of least power of 2 having exactly n consecutive 9's in its decimal representation.

Original entry on oeis.org

0, 12, 33, 50, 421, 422, 2187, 15554, 42483, 42485, 42486, 1522085, 2662514, 6855863, 6855865

Offset: 0

Shyam Sunder Gupta, Aug 26 2007

Similarly to A006889, the least power of 2 to contain at least n consecutive 9's will always contain exactly n consecutive 9's. The previous power of two will contain exactly n-1 consecutive 9's preceded by a 4. - Paul Geneau de Lamarlière, Jul 20 2024

No more terms < 28*10^6.

			a(3)=50 because 2^50 (i.e. 1125899906842624) is the smallest power of 2 to contain a run of 3 consecutive nines in its decimal form.

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

Cf. A006889.

a = ""; Do[ a = StringJoin[a, "9"]; b = StringJoin[a, "9"]; k = 1; While[ StringPosition[ ToString[2^k], a] == {} || StringPosition[ ToString[2^k], b] != {}, k++ ]; Print[k], {n, 1, 10} ]

a(11) from Sean A. Irvine, May 31 2010
a(12)-a(14) from Lars Blomberg, Jan 24 2013
a(0)=0 prepended by Paul Geneau de Lamarlière, Jul 20 2024

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

A131537 Exponent of least power of 2 having exactly n consecutive 3's in its decimal representation.

Original entry on oeis.org

0, 5, 25, 83, 219, 221, 2270, 11020, 18843, 192915, 271978, 743748, 1039315, 13873203, 14060685

Offset: 0

Shyam Sunder Gupta, Aug 26 2007

No more terms < 28*10^6.

			a(3) = 83 because 2^83 (= 9671406556917033397649408) is the smallest power of 2 to contain a run of exactly 3 consecutive threes in its decimal form.

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

Cf. A000079.

a = ""; Do[ a = StringJoin[a, "3"]; b = StringJoin[a, "3"]; k = 1; While[ StringPosition[ ToString[2^k], a] == {} || StringPosition[ ToString[2^k], b] != {}, k++ ]; Print[k], {n, 1, 9} ]

def a(n):
  k, n2, np2 = 1, '3'*n, '3'*(n+1)
  while True:
    while not n2 in str(2**k): k += 1
    if np2 not in str(2**k): return k
    k += 1
print([a(n) for n in range(1, 9)]) # Michael S. Branicky, May 25 2021

a(10)-a(12) from Sean A. Irvine, Jul 19 2010
a(13)-a(14) from Lars Blomberg, Jan 24 2013
a(0)=0 prepended by Paul Geneau de Lamarlière, Jul 20 2024

A131538 Exponent of least power of 2 having exactly n consecutive 4's in its decimal representation.

Original entry on oeis.org

0, 2, 18, 44, 192, 315, 3396, 8556, 13327, 81785, 279267, 865357, 1799674, 1727603, 8760851, 63416791, 106892452

Offset: 0

Shyam Sunder Gupta, Aug 26 2007

			a(3) = 44 because 2^44 (i.e. 17592186044416) is the smallest power of 2 to contain a run of 3 consecutive fours in its decimal form.

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

a = ""; Do[ a = StringJoin[a, "4"]; b = StringJoin[a, "4"]; k = 1; While[ StringPosition[ ToString[2^k], a] == {} || StringPosition[ ToString[2^k], b] != {}, k++ ]; Print[k], {n, 1, 10} ]

3 more terms from Sean A. Irvine, Jul 19 2010
a(14) from Lars Blomberg, Jan 24 2013
a(15) from Bert Dobbelaere, Feb 25 2019
a(16) from Paul Geneau de Lamarlière, Jun 26 2024
a(0)=0 prepended by Paul Geneau de Lamarlière, Jul 20 2024

A131539 Exponent of least power of 2 having exactly n consecutive 5's in its decimal representation.

Original entry on oeis.org

0, 8, 16, 76, 41, 1162, 973, 6838, 25265, 81782, 456686, 279270, 1727606, 6030753, 23157026, 106892455

Offset: 0

Shyam Sunder Gupta, Aug 26 2007

			a(3)=76 because 2^76 (i.e., 75557863725914323419136) is the smallest power of 2 to contain a run of 3 consecutive fives in its decimal form.

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

a = ""; Do[ a = StringJoin[a, "5"]; b = StringJoin[a, "5"]; k = 1; While[ StringPosition[ ToString[2^k], a] == {} || StringPosition[ ToString[2^k], b] != {}, k++ ]; Print[k], {n, 1, 10} ]

2 more terms from Sean A. Irvine, Jul 19 2010
a(13)-a(14) from Lars Blomberg, Jan 24 2013
a(0)=0 prepended by and a(15) from Paul Geneau de Lamarlière, Jul 19 2024

A131542 Exponent of least power of 2 having exactly n consecutive 8's in its decimal representation.

Original entry on oeis.org

0, 3, 19, 39, 180, 316, 971, 6836, 25263, 103825, 279274, 279268, 1727604, 18053496, 8760852, 106892453

Offset: 0

Shyam Sunder Gupta, Aug 26 2007

			a(3)=39 because 2^39 (i.e., 549755813888) is the smallest power of 2 to contain a run of 3 consecutive eights in its decimal form.

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

a = ""; Do[ a = StringJoin[a, "8"]; b = StringJoin[a, "8"]; k = 1; While[ StringPosition[ ToString[2^k], a] == {} || StringPosition[ ToString[2^k], b] != {}, k++ ]; Print[k], {n, 1, 10} ]

Two more terms from Sean A. Irvine, May 31 2010
a(13)-a(14) from Lars Blomberg, Jan 24 2013
a(0)=0 prepended by and a(15) from Paul Geneau de Lamarlière, Jul 19 2024

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

cp's OEIS Frontend

A259091 Smallest k such that 2^k contains two adjacent copies of n in its decimal expansion.

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A259092 Smallest k such that 2^k contains three adjacent copies of n in its decimal expansion.

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A131543 Exponent of least power of 2 having exactly n consecutive 9's in its decimal representation.

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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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A131537 Exponent of least power of 2 having exactly n consecutive 3's in its decimal representation.

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A131538 Exponent of least power of 2 having exactly n consecutive 4's in its decimal representation.

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A131539 Exponent of least power of 2 having exactly n consecutive 5's in its decimal representation.

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A131542 Exponent of least power of 2 having exactly n consecutive 8's in its decimal representation.

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