A378000 -id:A378000 - cp's OEIS frontend

A031443 Digitally balanced numbers: positive numbers that in base 2 have the same number of 0's as 1's.

2, 9, 10, 12, 35, 37, 38, 41, 42, 44, 49, 50, 52, 56, 135, 139, 141, 142, 147, 149, 150, 153, 154, 156, 163, 165, 166, 169, 170, 172, 177, 178, 180, 184, 195, 197, 198, 201, 202, 204, 209, 210, 212, 216, 225, 226, 228, 232, 240, 527, 535, 539, 541, 542, 551

Offset: 1

Clark Kimberling

Also numbers k such that the binary digital mean dm(2, k) = (Sum_{i=1..d} 2*d_i - 1) / (2*d) = 0, where d is the number of digits in the binary representation of k and d_i the individual digits. - Reikku Kulon, Sep 21 2008
From Reikku Kulon, Sep 29 2008: (Start)
Each run of values begins with 2^(2k + 1) + 2^(k + 1) - 2^k - 1. The initial values increase according to the sequence {2^(k - 1), 2^(k - 2), 2^(k - 3), ..., 2^(k - k)}.
After this, the values follow a periodic sequence of increases by successive powers of two with single odd values interspersed.
Each run ends with an odd increase followed by increases of {2^(k - k), ..., 2^(k - 2), 2^(k - 1), 2^k}, finally reaching 2^(2k + 2) - 2^(k + 1).
Similar behavior occurs in other bases. (End)
Numbers k such that A000120(k)/A070939(k) = 1/2. - Ctibor O. Zizka, Oct 15 2008
Subsequence of A053754; A179888 is a subsequence. - Reinhard Zumkeller, Jul 31 2010
A000120(a(n)) = A023416(a(n)); A037861(a(n)) = 0.
A001700 gives number of terms having length 2*n in binary representation: A001700(n-1) = #{m: A070939(a(m))=2*n}. - Reinhard Zumkeller, Jun 08 2011
The number of terms below 2^k is A079309(floor(k/2)) for k > 1. - Amiram Eldar, Nov 21 2020

			9 is a term because '1001' contains 2 '0's and 2 '1's.

Reikku Kulon, Table of n, a(n) for n = 1..10000
Jason Bell, Thomas F. Lidbetter and Jeffrey Shallit, Additive number theory via approximation by regular languages, International Journal of Foundations of Computer Science, Vol. 31, No. 6 (2020), pp. 667-687; arXiv preprint, arXiv:1804.07996 [cs.FL], 2018.
Thomas Finn Lidbetter, Counting, Adding, and Regular Languages, Master's Thesis, University of Waterloo, Ontario, Canada, 2018.
Reinhard Zumkeller, Haskell Programs for Binary Digitally Balanced Numbers.
Index entries for sequences related to binary expansion of n

Subsequences: A144798, A144799, A144800, A144801, A144812, A179888.
Subsequence of A053754.
Row n = 2 of A378000.
Cf. A049354-A049360, A000120, A001316, A006519, A023416, A070939, A144777, A145057, A145058, A145059, A145060, A144912, A145037, A191292, A090050, A014486, A061854, A037861, A079309.
Terms of binary width n are enumerated by A001700.

-- See link, showing that Ulrich Schimkes formula provides a very efficient algorithm. Reinhard Zumkeller, Jun 15 2011

[ n: n in [2..250] | Multiplicity({* z: z in Intseq(n,2) *}, 0) eq &+Intseq(n,2) ];  // Bruno Berselli, Jun 07 2011

a:=proc(n) local nn, n1, n0: nn:=convert(n,base,2): n1:=add(nn[i],i=1..nops(nn)): n0:=nops(nn)-n1: if n0=n1 then n else end if end proc: seq(a(n), n = 1..240); # Emeric Deutsch, Jul 31 2008

Select[Range[250],DigitCount[#,2,1]==DigitCount[#,2,0]&] (* Harvey P. Dale, Jul 22 2013 *)
FromDigits[#,2]&/@DeleteCases[Flatten[Permutations/@Table[PadRight[{},2n,{1,0}],{n,5}],1],?(#[[1]]==0&)]//Sort (* _Harvey P. Dale, May 30 2016 *)

for(n=1,100,b=binary(n); l=length(b); if(sum(i=1,l, component(b,i))==l/2,print1(n,",")))

is(n)=hammingweight(n)==hammingweight(bitneg(n,#binary(n))) \\ Charles R Greathouse IV, Mar 29 2013

