A031443 -id:A031443 - cp's OEIS frontend

A049354 Digitally balanced numbers in base 3: equal numbers of 0's, 1's, 2's.

11, 15, 19, 21, 260, 266, 268, 278, 290, 294, 302, 304, 308, 312, 316, 318, 332, 344, 348, 380, 384, 396, 410, 412, 416, 420, 424, 426, 434, 438, 450, 460, 462, 468, 500, 502, 508, 518, 520, 524, 528, 532, 534, 544, 550, 552, 572, 574, 578, 582, 586, 588, 596

Offset: 1

Harvey P. Dale

Alois P. Heinz, Table of n, a(n) for n = 1..2000

Cf. A031443, A031944.
Cf. A049354-A049360. See also A061854, A037861.
Cf. A062756, A077267, A081603.
Row n = 3 of A378000.

a049354 n = a049354_list !! (n-1)
a049354_list = filter f [1..] where
   f n = t0 == a062756 n && t0 == a081603 n where t0 = a077267 n
-- Reinhard Zumkeller, Aug 09 2014

Select[Range[600],Length[Union[DigitCount[#,3]]]== 1&]
FromDigits[#,3]&/@DeleteCases[Flatten[Permutations/@Table[PadRight[{},3n,{1,0,2}],{n,3}],1],?(#[[1]]==0&)]//Sort (* _Harvey P. Dale, May 30 2016 *)
Select[Range@5000, Differences@DigitCount[#,3]=={0,0}&] (* Hans Rudolf Widmer, Dec 11 2021 *)

from sympy.ntheory import count_digits
def ok(n): c = count_digits(n, 3); return c[0] == c[1] == c[2]
print([k for k in range(600) if ok(k)]) # Michael S. Branicky, Nov 15 2021

A062756(a(n)) = A077267(a(n)) and A081603(a(n)) = A077267(a(n)). - Reinhard Zumkeller, Aug 09 2014

A144812 Integers having ideal digital mean up to base 7.

Original entry on oeis.org

36990, 37230, 43350, 45390, 2149023720, 2149218300, 2149279740, 2149513020, 2149527540, 2149545960, 2151079740, 2151628020, 2151662460, 2151667320, 2152716540, 2152720860, 2152724280, 2153463540, 2154166200, 2154948600, 2155019220, 2155051980, 2155196340

Offset: 1

Reikku Kulon, Sep 21 2008

Subset of A031443, A144798, A144799, A144800 and A144801.
These numbers have digital mean dm(b, n) = (Sum_{i=1..d} 2*d_i - (b-1)) / (2*d) = 0, where d is the number of digits in the base b representation of n and d_i the individual digits, for 2 <= b <= 7.
There are no integers less than 2^32 for which this is true to base 8. It is believed there are either infinitely many starting at some larger n, or none. If they exist, it is conjectured that the set of all similar sequences continues at least to base ten, almost certainly to base 16 and likely to arbitrarily large b. Sequences for b at least ten have an intersection with A144777.

Jason Yuen, Table of n, a(n) for n = 1..10000 (first 283 terms from Pontus von Brömssen)

Cf. A007953, A008591, A031443, A144777, A144798, A144799, A144800, A144801, A364714.

A227870 Numbers with equal number of even and odd digits.

Original entry on oeis.org

10, 12, 14, 16, 18, 21, 23, 25, 27, 29, 30, 32, 34, 36, 38, 41, 43, 45, 47, 49, 50, 52, 54, 56, 58, 61, 63, 65, 67, 69, 70, 72, 74, 76, 78, 81, 83, 85, 87, 89, 90, 92, 94, 96, 98, 1001, 1003, 1005, 1007, 1009, 1010, 1012, 1014, 1016, 1018, 1021, 1023, 1025

Offset: 1

Jon Perry, Nov 02 2013

Numbers with an odd digit length cannot be in this sequence. - Alonso del Arte, Nov 02 2013

			1009 has 2 even digits (00) and 2 odd digits (19) and so is in the sequence.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Subsequence of A001637.

