A033015 -id:A033015 - cp's OEIS frontend

A175055 a(n) = decimal equivalent of {A033015(n) written in binary, and each run of 0's reduced in length by one digit, and each run of 1's reduced in length by one digit}.

1, 3, 2, 7, 4, 6, 15, 8, 5, 12, 14, 31, 16, 9, 11, 24, 13, 28, 30, 63, 32, 17, 19, 10, 23, 48, 25, 27, 56, 29, 60, 62, 127, 64, 33, 35, 18, 39, 20, 22, 47, 96, 49, 51, 26, 55, 112, 57, 59, 120, 61, 124, 126, 255, 128, 65, 67, 34, 71, 36, 38, 79, 40, 21, 44, 46, 95, 192, 97, 99

Offset: 1

Leroy Quet, Dec 08 2009

This is a permutation of the positive integers. Sequence A175056 is its inverse permutation.

Michael De Vlieger, Table of n, a(n) for n = 1..10945

Cf. A033015, A175056, A175057, A175058, A175059, A175060

FromDigits[Join @@ #, 2] & /@ Map[Drop[#, 1] &, Select[Array[Split@ IntegerDigits[#, 2] &, 905], Min[Length /@ #] > 1 &], {2}] (* Michael De Vlieger, Sep 03 2017 *)

Extended by Ray Chandler, Dec 18 2009

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

Original entry on oeis.org

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

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

A226227 Numbers m such that all lengths of runs of 0's in the binary representation of m are prime numbers.

Original entry on oeis.org

1, 3, 4, 7, 8, 9, 12, 15, 17, 19, 24, 25, 28, 31, 32, 35, 36, 39, 49, 51, 56, 57, 60, 63, 65, 68, 71, 72, 73, 76, 79, 96, 99, 100, 103, 113, 115, 120, 121, 124, 127, 128, 131, 136, 137, 140, 143, 145, 147, 152, 153, 156, 159, 193, 196, 199, 200, 201, 204, 207, 224, 227

Offset: 1

Alex Ratushnyak, May 31 2013

Numbers with no 0's in base 2 (that is, 2^k - 1) are included.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A033015, A226228, A226229.

Cf. A005811, A030308, A010051.

#include 
#include 
typedef unsigned long long U64;
U64 runs[] = {0,2,3,5,7,11,13,17,19,23,29,31,37,41,43}, *wr;
#define MemUSAGE 1ULL<<33
#define TOP (1ULL<<9)  // <<40 ok if MemUSAGE = 1ULL<<33
int compare64(const void *p1, const void *p2) {
  if (*(U64*)p1 == *(U64*)p2) return 0;
  return (*(U64*)p1 < *(U64*)p2) ? -1 : 1;
}
void rec(U64 bits) {
  for (U64 i = 0, b; (b = bits << runs[i]) < TOP; ++i)
    *wr++ = b, rec(b*2+1);
}
int main() {
  U64 *wr0 = (wr = (U64*)malloc(MemUSAGE));
  rec(1);
  //printf("%llu\n", wr-wr0);
  qsort(wr0, wr-wr0, 8, compare64);
  while (wr0

import Data.List (group, genericLength)
a226227 n = a226227_list !! (n-1)
a226227_list = filter (all (== 1) .
               map (a010051 . genericLength) .
                   other . tail . reverse . group . a030308_row) [1..]
   where other [] = []; other [x] = [x]; other (x:_:xs) = x : other xs
-- Reinhard Zumkeller, Jun 05 2013

Select[Range@227, And @@ PrimeQ[(Length /@ Split@ IntegerDigits[#, 2])[[2 ;; ;; 2]]] &] (* Giovanni Resta, Jun 01 2013 *)
Select[Range[250],AllTrue[Length/@Select[Split[IntegerDigits[#,2]],MemberQ[ #,0]&],PrimeQ]&] (* Harvey P. Dale, Aug 07 2021 *)

A226228 Numbers m such that all lengths of runs of 1's in the binary representation of m are prime numbers.

Original entry on oeis.org

3, 6, 7, 12, 14, 24, 27, 28, 31, 48, 51, 54, 55, 56, 59, 62, 96, 99, 102, 103, 108, 110, 112, 115, 118, 119, 124, 127, 192, 195, 198, 199, 204, 206, 216, 219, 220, 223, 224, 227, 230, 231, 236, 238, 248, 251, 254, 384, 387, 390, 391, 396, 398, 408, 411, 412, 415, 432

Offset: 1

Alex Ratushnyak, May 31 2013

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A033015, A226227, A226229.

