A049363 -id:A049363 - cp's OEIS frontend

A023811 Largest metadrome (number with digits in strict ascending order) in base n.

0, 1, 5, 27, 194, 1865, 22875, 342391, 6053444, 123456789, 2853116705, 73686780563, 2103299351334, 65751519677857, 2234152501943159, 81985529216486895, 3231407272993502984, 136146740744970718253, 6106233505124424657789, 290464265927977839335179

Offset: 1

Olivier Gérard

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

The smallest permutational number in A134640 in the n-positional system. - Artur Jasinski, Nov 07 2007

			a(5) = 1234[5] (in base 5) = 1*5^3 + 2*5^2 + 3*5 + 4 = 125 + 50 + 15 + 4 = 194.
a(10) = 123456789 (in base 10).

Vincenzo Librandi, Table of n, a(n) for n = 1..200
Christian Perfect, Integer sequence reviews on Numberphile (or vice versa), 2013.
Chai Wah Wu, Pandigital and penholodigital numbers, arXiv:2403.20304 [math.GM], 2024. See p. 1.

Cf. A049363, A051846, A058128, A060073.

Cf. A062813, A134640.

a023811 n = foldl (\val dig -> val * n + dig) 0 [0 .. n - 1]
-- Reinhard Zumkeller, Aug 29 2014

[0] cat [(n^n-n^2+n-1)/(n-1)^2: n in [2..20]]; // Vincenzo Librandi, May 22 2012

0, seq((n^n-n^2+n-1)/(n-1)^2, n=2..100); # Robert Israel, Dec 13 2015

Table[Total[(#1 n^#2) & @@@ Transpose@ {Range[n - 1], Reverse@ (Range[n - 1] - 1)}], {n, 20}] (* Michael De Vlieger, Jul 24 2015 *)
Table[Sum[(b - k)*b^(k - 1), {k, b - 1}], {b, 30}] (* Clark Kimberling, Aug 22 2015 *)
Table[FromDigits[Range[0, n - 1], n], {n, 20}] (* L. Edson Jeffery, Dec 13 2015 *)

{for(i=1,18,cuo=0; for(j=1,i-1,cuo=cuo+j*i^(i-j-1)); print1(cuo,", "))} \\\ Douglas Latimer, May 16 2012

A023811(n)=if(n>1,(n^n-n^2)\(n-1)^2+1)  \\ M. F. Hasler, Jan 22 2013

def a(n): return (n**n - n**2 + n - 1)//((n - 1)**2) if n > 1 else 0
print([a(n) for n in range(1, 21)]) # Michael S. Branicky, Apr 24 2023

a(n) = Sum_{j=1...n-1} j*n^(n-1-j).
lim_{n->infinity} a(n)/a(n-1) - a(n-1)/a(n-2) = exp(1). - Conjectured by Gerald McGarvey, Sep 26 2004. Follows from the formula below and lim_{n->infinity} (1+1/n)^n = e. - Franklin T. Adams-Watters, Jan 25 2010
a(n) = (n^n-n^2+n-1)/(n-1)^2 = A058128(n)-1 = n*A060073(n)-1 (for n>=2). - Henry Bottomley, Feb 21 2001

Edited by M. F. Hasler, Jan 22 2013

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]]

A061845 Numbers that have one of every digit in some base.

Original entry on oeis.org

2, 11, 15, 19, 21, 75, 78, 99, 108, 114, 120, 135, 141, 147, 156, 177, 180, 198, 201, 210, 216, 225, 228, 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

Offset: 2

Erich Friedman, Jun 23 2001

Also known as pandigital numbers, especially in base 10.

			Base 3 values are 102_3 = 11, 120_3 = 15, 201_3 = 19, 210_3 = 21.
Triangle begins:
    2;
   11,  15,  19,  21;
   75,  78,  99, 108, 114, 120, 135, 141, 147, 156, 177, 180,  198,  201, ...
  694, 698, 714, 722, 738, 742, 894, 898, 954, 970, 978, 990, 1014, 1022, ...
  ...

Alois P. Heinz, Rows n = 2..8, flattened

Column k=1 gives A049363 (for n>1).
Last elements of rows give A062813.
Cf. A050278, A134640, A001563 (row lengths).

dtn[ L_, base_ ] := Fold[ base*#1+#2&, 0, L ] f[ n_ ] := Map[ dtn[ #, n ]&, Select[ Permutations[ Range[ 0, n-1 ] ], First[ # ]>0& ] ] Flatten[ Join[ Table[ f[ i ], {i, 2, 5} ] ] ]

A258103 Number of pandigital squares (containing each digit exactly once) in base n.

