A043307 -id:A043307 - 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

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

A043308 a(n)=A033002(n)/5.

Original entry on oeis.org

1, 2, 3, 16, 18, 19, 32, 33, 35, 48, 49, 50, 257, 258, 259, 288, 289, 291, 304, 305, 306, 513, 514, 515, 528, 530, 531, 560, 561, 562, 769, 770, 771, 784, 786, 787, 800, 801, 803, 4112, 4114, 4115, 4128, 4129, 4131, 4144, 4145

Offset: 1

Clark Kimberling

Also: Numbers which, written in base 16, have all digits less than 4 and no two adjacent digits equal. - M. F. Hasler, Feb 03 2014

Cf. A043307 - A043320, A043291, A033001 - A033014, A033016 - A033029.

is_A043308(n)=(n=[n])&&!until(!n[1],((n=divrem(n[1],16))[2]<4 && n[1]%4!=n[2])||return) \\ M. F. Hasler, Feb 03 2014

A043312 a(n) = A033006(n)/9.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 64, 66, 67, 68, 69, 70, 71, 128, 129, 131, 132, 133, 134, 135, 192, 193, 194, 196, 197, 198, 199, 256, 257, 258, 259, 261, 262, 263, 320, 321, 322, 323, 324, 326, 327, 384, 385, 386, 387, 388, 389, 391, 448, 449

Offset: 1

Clark Kimberling

Also: Numbers which, written in base 64, have only digits 0 through 7, and no two adjacent digits equal. - M. F. Hasler, Feb 03 2014

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

Cf. A043307 - A043320, A043291, A033001 - A033014, A033016 - A033029.

f:= proc(n) local i;
      seq(64*n+i, i= subs(n mod 64 = NULL, [$0..7]))
end proc:
A:= $1..7: R:= [A]:
for d from 2 to 3 do
  R:= map(f, R);
  A:= A, op(R);
od:
A; # Robert Israel, Jun 11 2019

is_A043312(n)=(n=[n])&&!until(!n[1],((n=divrem(n[1],64))[2]<8 && n[1]%8!=n[2])||return) \\ M. F. Hasler, Feb 03 2014

A043317 a(n)=A033011(n)/14.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 169, 171, 172, 173, 174, 175, 176, 177, 178, 179, 180, 181, 338, 339, 341, 342, 343, 344, 345, 346, 347, 348, 349, 350, 507, 508, 509, 511, 512, 513, 514, 515, 516, 517, 518, 519, 676, 677, 678

Offset: 1

Clark Kimberling

Also: Numbers which, written in base 169, have all digits less than 13 and no two adjacent digits equal. - M. F. Hasler, Feb 03 2014

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A043307 - A043320, A043291, A033001 - A033014, A033016 - A033029.

Select[Range[700],Max[IntegerDigits[#,169]]<13&&SequenceCount[ IntegerDigits[ #,169],{x_,x_}]==0&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Mar 07 2018 *)

is_A043317(n)=(n=[n])&&!until(!n[1],((n=divrem(n,169))[2]<13 && n[2]!=n[1]%13)||return) \\ M. F. Hasler, Feb 03 2014

A043319 a(n)=A033013(n)/16.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 225, 227, 228, 229, 230, 231, 232, 233, 234, 235, 236, 237, 238, 239, 450, 451, 453, 454, 455, 456, 457, 458, 459, 460, 461, 462, 463, 464, 675, 676, 677, 679, 680, 681, 682, 683, 684

Offset: 1

Clark Kimberling

Also: Numbers which, written in base 225, have all digits less than 15 and no two adjacent digits equal. - M. F. Hasler, Feb 03 2014

Cf. A043307 - A043320, A043291, A033001 - A033014, A033016 - A033029.

is_A043319(n)=(n=[n])&&!until(!n[1], ((n=divrem(n[1], 225))[2]<15 && n[1]%15!=n[2])||return) \\ M. F. Hasler, Feb 03 2014

A043309 a(n)=A033003(n)/6.

