A049354 -id:A049354 - cp's OEIS frontend

A372146 The smallest number k for which exactly n of its divisors are digitally balanced numbers in base 3 (A049354).

1, 11, 105, 420, 924, 2772, 6240, 4620, 18480, 13860, 55440, 69300, 120120, 180180, 240240, 360360, 514800, 720720, 1029600, 1801800, 2162160, 2522520, 2282280, 5045040, 7207200, 4564560, 6846840, 12612600, 15135120, 11411400, 20540520, 29343600, 22822800, 49729680

Offset: 0

Marius A. Burtea, May 23 2024

			Since A049354(1) = 11 it follows that a(0) = 1.
The numbers 2 through 10 have no divisors in A049354 and A049354(1) = 11 = 102_3, so a(1) = 11.
105 has only two divisors in A049354, 15 = A049354(2) and 21 = A049354(4) and is the smallest with exactly two divisors in A049354, so a(2) = 105.

Cf. A049354, A357035.

bal:=func; a:=[]; for n in [0..34] do k:=1; while #[d:d in Divisors(k)|bal(d)] ne n do k:=k+1; end while; Append(~a,k); end for; a;

N:= 15: # for terms before the first term >= 3^(N+1)
db:= proc(n) option remember; local L,d,m;
  L:= convert(n,base,3);
  d:= nops(L);
  d mod 3 = 0 and 3*numboccur(0,L) = d and 3*numboccur(1,L) = d
end proc:
W:= Vector(3^(N+1),datatype=integer[4]):
for d from 3 to N by 3 do
  for t from 3^(d-1) to 3^d-1 do
    if db(t) then
          J:= [seq(i, i=t..3^(N+1), t)];
          W[J]:= W[J] +~ 1;
    fi
od od:
M:= max(W):
V:= Array(0..M): count:= 0:
for i from 1 to 3^(N+1) while count < M+1 do
  if V[W[i]] = 0 then V[W[i]]:= i; count:= count+1 fi;
od:
L:= convert(V,list):
if not member(0,L,'m') then m:= M+2 fi:
L[1..m-1]; # Robert Israel, Jun 03 2024

balQ[n_, b_] := balQ[n, b] = MinMax@ Differences@ DigitCount[n, b] == {0, 0}; f[n_] := DivisorSum[n, 1 &, balQ[#, 3] &]; seq[len_, nmax_] := Module[{s = Table[0, {len}], c = 0, n = 1, i}, While[c < len && n < nmax, i = f[n] + 1; If[i <= len && s[[i]] == 0, c++; s[[i]] = n]; n++]; s]; seq[12, 10^5] (* Amiram Eldar, Jun 03 2024 *)

A372147 Numbers k for which k and k^2 are digitally balanced numbers in base 3 (A049354).

Original entry on oeis.org

500, 544, 550, 552, 588, 622, 307172, 307402, 307888, 308920, 309480, 309522, 309604, 310348, 310422, 310658, 310754, 310964, 310974, 310980, 311206, 311214, 311342, 311466, 311582, 311788, 311790, 312050, 312070, 312100, 313022, 313050, 313336, 313502, 313512

Offset: 1

Marius A. Burtea, May 23 2024

			500 = 200112_3 and 500^2 = 250000 =110200221021_3, so 500 is a term.
544 = 202011_3 and 544^2 = 295936 =120000221121_3, so 544 is a term.

Cf. A049354, A353139, A353140.

bal:=func; [n:n in [1..320000]|bal(n) and bal(n*n)];

balQ[n_, b_] := MinMax@ Differences@ DigitCount[n, b] == {0, 0}; Select[Range[320000], balQ[#, 3] && balQ[#^2, 3] &] (* Amiram Eldar, Jun 03 2024 *)

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

Original entry on oeis.org

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)

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]]

A049364 Smallest number that is digitally balanced in all bases 2, 3, ... n.

Original entry on oeis.org

2, 572, 8410815, 59609420837337474

Offset: 2

Harvey P. Dale

Joe K. Crump, Joe's Number Theory Web.

Cf. A031443, A049354, A049355, etc.

See A065963 for another version.

a(5) from Joe K. Crump (joecr(AT)carolina.rr.com), Aug 29 2000, who reports that the next term is greater than 22185312344622607538702383625075608111920737991870030337.

a(6) > 5^434 and would be too large to include. This result comes from combining Joe K. Crump's algorithm with the observation that digitally balanced numbers in base 5 are all even and digitally balanced numbers in base 3 are even iff they are of even length. - Martin Fuller, Nov 27 2006

A031944 Numbers in which digits 0,1,2 all occur in base 3.