Cf. A005811, A030308, A010051.

#include 
#include 
typedef unsigned long long U64;
U64 runs[] = {0,2,3,5,7,11,13,17,19,23,29,31,37,41,43}, *wr;
#define MemUSAGE 1ULL<<33
#define TOP (1ULL<<9)  // <<40 ok if MemUSAGE = 1ULL<<33
int compare64(const void *p1, const void *p2) {
  if (*(U64*)p1 == *(U64*)p2) return 0;
  return (*(U64*)p1 < *(U64*)p2) ? -1 : 1;
}
void rec(U64 bits) {
  for (U64 i = 0, b; (b = ((bits*2+1) << runs[i])-1) < TOP; ++i)
    if (b) *wr++ = b, rec(b);
}
int main() {
  U64 *wr0 = (wr = (U64*)malloc(MemUSAGE));
  rec(0);
  //printf("%llu\n", wr-wr0);
  qsort(wr0, wr-wr0, 8, compare64);
  while (wr0

import Data.List (group, genericLength)
a226228 n = a226228_list !! (n-1)
a226228_list = filter (all (== 1) .
               map (a010051 . genericLength) .
                   other . reverse . group . a030308_row) [1..]
   where other [] = []; other [x] = [x]; other (x:_:xs) = x : other xs
-- Reinhard Zumkeller, Jun 05 2013

Select[Range@432, And @@ PrimeQ[(Length /@ Split@ IntegerDigits[#, 2])[[;; ;; 2]]] &] (* Giovanni Resta, Jun 01 2013 *)

A226229 Numbers m such that all lengths of runs of digits in base 2 representation of m are primes.

Original entry on oeis.org

3, 7, 12, 24, 28, 31, 51, 56, 96, 99, 103, 115, 124, 127, 199, 204, 224, 227, 231, 248, 384, 387, 396, 408, 412, 415, 455, 460, 499, 508, 775, 792, 796, 799, 819, 824, 896, 899, 908, 920, 924, 927, 992, 995, 999, 1016, 1539, 1548, 1587, 1592, 1632, 1635, 1639, 1651, 1660

Offset: 1

Alex Ratushnyak, May 31 2013

Intersection of A226227 and A226228.
From Emeric Deutsch, Jan 27 2018: (Start)
Also the indices of the compositions having only prime parts. For the definition of the index of a composition see A298644. For example, 387 is in the sequence since its binary form is 110000011 and the parts of the composition [2,5,2] are prime numbers. 540 is not in the sequence since its binary form is 1000011100 and not all the parts of the composition [1,4,3,2] are primes.
The command c(n) from the Maple program yields the composition having index n. (End)

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

Cf. A033015, A226227, A226228.

Cf. A005811, A030308, A010051.

import Data.List (group, genericLength)
a226229 n = a226229_list !! (n-1)
a226229_list = filter
   (all (== 1) . map (a010051 . genericLength) . group . a030308_row) [1..]
-- Reinhard Zumkeller, Jun 05 2013

Runs := proc (L) local j, r, i, k: npr: j := 1; r[j] := L[1]: for i from 2 to nops(L) do if L[i] = L[i-1] then r[j] := r[j], L[i] else j := j+1: r[j] := L[i] end if end do: [seq([r[k]], k = 1 .. j)] end proc: RunLengths := proc (L) map(nops, Runs(L)) end proc: c := proc (n) ListTools:-Reverse(convert(n, base, 2)): RunLengths(%) end proc: npr := proc (s) local q, j: q := 0: for j to nops(s) do if isprime(s[j]) = true then q := q+1 else  end if end do end proc: A := {}: for n to 1661 do if npr(c(n)) = nops(c(n)) then A := `union`(A, {n}) else  end if end do: A; # most of the Maple program is due to W. Edwin Clark. # Emeric Deutsch, Jan 27 2018

Select[Range@1660, And @@ PrimeQ[Length /@ Split@ IntegerDigits[#, 2]] &] (* Giovanni Resta, Jun 01 2013 *)

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

cp's OEIS Frontend

A175055 a(n) = decimal equivalent of {A033015(n) written in binary, and each run of 0's reduced in length by one digit, and each run of 1's reduced in length by one digit}.

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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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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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A226227 Numbers m such that all lengths of runs of 0's in the binary representation of m are prime numbers.

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A226228 Numbers m such that all lengths of runs of 1's in the binary representation of m are prime numbers.

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C

Haskell

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A226229 Numbers m such that all lengths of runs of digits in base 2 representation of m are primes.

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