A033014 -id:A033014 - cp's OEIS frontend

A033001 Every run of digits of n in base 3 has length 2.

4, 8, 36, 44, 72, 76, 328, 332, 396, 400, 652, 656, 684, 692, 2952, 2960, 2988, 2992, 3568, 3572, 3600, 3608, 5868, 5876, 5904, 5908, 6160, 6164, 6228, 6232, 26572, 26576, 26640, 26644, 26896, 26900, 26928, 26936, 32112, 32120

Offset: 1

Clark Kimberling

See A043291 for the base 2 version (which has a very simple formula), A033002 - A033014 for bases 4 through 16, A033015 - A033029 for the variants with runs of length >= 2. - M. F. Hasler, Feb 01 2014

Vincenzo Librandi, Table of n, a(n) for n = 1..250

Select[Range[10000], Union[Length/@Split[IntegerDigits[#, 3]]]=={2}&] (* Vincenzo Librandi, Feb 05 2014 *)

is_A033001(n)=!until(!n\=9,bittest(4588304,n%27)||return)

for(n=1,9999,is_A033001(n)&&print1(n",")) \\ (End)

a(n) = my(v=binary(n+1)); v[1]=0; for(i=2,#v, v[i]+=(v[i]>=v[i-1])); 4*fromdigits(v,9); \\ Kevin Ryde, Mar 13 2021

a(n)=4*A043307(n). - M. F. Hasler, Feb 01 2014

A043307 a(n) = A033001(n)/4.

Original entry on oeis.org

1, 2, 9, 11, 18, 19, 82, 83, 99, 100, 163, 164, 171, 173, 738, 740, 747, 748, 892, 893, 900, 902, 1467, 1469, 1476, 1477, 1540, 1541, 1557, 1558, 6643, 6644, 6660, 6661, 6724, 6725, 6732, 6734, 8028, 8030, 8037, 8038, 8101, 8102, 8118, 8119

Offset: 1

Clark Kimberling

Also: Numbers which, written in base 9, have only digits 0, 1 or 2, and no two adjacent digits equal. - M. F. Hasler, Feb 03 2014

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A043308 - A043320, A043291, A033001 - A033014, A033016 - A033029.

A[1]:= [1,2]:
for d from 2 to 6 do
  A[d]:= map(t -> seq(9*t+j,j=subs(t mod 9 = NULL, [0,1,2])), A[d-1])
od:
seq(op(A[d]),d=1..6); # Robert Israel, Jan 29 2017

Table[FromDigits[#,9]&/@Select[Tuples[{0,1,2},n],Min[Abs[Differences[#]]]>0&],{n,2,5}]// Flatten// Union (* Harvey P. Dale, May 27 2023 *)

is_A043307(n)=(n=[n])&&!until(!n[1],((n=divrem(n[1],9))[2]<3 && n[1]%3!=n[2])||return) \\ M. F. Hasler, Feb 03 2014

a(n) = my(v=binary(n+1)); v[1]=0; for(i=2,#v, v[i]+=(v[i]>=v[i-1])); fromdigits(v,9); \\ Kevin Ryde, Mar 13 2021

From Robert Israel, Jan 29 2017: (Start)
If a(n) == 0 (mod 3) then a(2*n+1) = 9*a(n) + 1 else a(2*n+1) = 9*a(n).
If a(n) == 2 (mod 3) then a(2*n+2) = 9*a(n) + 1 else a(2*n+1) = 9*a(n)+2.
a(4k+5) = 9*a(2k+2).
(End)

A043320 Numbers which, written in base 256, have all digits less than 16 and no two adjacent digits equal.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 256, 258, 259, 260, 261, 262, 263, 264, 265, 266, 267, 268, 269, 270, 271, 512, 513, 515, 516, 517, 518, 519, 520, 521, 522, 523, 524, 525, 526, 527, 768, 769, 770, 772, 773, 774

Offset: 1

Clark Kimberling

Sequence A033014 consists of the numbers that have all base 16 digits repeated *exactly* twice. (This is equivalent to say that the base-256 digits are 0x00, 0x11, 0x22,... or 0xFF, in hex notation, and no two adjacent base-256 digits are equal.) Thus, these numbers are divisible by 0x11 = 17, and the result of the division is a number which has no other base-256 digits than 0x00, 0x01,... or 0x0F, and no two adjacent digits equal. Conversely, it is clear that exactly these numbers are terms of A033014 when multiplied by 17 = 0x11. - M. F. Hasler, Feb 05 2014

Vincenzo Librandi, Table of n, a(n) for n = 1..1800

Cf. A043307 - A043319, A043291, A033001 - A033014, A033015 - A033029.