Original entry on oeis.org

0, 0, 1, 0, 1, 3, 4, 26, 87, 47, 87, 0, 547, 1303, 3402, 0, 24192, 187562

Offset: 2

Adam J.T. Partridge, May 20 2015

For n = 18, the smallest and largest pandigital squares are 2200667320658951859841 and 39207739576969100808801. For n = 19, they are 104753558229986901966129 and 1972312183619434816475625. For n = 20, they are 5272187100814113874556176 and 104566626183621314286288961. - Chai Wah Wu, May 20 2015
When n is even, (n-1) is a factor of the pandigital squares.  When n is odd, (n-1)/2 is a factor with the remaining factors being odd.  Therefore, when n is odd and (n-1)/2 has an odd number of 2s as prime factors there are no pandigital squares in base n (e.g. 5, 13, 17 and 21). - Adam J.T. Partridge, May 21 2015
If n is odd and (n-1)/2 has an odd 2-adic valuation, then there are no squares in base n using all the digits from 1 to n-1 once, or all the digits from 0 to n-2 once or all the digits from 1 to n-2 once. This can be proved using the same argument as in the linked blogposts. - Chai Wah Wu, Feb 25 2024

			For n=4 there is one pandigital square, 3201_4 = 225 = 15^2.
For n=6 there is one pandigital square, 452013_6 = 38025 = 195^2.
For n=10 there are 87 pandigital squares (A036745).
There are no pandigital squares in bases 2, 3, 5 or 13.
Hexadecimal has 3402 pandigital squares, the largest is FED5B39A42706C81.

A. J. T. Partridge, Why there are no pandigital squares in base 13
Chai Wah Wu, Square pandigital numbers
Chai Wah Wu, Pandigital and penholodigital numbers, arXiv:2403.20304 [math.GM], 2024. See p. 2.

Cf. A036745, A054038, A071519.

a(n) = if(n%2==1 && valuation(n-1,2)%2==0, 0, my(lim=sqrtint(n^n - (n^n-n)/(n-1)^2), count=0); for(m=sqrtint((n^n-n)/(n-1)^2 + n^(n-2)*(n-1) - 1), lim, if(#Set(digits(m^2,n))==n, count++)); count) \\ Jianing Song, Feb 23 2024. Note that the searching range for m is [sqrt(A049363(n)), sqrt(A062813(n))]

from gmpy2 import isqrt, mpz, digits
def A258103(n): # requires 2 <= n <= 62
    c, sm, sq = 0, mpz(''.join([digits(i, n) for i in range(n-1, -1, -1)]), n), mpz(''.join(['1', '0']+[digits(i, n) for i in range(2, n)]), n)
    m = isqrt(sq)
    sq = m*m
    m = 2*m+1
    while sq <= sm:
        if len(set(digits(sq, n))) == n:
            c += 1
        sq += m
        m += 2
    return c # Chai Wah Wu, May 20 2015

a(17)-a(19) from Giovanni Resta, May 20 2015

A260182 Smallest square that is pandigital in base n.

Original entry on oeis.org

4, 64, 225, 5329, 38025, 314721, 3111696, 61058596, 1026753849, 31529329225, 892067027049, 307197306432025, 803752551280900, 29501156485626049, 1163446635475467225, 830482914641378019961, 2200667320658951859841, 104753558229986901966129, 5272187100814113874556176

Offset: 2

Jon E. Schoenfield, Jul 17 2015

Compare this sequence with A260117, Smallest triangular number that is pandigital in base n. Presumably, lim_{n->infinity} A260117(n)/A049363(n) = 1, but the same cannot be true for this sequence: the sum of the base-n digits of a number that is pandigital in base n must be 0+1+2+...+n-1 = binomial(n,2), but there are certain values of n for which no n-digit square can have a digit sum of binomial(n,2); for such values of n, a(n) must have more than n digits in base n. [E.g., the base-13 expansion of every square has a digit sum s == {0,1,4,9} (mod 12) (cf. A096008), but a square that is pandigital in base 13 and has exactly 13 digits would have a digit sum s = 78 == 6 (mod 12), so no such number exists; a 14-digit base-13 pandigital square would have each of the digits 0..12 exactly once except for one duplicated digit, which would have to be 3, 6, 7, or 10 (to yield a digit sum of 81, 84, 85, or 88, whose residues modulo 12 are 9, 0, 1, and 4, respectively). - Jon E. Schoenfield, Mar 23 2019]
The values of n for which there exists no pandigital square that is exactly n digits long (in base n) begin with 2, 3, 5, 13, 17, 21, ...; presumably, for all such values of n, a(n) is exactly n+1 base-n digits long.
In base 2, there are no 2-digit squares at all, so a(2) must have more than 2 binary digits. For n = 3, 5, 13, 17, 21, ..., there exists no square, regardless of its number of digits, whose base-n digit sum equals binomial(n,2); see A260191.