Cf. A030141, A031443.

for (i = 1; i < 5000; i++) {
s = i.toString();
odds = 0; evens = 0;
for (j = 0; j < s.length; j++) if (s.charAt(j)%2 == 0) evens++; else odds++;
if (odds == evens) document.write(i + ", ");
}

Select[Range[1025], (d = Differences[Tally[Mod[IntegerDigits[#], 2]]]) != {} && d[[1, 2]] == 0 &] (* Amiram Eldar, Oct 01 2020 *)
eneodQ[n_]:=With[{id=IntegerDigits[n]},Count[id,?(OddQ[#]&)]==Count[id,?(EvenQ[#]&)]]; Select[Range[1100],eneodQ] (* Harvey P. Dale, Jul 19 2024 *)

isok(m) = my(d=digits(m)); #select(x->(x%2), d) == #select(x->!(x%2), d); \\ Michel Marcus, Oct 01 2020

def ok(i):
  stri = str(i)
  se = sum(1 for d in stri if d in "02468")
  so = sum(1 for d in stri if d in "13579")
  return se == so
def aupto(nn):
  alst, an = [None], 0
  for n in range(1, nn+1):
    while len(alst) < nn+1:
      if ok(an): alst.append(an)
      an += 1
  return alst[1:] # use alst[n] for a(n)
print(aupto(58))  # Michael S. Branicky, Dec 14 2020

A378000 Array read by ascending antidiagonals: T(n,k) is the k-th positive integer that is digitally balanced in base n.

Original entry on oeis.org

2, 11, 9, 75, 15, 10, 694, 78, 19, 12, 8345, 698, 99, 21, 35, 123717, 8350, 714, 108, 260, 37, 2177399, 123723, 8375, 722, 114, 266, 38, 44317196, 2177406, 123759, 8385, 738, 120, 268, 41, 1023456789, 44317204, 2177455, 123771, 8410, 742, 135, 278, 42

Offset: 2

Paolo Xausa, Nov 14 2024

A digitally balanced number in base b contains every digit from 0 to b-1 in equal amount.

			Array begins:
  n\k|           1            2            3            4            5  ...
  -------------------------------------------------------------------------
   2 |           2,           9,          10,          12,          35, ... = A031443
   3 |          11,          15,          19,          21,         260, ... = A049354
   4 |          75,          78,          99,         108,         114, ... = A049355
   5 |         694,         698,         714,         722,         738, ... = A049356
   6 |        8345,        8350,        8375,        8385,        8410, ... = A049357
   7 |      123717,      123723,      123759,      123771,      123807, ... = A049358
   8 |     2177399,     2177406,     2177455,     2177469,     2177518, ... = A049359
   9 |    44317196,    44317204,    44317268,    44317284,    44317348, ... = A049360
  10 |  1023456789,  1023456798,  1023456879,  1023456897,  1023456978, ...
  11 | 26432593615, 26432593625, 26432593725, 26432593745, 26432593845, ...
  ...         |                                                       \______ A378001 (main diagonal)
           A049363
T(2,4) = 12 = 1100_2 is the fourth number in base 2 containing an equal amount of zeros and ones.
T(9,5) = 44317348 = 102345867_9 is the fifth number in base 9 containing an equal amount of digits from 0 to 8.

Giovanni Resta, Digitally balanced numbers, Numbers Aplenty, 2013.

Cf. rows 2..9: A031443, A049354, A049355, A049356, A049357, A049358, A049359, A049360.
Cf. A049363 (first column, from n = 2), A378001 (main diagonal).
Cf. A049364, A065963, A378073, A378080, A378104.

Module[{dmax = 10, a, m}, a = Table[m = FromDigits[Join[{1, 0}, Range[2, n-1]], n] - 1; Table[While[!SameQ@@DigitCount[++m, n]]; m, dmax-n+2], {n, dmax+1, 2, -1}]; Array[Diagonal[a, # - dmax] &, dmax]]

A066196 Primes which have an equal number of zeros and ones in their binary expansion.