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

for my $half ( 1 .. 4 ) {
  my $N = 2 * $half;  # only even widths apply
  my $vector = (1 << ($N-1)) | ((1 << ($N/2-1)) - 1);  # first key
  my $n = 1; $n *= $_ for 2 .. $N;    # N!
  my $d = 1; $d *= $_ for 2 .. $N/2;  # (N/2)!
  for (1 .. $n/($d*$d*2)) {
    print "$vector, ";
    my ($v, $d) = ($vector, 0);
    until ($v & 1 or !$v) { $d = ($d << 1)|1; $v >>= 1 }
    $vector += $d + 1 + (($v ^ ($v + 1)) >> 2);  # next key
  }
} # Ruud H.G. van Tol, Mar 30 2014

from sympy.utilities.iterables import multiset_permutations
A031443_list = [int('1'+''.join(p),2) for n in range(1,10) for p in multiset_permutations('0'*n+'1'*(n-1))] # Chai Wah Wu, Nov 15 2019

a(n+1) = a(n) + 2^k + 2^(m-1) - 1 + floor((2^(k+m) - 2^k)/a(n))*(2^(2*m) + 2^(m-1)) where k is the largest integer such that 2^k divides a(n) and m is the largest integer such that 2^m divides a(n)/2^k+1. - Ulrich Schimke (UlrSchimke(AT)aol.com)

A145037(a(n)) = 0. - Reikku Kulon, Oct 02 2008

A049363 a(1) = 1; for n > 1, smallest digitally balanced number in base n.

Original entry on oeis.org

1, 2, 11, 75, 694, 8345, 123717, 2177399, 44317196, 1023456789, 26432593615, 754777787027, 23609224079778, 802772380556705, 29480883458974409, 1162849439785405935, 49030176097150555672, 2200618769387072998445, 104753196945250864004691, 5271200265927977839335179

Offset: 1

Harvey P. Dale

A037968(a(n)) = n and A037968(m) < n for m < a(n). - Reinhard Zumkeller, Oct 27 2003

Also smallest pandigital number in base n. - Franklin T. Adams-Watters, Nov 15 2006

			a(6) = 102345_6 = 1*6^5 + 2*6^3 + 3*6^2 + 4*6^1 + 5*6^0 = 8345.

Reinhard Zumkeller, Table of n, a(n) for n = 1..250
Eric Weisstein's World of Mathematics, Pandigital Number
Wikipedia, Pandigital number
Chai Wah Wu, Pandigital and penholodigital numbers, arXiv:2403.20304 [math.GM], 2024. See p. 1.

Column k=1 of A061845 and A378000 (for n>1).

Cf. A031443, A049354, A049355, A023811.

a049363 n = foldl (\v d -> n * v + d) 0 (1 : 0 : [2..n-1])
-- Reinhard Zumkeller, Apr 04 2012

a:= n-> n^(n-1)+add((n-i)*n^(i-1), i=1..n-2):
seq(a(n), n=1..23);  # Alois P. Heinz, May 02 2020

Table[FromDigits[Join[{1,0},Range[2,n-1]],n],{n,20}] (* Harvey P. Dale, Oct 12 2012 *)

A049363(n)=n^(n-1)+sum(i=1,n-2,n^(i-1)*(n-i))  \\ M. F. Hasler, Jan 10 2012

A049363(n)=if(n>1,(n^n-n)/(n-1)^2+n^(n-2)*(n-1)-1,1)  \\ M. F. Hasler, Jan 12 2012

def A049363(n): return (n**n-n)//(n-1)**2+n**(n-2)*(n-1)-1 if n>1 else 1 # Chai Wah Wu, Mar 13 2024

a(n) = (102345....n-1) in base n. - Ulrich Schimke (ulrschimke(AT)aol.com)
For n > 1, a(n) = (n^n-n)/(n-1)^2 + n^(n-2)*(n-1) - 1 = A023811(n) + A053506(n). - Franklin T. Adams-Watters, Nov 15 2006
a(n) = n^(n-1) + Sum_{m=2..n-1} m * n^(n - 1 - m). - Alexander R. Povolotsky, Sep 18 2022

More terms from Ulrich Schimke (ulrschimke(AT)aol.com)

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

Original entry on oeis.org

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

A049355 Digitally balanced numbers in base 4: equal numbers of 0's, 1's, ... 3's.