			Using the letters a, b, c, ... to represent digit values 10, 11, 12, ..., the terms begin as follows:
.
   n            a(n) in base 10          a(n) in base n
  ==  =========================  ======================
   2                          4                   100_2
   3                         64                  2101_3
   4                        225                  3201_4
   5                       5329                132304_5
   6                      38025                452013_6
   7                     314721               2450361_7
   8                    3111696              13675420_8
   9                   61058596             136802574_9
  10                 1026753849            1026753849_10
  11                31529329225           1240a536789_11
  12               892067027049          124a7b538609_12
  13            307197306432025        10254773ca86b9_13
  14            803752551280900        10269b8c57d3a4_14
  15          29501156485626049       102597bace836d4_15
  16        1163446635475467225      1025648cfea37bd9_16
  17      830482914641378019961    101246a89cgfb357ed_17
  18     2200667320658951859841    10236b5f8eg4ad9ch7_18
  19   104753558229986901966129   10234dhbg7ci8f6a9e5_19
  20  5272187100814113874556176  1024e7cdi3hb695fja8g_20

Chai Wah Wu, Table of n, a(n) for n = 2..28 (n = 2..22 from Jon E. Schoenfield)
Rosetta Code, First perfect square in base N with N unique digits, lists a(n) for n = 2..39.

Cf. A007953, A049363, A096008, A249034, A260117, A260191.

A260117 Smallest triangular number that is pandigital in base n.

Original entry on oeis.org

6, 15, 78, 990, 8385, 128271, 2293011, 46923828, 1062489753, 27403863105, 757016521030, 24028339652778, 807863408487460, 29499468896141965, 1162871296355724735, 49093065731151773880, 2200689210818047715703, 104755000778178115071015, 5271254575974180914006953

Offset: 2

Jon E. Schoenfield, Jul 17 2015

Presumably, lim_{n->infinity} a(n)/A049363(n) = 1.

			Using the letters a, b, c, ... to represent digit values 10, 11, 12, ..., the terms are as follows:
.
   n            a(n) in base 10  a(n) in base n
  ==  =========================  ==============
   2                          6  110_2
   3                         15  120_3
   4                         78  1032_4
   5                        990  12430_5
   6                       8385  102453_6
   7                     128271  1042653_7
   8                    2293011  10576423_8
   9                   46923828  107258346_9
  10                 1062489753  1062489753_10
  11                27403863105  10692847a53_11
  12               757016521030  10286b37459a_12
  13             24028339652778  1053b2a49c786_13
  14            807863408487460  1036cb2487d59a_14
  15          29499468896141965  102568d3be749ca_15
  16        1162871296355724735  102359486ac7edbf_16
  17       49093065731151773880  1029a46d78g53cbef_17
  18     2200689210818047715703  10237c486geh5bdaf9_18
  19   104755000778178115071015  10236a47589cgdfeibh_19
  20  5271254575974180914006953  10235i96e4jb8gfcah7d_20

Chai Wah Wu, Table of n, a(n) for n = 2..28 (n = 2..22 from Jon E. Schoenfield)

Cf. A049363.

A049363(n)=if(n>1, (n^n-n)/(n-1)^2+n^(n-2)*(n-1)-1, 1)
a(n)=my(k=ceil((sqrt(8*A049363(n)+1)-1)/2),t); while(#Set(digits(t=binomial(k+1,2),n))Charles R Greathouse IV, Jul 17 2015

A037968 Greatest number of distinct digits of n in any base b>=2.

Original entry on oeis.org

1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 3, 2, 2, 3, 3, 2, 3, 2, 2, 3, 2, 2, 3, 3, 3, 3, 2, 3, 3, 3, 3, 3, 2, 3, 3, 3, 3, 3, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 3, 3, 4, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 3, 3, 3, 3, 3, 3

Offset: 1

Clark Kimberling

a(A049363(n)) = n and a(m) < n for m < A049363(n). - Reinhard Zumkeller, Oct 27 2003
From Robert G. Wilson v, Aug 29 2014: (Start)
First occurrence of k: 1, 2, 11, 75, 694, 8345, 123717, …, . A049363
Last occurrence of k: 1, 43, 2462, 140081, 20338085, …, . A246535
Count of f(j) = k: 1, 22, 340, 8888, …, . not in the OEIS data base.
(End)

Robert G. Wilson v, Table of n, a(n) for n = 1..1000

See A037914 for the least corresponding bases.

f[n_] := Max@ Table[ Length@ Union@ IntegerDigits[n, b], {b, 2, n }]; f[1] = 1; Array[f, 105] (* Robert G. Wilson v, Aug 29 2014 *)

a(n) = {nb = 1; for (k=2, n, nb = max (nb, #Set(digits(n, k)));); nb;} \\ Michel Marcus, Sep 13 2014

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

A171744 a(n) is the smallest exponent such that prime(n)^k is pandigital in base 10.