Original entry on oeis.org

2, 37, 41, 139, 149, 163, 197, 541, 557, 563, 569, 587, 601, 613, 617, 647, 653, 659, 661, 677, 709, 787, 809, 929, 2141, 2203, 2221, 2251, 2281, 2333, 2347, 2357, 2381, 2389, 2393, 2417, 2467, 2473, 2617, 2659, 2699, 2707, 2713, 2729, 2837, 2851, 2857

Offset: 1

Robert G. Wilson v, Dec 15 2001

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harry J. Smith)

Cf. A000040, A031443.

Prime[ Select[ Range[ 10^3 ], Count[ IntegerDigits[ Prime[ # ], 2 ], 0 ] == Count[ IntegerDigits[ Prime[ # ], 2 ], 1 ] & ] ]
digBalQ[n_] := Module[{d = IntegerDigits[n, 2], m}, EvenQ@(m = Length@d) && Count[d, 1] == m/2]; Select[Range[3000], PrimeQ[#] && digBalQ[#] &] (* Amiram Eldar, Nov 21 2020 *)
Select[Prime[Range[500]],DigitCount[#,2,1]==DigitCount[#,2,0]&] (* Harvey P. Dale, Jun 24 2025 *)

isok(p) = isprime(p) && (2*hammingweight(p) == #binary(p)); \\ Michel Marcus, May 16 2022

from itertools import count, islice
from sympy import isprime
from sympy.utilities.iterables import multiset_permutations
def agen():
    yield from filter(isprime, (int("1"+"".join(p), 2) for n in count(1) for p in multiset_permutations("0"*n+"1"*(n-1))))
print(list(islice(agen(), 50))) # Michael S. Branicky, May 15 2022

A000040 INTERSECT A031443. - R. J. Mathar, Jun 01 2011

A072600 Numbers which in base 2 have fewer 0's than 1's.

Original entry on oeis.org

1, 3, 5, 6, 7, 11, 13, 14, 15, 19, 21, 22, 23, 25, 26, 27, 28, 29, 30, 31, 39, 43, 45, 46, 47, 51, 53, 54, 55, 57, 58, 59, 60, 61, 62, 63, 71, 75, 77, 78, 79, 83, 85, 86, 87, 89, 90, 91, 92, 93, 94, 95, 99, 101, 102, 103, 105, 106, 107, 108, 109, 110, 111, 113, 114, 115

Offset: 1

Reinhard Zumkeller, Jun 23 2002

A037861(a(n)) < 0.

b_k = {a(n) | for all n s.t. a(n) contains k binary digits equal to 1} is the list of all valid win/loss round sequences in a "best of 2k-1" two player game, where 1 is a win and 0 is a loss. For example 19 = 10011b represents a game where the winner won the first two rounds, lost the next two, and won the last one. |b_k| = A001700(k). - Philippe Beaudoin, May 14 2014

			11 is present because '1011' contains 1 '0' and 3 '1's: 1<3.

T. D. Noe, Table of n, a(n) for n = 1..4733 ( numbers < 2^13)
Jason Bell, Thomas Finn Lidbetter, Jeffrey Shallit, Additive Number Theory via Approximation by Regular Languages, arXiv:1804.07996 [cs.FL], 2018.
Thomas Finn Lidbetter, Counting, Adding, and Regular Languages, Master's Thesis, University of Waterloo, Ontario, Canada, 2018.
Index entries for sequences related to binary expansion of n

Cf. A007088, A000120, A023416, A072601, A031443, A072602, A072603, A037861

a072600 n = a072600_list !! (n-1)
a072600_list = filter ((< 0) . a037861) [0..]
-- Reinhard Zumkeller, Mar 31 2015

Select[Range[130],DigitCount[#,2,0]Harvey P. Dale, Jan 12 2011 *)

is(n)=2*hammingweight(n)>exponent(n)+1 \\ Charles R Greathouse IV, Apr 18 2020

A144912 Unreduced numerators of digital mean, dm_num(b, n), with rows n in {2, 3, 4, ...} and columns b in {2, 3, 4, ..., n}.