Select[Range[20000], Union[Length/@Split[IntegerDigits[#, 16]]]=={2}&]/17 (* Vincenzo Librandi, Feb 06 2014 *)

is_A043320(n)={(n=[n])&&!until(!n[1], ((n=divrem(n[1], 256))[2]<16 && n[1]%16!=n[2])||return)} \\ M. F. Hasler, Feb 03 2014

from itertools import count, islice, groupby
def A043320_gen(startvalue=1): # generator of terms >= startvalue
    return filter(lambda n:set(len(list(g)) for k, g in groupby(hex(17*n)[2:]))=={2},count(max(startvalue,1)))
A043320_list = list(islice(A043320_gen(),20)) # Chai Wah Wu, Mar 10 2023

a(n) = A033014(n)/17. [This was initially the definition of the sequence. - M. F. Hasler, Feb 03 2014]

New definition by M. F. Hasler, Feb 03 2014

A033002 Every run of digits of n in base 4 has length 2.

Original entry on oeis.org

5, 10, 15, 80, 90, 95, 160, 165, 175, 240, 245, 250, 1285, 1290, 1295, 1440, 1445, 1455, 1520, 1525, 1530, 2565, 2570, 2575, 2640, 2650, 2655, 2800, 2805, 2810, 3845, 3850, 3855, 3920, 3930, 3935, 4000, 4005, 4015, 20560

Offset: 1

Clark Kimberling

See A043291 and A033001 through A033014 for the analog in other bases, A033015 - A033029 for the variants with run lengths >= 2. - M. F. Hasler, Feb 04 2014

Vincenzo Librandi, Table of n, a(n) for n = 1..600

Select[Range[10000], Union[Length/@Split[IntegerDigits[#, 4]]]=={2}&] (* Vincenzo Librandi, Feb 05 2014 *)

a(n) = 5*A043308(n) (= 5*n for n<4). - M. F. Hasler, Feb 04 2014

A033008 Every run of digits of n in base 10 has length 2.

Original entry on oeis.org

11, 22, 33, 44, 55, 66, 77, 88, 99, 1100, 1122, 1133, 1144, 1155, 1166, 1177, 1188, 1199, 2200, 2211, 2233, 2244, 2255, 2266, 2277, 2288, 2299, 3300, 3311, 3322, 3344, 3355, 3366, 3377, 3388, 3399, 4400, 4411, 4422, 4433

Offset: 1

Clark Kimberling

See A043291 and A033001 through A033014 for the analog in other bases, A033015 - A033029 for the variants with run lengths >= 2. - M. F. Hasler, Feb 02 2014

Vincenzo Librandi, Table of n, a(n) for n = 1..800

Select[Range[10000], Union[Length/@Split[IntegerDigits[#, 10]]]=={2}&] (* Vincenzo Librandi, Feb 05 2014 *)

a(n) = 11*A043314(n) (= 11*n for n<10). - M. F. Hasler, Feb 02 2014

A033003 Every run of digits of n in base 5 has length 2.

Original entry on oeis.org

6, 12, 18, 24, 150, 162, 168, 174, 300, 306, 318, 324, 450, 456, 462, 474, 600, 606, 612, 618, 3756, 3762, 3768, 3774, 4050, 4056, 4068, 4074, 4200, 4206, 4212, 4224, 4350, 4356, 4362, 4368, 7506, 7512, 7518, 7524, 7650, 7662

Offset: 1

Clark Kimberling

See A043291 and A033001 through A033014 for the analog in other bases, A033015 - A033029 for the variants with run lengths >= 2. - M. F. Hasler, Feb 02 2014

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Select[Range[10000],Union[Length/@Split[IntegerDigits[#, 5]]]=={2}&] (* Vincenzo Librandi, Feb 05 2014 *)

a(n) = 6*A043309(n) (= 6*n for n<5). - M. F. Hasler, Feb 02 2014

A033007 Every run of digits of n in base 9 has length 2.

Original entry on oeis.org

10, 20, 30, 40, 50, 60, 70, 80, 810, 830, 840, 850, 860, 870, 880, 890, 1620, 1630, 1650, 1660, 1670, 1680, 1690, 1700, 2430, 2440, 2450, 2470, 2480, 2490, 2500, 2510, 3240, 3250, 3260, 3270, 3290, 3300, 3310, 3320, 4050

Offset: 1

Clark Kimberling

See A043291 and A033001 through A033014 for the analog in other bases, A033015 - A033029 for the variants with run lengths >= 2. - M. F. Hasler, Feb 02 2014

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Select[Range[10000], Union[Length/@Split[IntegerDigits[#, 9]]]=={2}&] (* Vincenzo Librandi, Feb 05 2014 *)

a(n) = 10*A043313(n) (= 10*n for n<9). - M. F. Hasler, Feb 02 2014

A033010 Numbers each of whose runs of digits in base 12 has length 2.