Original entry on oeis.org

68, 39, 19, 18, 23, 22, 14, 17, 14, 12, 11, 13, 11, 13, 12, 13, 11, 14, 10, 15, 14, 13, 9, 11, 13, 9, 15, 14, 13, 12, 11, 15, 10, 7, 12, 9, 12, 10, 11, 8, 11, 8, 12, 11, 13, 13, 10, 12, 10, 8, 11, 12, 9, 7, 6, 7, 8, 12, 8, 8, 7, 7, 10, 9, 9, 6, 9, 10, 9, 10

Offset: 1

Eva-Maria Zschorn (e-m.zschorn(AT)zaschendorf.km3.de), Dec 17 2009

A pandigital number is an integer that in a given base has among its significant digits each digit used in the base at least once.

			2^68 = 295147905179352825856 (21 digits), 3^39 = 4052555153018976267 (19)
5^19 = 19073486328125 (14), 7^18 = 1628413597910449 (16), 11^23 = 895430243255237372246531 (24)
13^22 = 3211838877954855105157369 (25), 17^14 = 168377826559400929 (18)
19^17 = 5480386857784802185939 (22), 23^14 = 11592836324538749809 (20)
29^12 = 353814783205469041 (18), 31^11 = 25408476896404831 (17)
37^13 = 243569224216081305397 (21), 41^11 = 550329031716248441 (18)
43^13 = 1718264124282290785243 (22), 47^12 = 116191483108948578241 (21)
53^13 = 26036721925606486195973 (23), 59^11 = 30155888444737842659 (20)
61^14 = 9876832533361318095112441 (25), 67^10 = 1822837804551761449 (19)
71^15 = 5873205959385493353867330551 (28), 73^14 = 122045014039746588673695409 (23)
79^13 = 4668229371502258117133839 (25), 83^9 = 186940255267540403 (18)
89^11 = 2775173073766990340489 (22), 97^13 = 67302709016557486028618977 (26)
101^9 = 1093685272684360901 (19), 103^15 = 1557967416600764580522382952407 (31)
107^14 = 25785341502012466393542552649 (29), 109^13 = 306580461214335498944273629 (27)
113^12 = 4334523100191686738306881 (25), 127^11 = 138624799340320978519423 (24)

E.I. Ignatjew, Mathematische Spielereien, Urania Verlag Leipzig-Jena-Berlin, 2. Auflage 1982.
Helmut Kracke, Mathe-musische Knobelisken, Duemmler Bonn, 2. Auflage 1983.

David A. Corneth, Table of n, a(n) for n = 1..10000

Cf. A049363, A050278, A050288, A171132.

sepan[n_]:=Module[{p=Prime[n],k=1},While[Min[DigitCount[p^k]]==0,k++];k]; Array[sepan,100] (* Harvey P. Dale, Aug 03 2019 *)

a(n) = {my(k=1, p=prime(n)); while(#Set(digits(p^k))<10, k++); k; } \\ Jinyuan Wang, Aug 14 2020

Edited by Charles R Greathouse IV, Aug 02 2010

Corrected and extended by Harvey P. Dale, Aug 03 2019

A277055 Irregular array by rows: A(n,m) is the least number which gives a pandigital product when multiplied by the m-th repunit in base n; each row is truncated when it reaches its stationary point.

Original entry on oeis.org

2, 11, 8, 5, 75, 15, 7, 694, 119, 34, 8345, 1505, 195, 123717, 105803, 2217, 2134, 727, 2177399, 241934, 37303, 3724, 44317196, 4431858, 487068, 54771, 9124, 1023456789, 93125079, 9222117, 1110789, 92115, 26432593615

Offset: 2

Andrey Zabolotskiy and Altug Alkan, Sep 26 2016

For row n, the initial number is A049363(n) and the trailing number is A277056(n). Row 10 is A277054(1)-A277054(5) [note that the initial row is row 2].

			The first rows of the array are:
2, (2, 2...)
11, 8, 5, (5, 5...)
75, 15, 7, (7, 7...)
694, 119, 34,
8345, 1505, 195,
123717, 105803, 2217, 2134, 727,
2177399, 241934, 37303, 3724,
44317196, 4431858, 487068, 54771, 9124,
1023456789, 93125079, 9222117, 1110789, 92115

Cf. A002275, A171102, A049363, A277054, A277056, A277058.

cp's OEIS Frontend

A023811 Largest metadrome (number with digits in strict ascending order) in base n.

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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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A061845 Numbers that have one of every digit in some base.

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A258103 Number of pandigital squares (containing each digit exactly once) in base n.

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A260182 Smallest square that is pandigital in base n.

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A260117 Smallest triangular number that is pandigital in base n.

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A037968 Greatest number of distinct digits of n in any base b>=2.

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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.