Original entry on oeis.org

0, 2, -2, -1, 0, -4, 1, 2, -2, -6, 1, 0, 0, -4, -8, 3, 2, 2, -2, -6, -10, -2, 4, -2, 0, -4, -8, -12, 0, -4, 0, 2, -2, -6, -10, -14, 0, -2, 2, -4, 0, -4, -8, -12, -16, 2, 0, 4, -2, 2, -2, -6, -10, -14, -18, 0, -2, 0, 0, -6, 0, -4, -8, -12, -16, -20

Offset: 2

Reikku Kulon, Sep 25 2008, Oct 03 2008

The unreduced numerator of dm(b, n) is Sum_{i=1..d} (2*d_i - (b-1)), where d is the number of digits in the base b representation of n and d_i the individual digits. The corresponding denominator is 2 * d, giving a value in (-(b - 1) / 2, (b - 1) / 2] for n > 0.
dm_num(b, n) = d(b - 1) iff all the digits in n are b - 1.
dm_num(b, n) = -2(b - 2) for b = n, because n in base n is 10, giving dm_num(n, n) = 2 - n + 1 + 0 - n + 1 = 4 - 2 * n = -2(n - 2).
dm_num(b, n) = 0 for odd b and n having all digits equal to (b - 1) / 2, as well as for many other (b, n).
Defining m = ceiling((n + 1) / 2):
dm_num(b, n) = dm_num(b - 1, n) - 4 for b in [m + 1, n].
dm_num(m, n) = 0 for even n and 2 for odd n.
dm_num(m - 1, n) = 6 - n for even n > 4 and 9 - n for odd n > 5, producing a sequence of first differences {+2, -4, +2, -4, ...}.
Triangular patterns become clearly visible for large n, defined by additive periodicities along rational slopes. Zeros along the triangle borders correspond to ones in the Redheffer matrix until odd values become dominant. The line along m is the border between the two largest triangles. This pattern is masked by aliasing effects for small bases, notably including base 10, due to the thinness of the triangles which dominate at small b. Odd values may represent "artifacts" caused by "interference".

			Triangle begins:
   0;
   2, -2;
  -1,  0, -4;
   1,  2, -2, -6;
   1,  0,  0, -4, -8;
   3,  2,  2, -2, -6, -10;
   ...

Reikku Kulon, Rows of triangle for b in [2, 142]
Reikku Kulon, C99 source to produce the triangle
Reikku Kulon, Triangle as 2048x2048 PNG image (zero is white, primes are black and darker grays indicate fewer prime factors)
Reikku Kulon, Triangle as 2048x2048 PNG image, extended to b in [2, 2 * n + 1]
Reikku Kulon, Prime band as 16384x256 PNG image (note the curves coincident with new strips of primes, as well as the second band which appears at 4096 and corresponds to the 637/638 gap in A031443)
Reikku Kulon, Prime band as 16384x256 PNG image, starting from n = 57344
Eric Weisstein's World of Mathematics, Redheffer Matrix

Cf. A002321, A031443, A083058, A144777, A144798, A144799, A144800, A144801, A144812, A144923, A240236.

dmnum[b_,n_]:=2Total[IntegerDigits[n,b]]-(b-1)Floor[Log[b,n*b]]; (* after Jinyuan Wang *)
Table[dmnum[b,n],{n,2,10},{b,2,n}] (* Paolo Xausa, Sep 26 2023 *)

dm(b, n) = 2*sumdigits(n, b) - (b-1)*logint(n*b, b); \\ Jinyuan Wang, Jul 21 2020

A061854 Nondiving binary sequences: numbers which in base 2 have at least the same number of 1's as 0's and reading the binary expansion from left (msb) to right (least significant bit), the number of 0's never exceeds the number of 1's.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 10, 11, 12, 13, 14, 15, 21, 22, 23, 25, 26, 27, 28, 29, 30, 31, 42, 43, 44, 45, 46, 47, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 85, 86, 87, 89, 90, 91, 92, 93, 94, 95, 101, 102, 103, 105, 106, 107, 108, 109, 110, 111, 113, 114, 115, 116

