A049356 -id:A049356 - cp's OEIS frontend

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

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]]

A145100 Integers in which no more than half the digits (rounded up) are the same, for all bases up to ten.

Original entry on oeis.org

1, 2, 17, 19, 25, 38, 52, 56, 75, 76, 82, 83, 90, 92, 97, 98, 100, 102, 104, 105, 108, 113, 116, 135, 139, 141, 142, 147, 150, 153, 163, 165, 177, 178, 180, 184, 195, 197, 198, 201, 204, 209, 210, 212, 225, 226, 232, 267, 269, 275, 278, 279, 291, 293, 294, 298

Offset: 1

Reikku Kulon, Oct 01 2008

			267 in bases [2, 10] is 100001011, 100220, 10023, 2032, 1123, 531, 413, 326, 267. There are five zeros out of nine digits in its binary representation and no more than half the digits in the other bases are identical.

Cf. A049354, A049355, A049356, A049357, A049358, A049359, A049360, A049361, A049362, A049363, A049364

A031946 Numbers whose base-5 expansions have 5 distinct digits.

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

Offset: 1

Clark Kimberling

In other words, numbers in which 0,1,2,3,4 all occur in base 5.

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

A supersequence of A049356. The first 96 (=4*4!) terms of this sequence and of A049356 are identical. a(97) = 100234_5 = 3194.

Edited by Rick L. Shepherd, Jun 22 2003

A145101 Integers in which no digit occurs more than once more often than any other digit and not all repeated digits are identical, for all bases up to ten.

Original entry on oeis.org

1, 2, 17, 19, 25, 38, 52, 56, 75, 76, 82, 90, 92, 98, 100, 102, 104, 105, 108, 116, 141, 142, 150, 153, 177, 178, 180, 184, 195, 198, 204, 210, 212, 225, 226, 232, 294, 308, 316, 332, 395, 396, 410, 412, 420, 434, 450, 460, 481, 542, 572, 611, 689, 752, 818

Offset: 1

Reikku Kulon, Oct 01 2008

Subset of A145100. The first number not in both sequences is 83.

			97 is in A145100 but not in this sequence: in base 3 it is 10121 and 1 occurs two times more often than either 0 or 2.
98 is in this sequence: in bases [2, 10] it is 1100010, 10122, 1202, 343, 242, 200, 142, 118, 98.

Cf. A049354, A049355, A049356, A049357, A049358, A049359, A049360, A049361, A049362, A049363, A049364, A145100

A145104 Digitally fair numbers: integers n such that in all bases b = 2..10 no digit occurs more often than ceiling(d/b) times, where d is the number of digits of n in base b.

Original entry on oeis.org

1, 2, 19, 198, 25410896, 31596420, 10601629982, 10753657942, 11264883970, 11543640378, 11553029646, 11665278790, 12034384190, 12038440382, 12366849814, 12519032774, 12781964290, 12971872086, 13156400486

Offset: 1

Reikku Kulon, Oct 01 2008

Presumed infinite. Next term >= 3^20.

Hagen von Eitzen, Table of n, a(n) for n = 1..10000

Cf. A049354, A049355, A049356, A049357, A049358, A049359, A049360, A049361, A049362, A049363, A049364, A145100, A145101.

More terms from Hagen von Eitzen, Jun 20 2009

cp's OEIS Frontend

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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A145100 Integers in which no more than half the digits (rounded up) are the same, for all bases up to ten.

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A031946 Numbers whose base-5 expansions have 5 distinct digits.

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A145101 Integers in which no digit occurs more than once more often than any other digit and not all repeated digits are identical, for all bases up to ten.

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A145104 Digitally fair numbers: integers n such that in all bases b = 2..10 no digit occurs more often than ceiling(d/b) times, where d is the number of digits of n in base b.

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