Original entry on oeis.org

1, 2, 3, 4, 25, 27, 28, 29, 50, 51, 53, 54, 75, 76, 77, 79, 100, 101, 102, 103, 626, 627, 628, 629, 675, 676, 678, 679, 700, 701, 702, 704, 725, 726, 727, 728, 1251, 1252, 1253, 1254, 1275, 1277, 1278, 1279, 1325, 1326, 1327, 1329

Offset: 1

Clark Kimberling

Also: Numbers which, written in base 25, have all digits less than 5 and no two adjacent digits equal. - M. F. Hasler, Feb 03 2014

Cf. A043307 - A043320, A043291, A033001 - A033014, A033016 - A033029.

is_A043309(n)=(n=[n])&&!until(!n[1],((n=divrem(n[1],25))[2]<5 && n[1]%5!=n[2])||return) \\ M. F. Hasler, Feb 03 2014

A043310 a(n)=A033004(n)/7.

Original entry on oeis.org

1, 2, 3, 4, 5, 36, 38, 39, 40, 41, 72, 73, 75, 76, 77, 108, 109, 110, 112, 113, 144, 145, 146, 147, 149, 180, 181, 182, 183, 184, 1297, 1298, 1299, 1300, 1301, 1368, 1369, 1371, 1372, 1373, 1404, 1405, 1406, 1408, 1409, 1440, 1441, 1442, 1443, 1445, 1476, 1477

Offset: 1

Clark Kimberling

Also: Numbers which, written in base 36, have all digits less than 6 and no two adjacent digits equal. - M. F. Hasler, Feb 03 2014

Cf. A043307 - A043320, A043291, A033001 - A033014, A033016 - A033029.

is_A043310(n)=(n=[n])&&!until(!n[1],((n=divrem(n[1],36))[2]<6 && n[1]%6!=n[2])||return) \\ M. F. Hasler, Feb 03 2014

Definition corrected by and more terms from Georg Fischer, Mar 04 2021

A043311 a(n)=A033005(n)/8.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 49, 51, 52, 53, 54, 55, 98, 99, 101, 102, 103, 104, 147, 148, 149, 151, 152, 153, 196, 197, 198, 199, 201, 202, 245, 246, 247, 248, 249, 251, 294, 295, 296, 297, 298, 299, 2402, 2403, 2404, 2405, 2406, 2407, 2499

Offset: 1

Clark Kimberling

Numbers which, written in base 49, have only digits 0 through 6 and no two adjacent digits equal. - M. F. Hasler, Feb 03 2014

Cf. A043307 - A043320, A043291, A033001 - A033014, A033016 - A033029. ~

is_A043311(n)=(n=[n])&&!until(!n[1],((n=divrem(n[1],49))[2]<7 && n[1]%7!=n[2])||return) \\ M. F. Hasler, Feb 03 2014

A043313 a(n)=A033007(n)/10.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 81, 83, 84, 85, 86, 87, 88, 89, 162, 163, 165, 166, 167, 168, 169, 170, 243, 244, 245, 247, 248, 249, 250, 251, 324, 325, 326, 327, 329, 330, 331, 332, 405, 406, 407, 408, 409, 411, 412, 413, 486, 487, 488

Offset: 1

Clark Kimberling

Also: Numbers which, written in base 81, have only digits 0,...,8, and no two adjacent digits equal. - M. F. Hasler, Feb 03 2014

Cf. A043307 - A043320, A043291, A033001 - A033014, A033016 - A033029.

is_A043313(n)=(n=[n])&&!until(!n[1],((n=divrem(n[1],81))[2]<9 && n[1]%9!=n[2])||return) \\ M. F. Hasler, Feb 03 2014

cp's OEIS Frontend

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

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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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A043308 a(n)=A033002(n)/5.

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A043312 a(n) = A033006(n)/9.

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A043317 a(n)=A033011(n)/14.

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A043319 a(n)=A033013(n)/16.

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A043309 a(n)=A033003(n)/6.

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A043310 a(n)=A033004(n)/7.

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A043311 a(n)=A033005(n)/8.