Offset: 1

Antti Karttunen, May 11 2001

"msb" = "most significant bit", A053644.
These encode lattice walks using steps (+1,+1) (= 1's in binary expansion) and (+1,-1) (= 0's in binary expansion) that start from the origin (0,0) and never "dive" under the "sea-level" y=0.
The number of such walks of length n (here: the terms of binary width n) is given by C(n,floor(n/2)) = A001405, which is based on the fact mentioned in Guy's article that the shallow diagonals of the Catalan triangle A009766 sum to A001405.
From Jason Kimberley, Feb 08 2013: (Start)
This sequence is a subsequence of A072601.
Define a map from this set onto the nonnegative integers as follows: set the output bit string to be empty, representing zero; process the input string from left to right; when 1 occurs, change the rightmost 0 in the output to 1; if there is no 0 in the output, prepend a 1; when 0 occurs in the input, change the rightmost 1 in the output to 1. The definition of this sequence ensures that we always have a 1 in the output when a 0 occurs in the input. We this map is onto by showing the restriction to the subset Asubsequence is onto. (End)
The binary representation of a(n) is the numeric representation of the left half of a symmetric balanced string of parentheses with "(" representing 1 and ")" representing 0 (see comments and examples in A001405). Some of the numbers in this sequence cannot be realized as the 1-0-pattern of the odd/even positions of 1's in any row n of A237048 that determines the parts and their widths in the symmetric representation of sigma(n), see A352696. - Hartmut F. W. Hoft, Mar 29 2022

			From _Hartmut F. W. Hoft_, Mar 29 2022: (Start)
The columns in the table are the numbers n, the base-2 representation of n, the left half of the symmetric balanced string of parentheses corresponding to n, validity of the nondiving property for n, and associated number a(n):
1   1      (      True    a(1)
2   10     ()     True    a(2)
3   11     ((     True    a(3)
4   100    ())    False    -
5   101    ()(    True    a(4)
6   110    (()    True    a(5)
7   111    (((    True    a(6)
8   1000   ()))   False    -
9   1001   ())(   False    -
10  1010   ()()   True    a(7)
...
20  10100  ()())  False    -
21  10101  ()()(  True    a(13)
...
(End)

Harvey P. Dale, Table of n, a(n) for n = 1..1000
R. K. Guy, Catwalks, sandsteps and Pascal pyramids, J. Integer Sequences, Vol. 3 (2000), Article #00.1.6.
Antti Karttunen, Some notes on Catalan's Triangle
Thomas Finn Lidbetter, Counting, Adding, and Regular Languages, Master's Thesis, University of Waterloo, Ontario, Canada, 2018.
Index entries for sequences related to binary expansion of n

Cf. A001405, A031443, A036990, A036991, A061855, A014486.

Cf. A237593, A237048, A237591, A237593, A352696.

# We use a simple backtracking algorithm: map(op,[seq(NonDivingLatticeSequences(j),j=1..10)]);
NDLS_GLOBAL := []; NonDivingLatticeSequences := proc(n) global NDLS_GLOBAL; NDLS_GLOBAL := []; NonDivingLatticeSequencesAux(0,0,n); RETURN(NDLS_GLOBAL); end;
NonDivingLatticeSequencesAux := proc(x,h,i) global NDLS_GLOBAL; if(0 = i) then NDLS_GLOBAL := [op(NDLS_GLOBAL),x]; else if(h > 0) then NonDivingLatticeSequencesAux((2*x),h-1,i-1); fi; NonDivingLatticeSequencesAux((2*x)+1,h+1,i-1); fi; end;

a061854[n_] := Select[Range[n], !MemberQ[FoldList[#1+If[#2>0, 1, -1]&, 0, IntegerDigits[n, 2]], -1]]
a061854[116] (* Hartmut F. W. Hoft, Mar 29 2022 *)
Select[Range[120],Min[Accumulate[IntegerDigits[#,2]/.(0->-1)]]>=0&] (* Harvey P. Dale, Sep 11 2023 *)

A090050 Numbers having equal length of longest contiguous block of zeros and ones in binary expansion.