Original entry on oeis.org

75, 78, 99, 108, 114, 120, 135, 141, 147, 156, 177, 180, 198, 201, 210, 216, 225, 228, 16815, 16827, 16830, 16875, 16878, 16890, 17007, 17019, 17022, 17055, 17079, 17085, 17115, 17118, 17127, 17133, 17142, 17145, 17259, 17262, 17274, 17307, 17310, 17319, 17325

Offset: 1

Harvey P. Dale

The sum of reciprocals, Sum_{n>=1} 1/a(n), converges. In general, the sum of the reciprocals of balanced numbers in base b converges for all b >= 4, and diverges for b = 2 or 3 (Papanicolaou, 2013). Grivaux (2015) gives 3 * Sum_{k>=1} (4*k)!/(k!^4 * 4^(4*k)) = 0.857... as an upper bound for this sum. The sum is converging slowly: the sums of the reciprocals of the terms with no more than 4*k digits in base 4, for k = 1, 2, ..., are 0.129.., 0.183..., 0.213..., 0.233..., 0.248..., 0.260..., 0.269..., 0.276..., 0.282..., 0.288..., ... . - Amiram Eldar, Feb 15 2024

Harvey P. Dale, Table of n, a(n) for n = 1..1001
Vassilis Papanicolaou, Problem 11729, The American Mathematical Monthly, Vol. 120, No. 8 (2013), p. 754; Summing the Reciprocals of Normal Numbers Base b, Solution to Problem 11729 by Nicole Grivaux, ibid., Vol. 122, No. 8 (2015), p. 806.
Roberto Tauraso, Problem 11729.

Cf. A031443.

Row n = 4 of A378000.

Select[Range[20000],Length[Union[DigitCount[#,4]]]==1&] (* Harvey P. Dale, Mar 19 2013 *)
FromDigits[#,4]&/@DeleteCases[Flatten[Permutations/@Table[PadRight[{},4n,{1,0,2,3}],{n,2}],1],?(#[[1]]==0&)]//Sort (* _Harvey P. Dale, May 30 2016 *)

is(n) = {my(c = matreduce(digits(n,4))[,2]); #c == 4 && #Set(c) == 1;} \\ Amiram Eldar, Feb 15 2024

Offset corrected by Amiram Eldar, Feb 15 2024

A049360 Digitally balanced numbers in base 9: equal numbers of 0's, 1's, ..., 8's.

Original entry on oeis.org

44317196, 44317204, 44317268, 44317284, 44317348, 44317356, 44317844, 44317852, 44317988, 44318012, 44318068, 44318084, 44318564, 44318580, 44318636, 44318660, 44318796, 44318804, 44319292, 44319300, 44319364

Offset: 1

Harvey P. Dale

Paolo Xausa, Table of n, a(n) for n = 1..10000

Cf. A031443.

Row n = 9 of A378000.

A049356 Digitally balanced numbers in base 5: equal numbers of 0's, 1's, ..., 4's.

Original entry on oeis.org

694, 698, 714, 722, 738, 742, 894, 898, 954, 970, 978, 990, 1014, 1022, 1054, 1070, 1102, 1110, 1138, 1142, 1178, 1190, 1202, 1210, 1294, 1298, 1334, 1346, 1358, 1366, 1394, 1398, 1454, 1470, 1478, 1490, 1634, 1646, 1654, 1670, 1726, 1730, 1758, 1766, 1778

Offset: 1

Harvey P. Dale

The first 96 (=4*4!) terms of this sequence and of A031946 are identical. a(97) = 1001223344_5 = 1976724.

			a(1) = 10234_5 = 694.
a(96) = 43210_5 = 2930.

Alois P. Heinz, Table of n, a(n) for n = 1..10000 (first 96 terms from Vincenzo Librandi)

Cf. A031443.
Subsequence of A031946.
Row n = 5 of A378000.

Select[Range[5000], Length[Union[DigitCount[#, 5]]]==1&] (* Vincenzo Librandi, Apr 18 2013 *)

Edited by Rick L. Shepherd, Jun 22 2003

A049357 Digitally balanced numbers in base 6: equal numbers of 0's, 1's, ..., 5's.

Original entry on oeis.org

8345, 8350, 8375, 8385, 8410, 8415, 8525, 8530, 8585, 8600, 8620, 8630, 8735, 8745, 8765, 8780, 8835, 8840, 8950, 8955, 8980, 8990, 9015, 9020, 10505, 10510, 10535, 10545, 10570, 10575, 11045, 11050, 11165, 11190, 11200, 11220, 11255

Offset: 1

Harvey P. Dale

Subset of A031947. [R. J. Mathar, Oct 15 2008]

			8345 is 102345 in base 6.

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

Cf. A031443.

