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
a(6) = 102345_6 = 1*6^5 + 2*6^3 + 3*6^2 + 4*6^1 + 5*6^0 = 8345.
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a049363 n = foldl (\v d -> n * v + d) 0 (1 : 0 : [2..n-1])
-- Reinhard Zumkeller, Apr 04 2012
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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
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Table[FromDigits[Join[{1,0},Range[2,n-1]],n],{n,20}] (* Harvey P. Dale, Oct 12 2012 *)
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A049363(n)=n^(n-1)+sum(i=1,n-2,n^(i-1)*(n-i)) \\ M. F. Hasler, Jan 10 2012
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A049363(n)=if(n>1,(n^n-n)/(n-1)^2+n^(n-2)*(n-1)-1,1) \\ M. F. Hasler, Jan 12 2012
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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
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
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.
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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
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
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
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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 *)
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
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
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
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
Cf.
A049354,
A049355,
A049356,
A049357,
A049358,
A049359,
A049360,
A049361,
A049362,
A049363,
A049364,
A145100,
A145101.
Showing 1-7 of 7 results.
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