Original entry on oeis.org

13, 26, 39, 52, 65, 78, 91, 104, 117, 130, 143, 1872, 1898, 1911, 1924, 1937, 1950, 1963, 1976, 1989, 2002, 2015, 3744, 3757, 3783, 3796, 3809, 3822, 3835, 3848, 3861, 3874, 3887, 5616, 5629, 5642, 5668, 5681, 5694, 5707, 5720, 5733, 5746, 5759, 7488, 7501

Offset: 1

Clark Kimberling

See A043291 and A033001 through A033014 for the analog in other bases, A033015 - A033029 for the variants with run lengths >= 2. - M. F. Hasler, Feb 02 2014

Numbers without repeating adjacent digits for which all digits are divisible by 13, in base 144. Consequently there are 11^n n-digit members of this sequence (base 144) and so (11^(n+1)-1)/10 members of this sequence below 144^n. - Charles R Greathouse IV, Feb 02 2014

Vincenzo Librandi, Table of n, a(n) for n = 1..1400

Select[Range[10000], Union[Length/@Split[IntegerDigits[#, 12]]]=={2}&] (* Vincenzo Librandi, Feb 05 2014 *)

from sympy.ntheory import digits
from itertools import groupby
def ok(n):
  return all(len(list(g))==2 for k, g in groupby(digits(n, 12)[1:]))
print(list(filter(ok, range(1, 7502)))) # Michael S. Branicky, Apr 27 2021

a(n) = 13*A043316(n) (= 13*n for n < 12). - M. F. Hasler, Feb 02 2014

A033004 Every run of digits of n in base 6 has length 2.

Original entry on oeis.org

7, 14, 21, 28, 35, 252, 266, 273, 280, 287, 504, 511, 525, 532, 539, 756, 763, 770, 784, 791, 1008, 1015, 1022, 1029, 1043, 1260, 1267, 1274, 1281, 1288, 9079, 9086, 9093, 9100, 9107, 9576, 9583, 9597, 9604, 9611, 9828, 9835

Offset: 1

Clark Kimberling

See A043291 and A033001 through A033014 for the analog in other bases, A033015 - A033029 for the variants with run lengths >= 2. - M. F. Hasler, Feb 02 2014

Vincenzo Librandi, Table of n, a(n) for n = 1..700

Cf. A033007, A033010.

Select[Range[10000], Union[Length/@Split[IntegerDigits[#, 6]]]=={2}&] (* Vincenzo Librandi, Feb 05 2014 *)

a(n) = 7*A043310(n) (= 7*n for n<6). - M. F. Hasler, Feb 02 2014

A033005 Every run of digits of n in base 7 has length 2.

Original entry on oeis.org

8, 16, 24, 32, 40, 48, 392, 408, 416, 424, 432, 440, 784, 792, 808, 816, 824, 832, 1176, 1184, 1192, 1208, 1216, 1224, 1568, 1576, 1584, 1592, 1608, 1616, 1960, 1968, 1976, 1984, 1992, 2008, 2352, 2360, 2368, 2376, 2384, 2392

Offset: 1

Clark Kimberling

See A043291 and A033001 through A033014 for the analog in other bases, A033015 - A033029 for the variants with run lengths >= 2. - M. F. Hasler, Feb 02 2014

Vincenzo Librandi, Table of n, a(n) for n = 1..1500

Select[Range[2500],Union[Length/@Split[IntegerDigits[#,7]]]=={2}&] (* Harvey P. Dale, Oct 24 2011 *)

a(n) = 8*A043311(n) (= 8*n for n<7). - M. F. Hasler, Feb 02 2014

cp's OEIS Frontend

A033001 Every run of digits of n in base 3 has length 2.

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A043307 a(n) = A033001(n)/4.

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A043320 Numbers which, written in base 256, have all digits less than 16 and no two adjacent digits equal.

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A033002 Every run of digits of n in base 4 has length 2.

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A033008 Every run of digits of n in base 10 has length 2.

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A033003 Every run of digits of n in base 5 has length 2.

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A033007 Every run of digits of n in base 9 has length 2.

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A033010 Numbers each of whose runs of digits in base 12 has length 2.

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