Row n = 6 of A378000.

isA049357 := proc(n)
    local ct60, ct61,ct62,ct63,ct64,ct65,d ;
    ct60 := 0 ; ct61 := 0 ; ct62 := 0 ;
    ct63 := 0 ; ct64 := 0 ; ct65 := 0 ;
    for d in convert(n,base,6) do
        if d = 0 then
            ct60 := ct60+1 ;
        elif d = 1 then
            ct61 := ct61+1 ;
        elif d =2 then
            ct62 := ct62+1 ;
        elif d =3 then
            ct63 := ct63+1 ;
        elif d =4 then
            ct64 := ct64+1 ;
        elif d =5 then
            ct65 := ct65+1 ;
        end if ;
    end do:
    if ct60 = ct61 and ct61 = ct62 and ct62 = ct63 and ct63 = ct64 and ct64 = ct65 then
        return true ;
    else
        return false;
    end if;
end proc:
for n from 1 to 12000 do
    if isA04957(n) then
        print(n);
    end if;
end do: # R. J. Mathar, Aug 24 2023

Select[Range[6000], Length[Union[DigitCount[#, 6]]] == 1 &] (* Vincenzo Librandi, Apr 18 2013 *)

A049358 Digitally balanced numbers in base 7: equal numbers of 0's, 1's, ..., 6's.

Original entry on oeis.org

123717, 123723, 123759, 123771, 123807, 123813, 124011, 124017, 124095, 124113, 124143, 124155, 124347, 124359, 124389, 124407, 124485, 124491, 124689, 124695, 124731, 124743, 124779, 124785, 125775, 125781, 125817, 125829, 125865, 125871, 126363, 126369

Offset: 1

Harvey P. Dale

There are 4,320 terms with single numbers of the zero-to-six digits. - Harvey P. Dale, May 02 2025

			123717 is 1023456 in base 7.

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

Cf. A031443.

Row n = 7 of A378000.

Select[Range[126000], Length[Union[DigitCount[#, 7]]]==1&] (* Vincenzo Librandi, Apr 18 2013 *)
FromDigits[IntegerDigits[#],7]&/@Select[FromDigits/@Permutations[Range[0,6]],IntegerLength[#]==7&] (* Harvey P. Dale, May 02 2025 *)

A049359 Digitally balanced numbers in base 8: equal numbers of 0's, 1's, ..., 7's.

Original entry on oeis.org

2177399, 2177406, 2177455, 2177469, 2177518, 2177525, 2177847, 2177854, 2177959, 2177980, 2178022, 2178036, 2178351, 2178365, 2178407, 2178428, 2178533, 2178540, 2178862, 2178869, 2178918, 2178932, 2178981, 2178988

Offset: 1

Harvey P. Dale

			2177399 is 10234567 in base 8.

Vincenzo Librandi, Table of n, a(n) for n = 1..720

Cf. A031443.

Row n = 8 of A378000.

Select[Range[2100000,2200000],Union[DigitCount[#,8]]=={1}&] (* Harvey P. Dale, Sep 14 2012 *)

A378073 Positive integers that are digitally balanced in some integer base b >= 2.

Original entry on oeis.org

2, 9, 10, 11, 12, 15, 19, 21, 35, 37, 38, 41, 42, 44, 49, 50, 52, 56, 75, 78, 99, 108, 114, 120, 135, 139, 141, 142, 147, 149, 150, 153, 154, 156, 163, 165, 166, 169, 170, 172, 177, 178, 180, 184, 195, 197, 198, 201, 202, 204, 209, 210, 212, 216, 225, 226, 228

Offset: 1

Paolo Xausa, Nov 16 2024

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

This is the set of all of the distinct terms in A378000.

			99 is a term because it's a digitally balanced number in base 5 (99 = 12034_5).
135 is a term because it's a digitally balanced number in two bases (135 = 10000111_2 = 2013_4).

Paolo Xausa, Table of n, a(n) for n = 1..10000
Giovanni Resta, Digitally balanced numbers, Numbers Aplenty, 2013.

Cf. A049364, A061845, A065963, A378000, A378080 (complement).
Supersequence of A378104.
Positions of positive terms in A378191.

A378073Q[n_] := Module[{b = 1, len}, While[(len = IntegerLength[n, ++b]) >= b && !(Divisible[len, b] && SameQ @@ DigitCount[n, b])]; len >= b];
Select[Range[500], A378073Q]

cp's OEIS Frontend

A031443 Digitally balanced numbers: positive numbers that in base 2 have the same number of 0's as 1's.

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A049363 a(1) = 1; for n > 1, smallest digitally balanced number in base n.

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

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A049355 Digitally balanced numbers in base 4: equal numbers of 0's, 1's, ... 3's.

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A049360 Digitally balanced numbers in base 9: equal numbers of 0's, 1's, ..., 8's.

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A049356 Digitally balanced numbers in base 5: equal numbers of 0's, 1's, ..., 4's.

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A049357 Digitally balanced numbers in base 6: equal numbers of 0's, 1's, ..., 5's.

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