Original entry on oeis.org

11, 15, 19, 21, 29, 32, 33, 34, 35, 38, 42, 45, 46, 47, 48, 51, 55, 57, 58, 59, 61, 63, 64, 65, 66, 69, 73, 75, 83, 86, 87, 88, 89, 92, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 110, 113, 114, 115, 116, 119, 123, 126, 127, 128, 129, 132, 135

Offset: 1

Clark Kimberling

A043530(a(n)) = 3; complement of A154314. - Reinhard Zumkeller, Jan 07 2009

A212193(a(n)) = 3. - Reinhard Zumkeller, May 04 2012

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Ternary
Eric Weisstein's World of Mathematics, Pandigital Number
Wikipedia, Ternary numeral system
Wikipedia, Pandigital number

Cf. A043530, A049354, A154314, A212193.

import Data.List (elemIndices)
a031944 n = a031944_list !! (n-1)
a031944_list = elemIndices 3 a212193_list
-- Reinhard Zumkeller, May 04 2012

Select[Range[200],Min[DigitCount[#,3]]>0&] (* Harvey P. Dale, Nov 21 2015 *)

from sympy.ntheory import count_digits
def ok(n): c = count_digits(n, 3); return all(c[d] > 0 for d in [0, 1, 2])
print([k for k in range(136) if ok(k)]) # Michael S. Branicky, Nov 15 2021

A065963 Smallest number which is digitally balanced in n bases, for 2 <= base b <= a(n).

Original entry on oeis.org

2, 135, 13575, 8451765, 71749827664620

Offset: 1

Naohiro Nomoto, Dec 08 2001

See A049364 for another version.

			For n=3, a (3) = 13575. 11010100000111 (base 2), 200121210 (base 3), 142503 (base 6). 13575 is the minimum number which is digitally balanced in three bases.

Cf. A031443, A049354-A049364.

71749827664620 from Martin Fuller, Oct 16 2006

A049361 Digitally balanced numbers in both bases 2 and 3.

Original entry on oeis.org

572, 678, 684, 8287, 8351, 8367, 8399, 8429, 8435, 8479, 8527, 8539, 8551, 8563, 8569, 8591, 8605, 8615, 8619, 8677, 8819, 8825, 8855, 8903, 8907, 8933, 8945, 8975, 8983, 9017, 9065, 9161, 9295, 9303, 9307, 9309, 9323, 9371, 9387, 9401

Offset: 1

Harvey P. Dale

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

Intersection of A031443 and A049354.

With[{digMax = 3, termsNumber = 100}, Select[Union @@ Table[(FromDigits[#, 3] &) /@ Select[Permutations[Flatten[Table[Range[0, 2], k]]], #[[1]] > 0 &], {k, 1, digMax}], SameQ @@ DigitCount[#, 2, Range[0, 1]] &][[1 ;; termsNumber]]] (* Amiram Eldar, Feb 15 2024 *)

A049362 Digitally balanced numbers in bases 2, 3 and 4.

Original entry on oeis.org

8410815, 8411067, 8411631, 8411871, 8411895, 8412153, 8412855, 8413083, 8413161, 8416095, 8416119, 8416125, 8417133, 8417205, 8418795, 8418999, 8419035, 8419065, 8419743, 8419815, 8420055, 8420217, 8420253, 8427135

Offset: 1

Harvey P. Dale

Is this sequence infinite?

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

Intersection of A031443, A049354 and A049355.

dbQ[n_, b_] := SameQ @@ DigitCount[n, b, Range[0, b - 1]]; With[{digMax = 3, termsNumber = 24}, Select[Union @@ Table[(FromDigits[#, 4] &) /@ Select[Permutations[Flatten[Table[Range[0, 3], k]]], #[[1]] > 0 &], {k, 1, digMax}], dbQ[#, 2] && dbQ[#, 3] &][[1 ;; termsNumber]]] (* Amiram Eldar, Feb 15 2024 *)

cp's OEIS Frontend

A372146 The smallest number k for which exactly n of its divisors are digitally balanced numbers in base 3 (A049354).

Views

Author

Keywords

Examples

Crossrefs

Programs

Magma

Maple

Mathematica

A372147 Numbers k for which k and k^2 are digitally balanced numbers in base 3 (A049354).

Views

Author

Keywords

Examples

Crossrefs

Programs

Magma

Mathematica

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Magma

Maple

Mathematica

PARI

PARI

PARI

Perl

Python

Formula

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

PARI

PARI

Python

Formula

Extensions

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A049364 Smallest number that is digitally balanced in all bases 2, 3, ... n.

Views

Author

Keywords

Links

Crossrefs

Extensions

A031944 Numbers in which digits 0,1,2 all occur in base 3.

Views

Author

Keywords

Comments

Links

Crossrefs