Original entry on oeis.org

2, 5, 10, 12, 19, 21, 25, 38, 42, 44, 50, 51, 52, 56, 71, 75, 76, 77, 83, 85, 89, 100, 101, 102, 105, 108, 113, 142, 147, 150, 153, 154, 155, 166, 170, 172, 178, 179, 180, 184, 199, 201, 202, 203, 204, 205, 210, 211, 212, 217, 226, 227, 232, 240, 271, 279, 284

Offset: 1

Reinhard Zumkeller, Nov 20 2003

A087117(a(n)) = A038374(a(n)), see also A000975.

			180 -> '10110100' with A087117(180)=2 and A038374(180)=2, therefore 180 is a term.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Index entries for sequences related to binary expansion of n

Cf. A031443 (binary digitally balanced).

Cf. A000975, A038374, A087117.

a090050 n = a090050_list !! (n+1)
a090050_list = [x | x <- [1..], a087117 x == a038374 x]
-- Reinhard Zumkeller, May 01 2012

zobQ[n_]:=Module[{s=Split[IntegerDigits[n,2]]},Max[Length/@Select[ s, MemberQ[ #,0]&]] == Max[Length/@Select[s,MemberQ[#,1]&]]]; Select[ Range[ 300],zobQ] (* Harvey P. Dale, Aug 25 2019 *)
Select[Range@1000, (s=Split@IntegerDigits[#,2]; Length@s>1 && Last@Differences@(Length@# & /@ Union@s) == 0) &] (* Hans Rudolf Widmer, Oct 10 2023 *)

Definition corrected, thanks to Leroy Quet. - Sep 17 2008

A072603 Numbers which in base 2 have more 0's than 1's.

Original entry on oeis.org

4, 8, 16, 17, 18, 20, 24, 32, 33, 34, 36, 40, 48, 64, 65, 66, 67, 68, 69, 70, 72, 73, 74, 76, 80, 81, 82, 84, 88, 96, 97, 98, 100, 104, 112, 128, 129, 130, 131, 132, 133, 134, 136, 137, 138, 140, 144, 145, 146, 148, 152, 160, 161, 162, 164, 168, 176, 192, 193

Offset: 1

Reinhard Zumkeller, Jun 23 2002

			8 is present because '1000' contains 3 '0's and 1 '1': 3>1.

T. D. Noe, Table of n, a(n) for n = 1..5202 (numbers up to 2^14)
Jason Bell, Thomas Finn Lidbetter, Jeffrey Shallit, Additive Number Theory via Approximation by Regular Languages, arXiv:1804.07996 [cs.FL], 2018.
Index entries for sequences related to binary expansion of n

Cf. A007088, A000120, A023416, A072600, A072601, A031443, A072602.

Cf. A037861(a(n)) > 0.

a072603 n = a072603_list !! (n-1)
a072603_list = filter ((> 0) . a037861) [1..]
-- Reinhard Zumkeller, Mar 31 2015

gtQ[n_] := Module[{a, b}, {a, b} = DigitCount[n, 2]; b > a]; Select[Range[2^8], gtQ] (* T. D. Noe, Apr 20 2013 *)
Select[Range[200],DigitCount[#,2,0]>DigitCount[#,2,1]&] (* Harvey P. Dale, Feb 26 2023 *)

is(n)=2*hammingweight(n)Charles R Greathouse IV, Apr 18 2020

cp's OEIS Frontend

A049354 Digitally balanced numbers in base 3: equal numbers of 0's, 1's, 2's.

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A144812 Integers having ideal digital mean up to base 7.

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A227870 Numbers with equal number of even and odd digits.

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A378000 Array read by ascending antidiagonals: T(n,k) is the k-th positive integer that is digitally balanced in base n.

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A066196 Primes which have an equal number of zeros and ones in their binary expansion.

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A072600 Numbers which in base 2 have fewer 0's than 1's.

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A144912 Unreduced numerators of digital mean, dm_num(b, n), with rows n in {2, 3, 4, ...} and columns b in {2, 3, 4, ..., n}.

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A061854 Nondiving binary sequences: numbers which in base 2 have at least the same number of 1's as 0's and reading the binary expansion from left (msb) to right (least significant bit), the number of 0's never exceeds the number of 1's.

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