A049363 -id:A049363 - cp's OEIS frontend

A318725 a(n) is the smallest number k such that k! is pandigital in base n.

2, 8, 5, 11, 17, 14, 15, 23, 23, 24, 36, 30, 35, 46, 50, 43, 50, 40, 59, 62, 54, 69, 75, 70, 65, 79, 83, 68, 97, 99, 86, 89, 93, 97, 118, 94, 106, 126, 128, 116, 145, 127, 134, 151, 143, 124, 141, 124, 141, 170, 194, 169, 190, 183, 181, 180, 195, 195, 210, 163

Offset: 2

Jon E. Schoenfield, Sep 02 2018

			a(2) = 2! = 2_10 = 10_2;
a(3) = 8! = 40320_10 = 2001022100_3;
a(4) = 5! = 120_10 = 1320_4.
a(5) = 11! = 39916800_10 = 4020431420_5;
a(6) = 17! = 355687428096000_10 = 3300252314304000000_6.

Chai Wah Wu, Table of n, a(n) for n = 2..1348 (terms 2..1000 from Seiichi Manyama)

Cf. A049363 (smallest pandigital number in base n), A185122 (smallest pandigital prime in base n), A260182 (smallest square that is pandigital in base n), A260117 (smallest triangular number that is pandigital in base n).

a(n) = {my(k=1); while (#Set(digits(k!, n)) != n, k++); k;} \\ Michel Marcus, Sep 02 2018

from itertools import count
from sympy.ntheory import digits
def A318725(n):
    c, flag = 1, False
    for k in count(1):
        m = k
        if flag:
            a, b = divmod(m,n)
            while not b:
                m = a
                a, b = divmod(m,n)
        c *= m
        if len(set(digits(c,n)[1:]))==n:
            return k
        if not (flag or c%n):
            flag = True # Chai Wah Wu, Mar 13 2024

A318779 Smallest n-th power that is pandigital in base n.

Original entry on oeis.org

4, 64, 625, 248832, 11390625, 170859375, 1406408618241, 3299763591802133, 3656158440062976, 550329031716248441, 766217865410400390625, 15791096563156692195651, 6193386212891813387462761, 243008175525757569678159896851, 3433683820292512484657849089281

Offset: 2

Jon E. Schoenfield, Sep 03 2018

For the corresponding n-th roots a(n)^(1/n), see A318780.

			a(2)=4 because 1^2 = 1 = 1_2 (not pandigital in base 2, since it contains no 0 digit), but 2^2 = 4 = 100_2.
a(3)=64 because 1^3 = 1 = 1_3, 2^3 = 8 = 22_3, and 3^3 = 27 = 1000_3 are all nonpandigital in base 3, but 4^3 = 64 = 2101_3.
a(16) = 81^16 = 3433683820292512484657849089281 = 2b56d4af8f7932278c797ebd01_16.

Jon E. Schoenfield, Table of n, a(n) for n = 2..165

Cf. A049363 (smallest pandigital number in base n), A185122 (smallest pandigital prime in base n), A260182 (smallest square that is pandigital in base n), A260117 (smallest triangular number that is pandigital in base n), A318725 (smallest k such that k! is pandigital in base n), A318780 (smallest k such that k^n is pandigital in base n).

from itertools import count
from sympy import integer_nthroot
from sympy.ntheory import digits
def A318779(n): return next(k for k in (k**n for k in count(integer_nthroot((n**n-n)//(n-1)**2+n**(n-2)*(n-1)-1,n)[0])) if len(set(digits(k,n)[1:]))==n) # Chai Wah Wu, Mar 13 2024

a(n) = A318780(n)^n.

A318780 a(n) is the smallest positive integer k such that k^n is pandigital in base n.

Original entry on oeis.org

2, 4, 5, 12, 15, 15, 33, 53, 36, 41, 55, 51, 59, 91, 81, 60, 131, 167, 173, 312, 213, 394, 309, 222, 356, 868, 351, 704, 526, 1190, 1314, 847, 1435, 1148, 1755, 1499, 1797, 1455, 2311, 1863, 1838, 2120, 2859, 3219, 3463, 2833, 1723, 3009, 3497, 5886, 3746

Offset: 2

Jon E. Schoenfield, Sep 03 2018

For the corresponding values of k^n, see A318779.

			a(2)=2 because 1^2 = 1 = 1_2 (not pandigital in base 2, since it contains no 0 digit), but 2^2 = 4 = 100_2.
a(3)=4 because 1^3 = 1 = 1_3, 2^3 = 8 = 22_3, and 3^3 = 27 = 1000_3 are all nonpandigital in base 3, but 4^3 = 64 = 2101_3.
a(16)=81: 81^16 = 3433683820292512484657849089281 = 2b56d4af8f7932278c797ebd01_16.

Chai Wah Wu, Table of n, a(n) for n = 2..229 (terms 2..165 from Jon E. Schoenfield)

Cf. A049363 (smallest pandigital number in base n), A185122 (smallest pandigital prime in base n), A260182 (smallest square that is pandigital in base n), A260117 (smallest triangular number that is pandigital in base n), A318725 (smallest k such that k! is pandigital in base n), A318779 (smallest n-th power that is pandigital in base n).

from itertools import count
from sympy import integer_nthroot
from sympy.ntheory import digits
def A318780(n): return next(k for k in count(integer_nthroot((n**n-n)//(n-1)**2+n**(n-2)*(n-1)-1,n)[0]) if len(set(digits(k**n,n)[1:]))==n) # Chai Wah Wu, Mar 13 2024

a(n) = A318779(n)^(1/n).

A350510 Square array read by descending antidiagonals: A(n,k) is the least number m such that the base-n expansion of m contains the base-n expansions of 1..k as substrings.

Original entry on oeis.org

1, 2, 1, 6, 5, 1, 12, 11, 6, 1, 44, 38, 27, 7, 1, 44, 95, 75, 38, 8, 1, 92, 285, 331, 194, 51, 9, 1, 184, 933, 1115, 694, 310, 66, 10, 1, 1208, 2805, 4455, 3819, 1865, 466, 83, 11, 1, 1256, 7179, 17799, 16444, 8345, 3267, 668, 102, 12, 1

Offset: 2

Davis Smith, Jan 02 2022

			Square array begins:
n/k|| 1 |  2 |   3 |    4 |     5 |      6 |       7 |        8 |
================================================================|
2  || 1 |  2 |   6 |   12 |    44 |     44 |      92 |      184 |
3  || 1 |  5 |  11 |   38 |    95 |    285 |     933 |     2805 |
4  || 1 |  6 |  27 |   75 |   331 |   1115 |    4455 |    17799 |
5  || 1 |  7 |  38 |  194 |   694 |   3819 |   16444 |    82169 |
6  || 1 |  8 |  51 |  310 |  1865 |   8345 |   55001 |   289577 |
7  || 1 |  9 |  66 |  466 |  3267 |  22875 |  123717 |   947260 |
8  || 1 | 10 |  83 |  668 |  5349 |  42798 |  342391 |  2177399 |
9  || 1 | 11 | 102 |  922 |  8303 |  74733 |  672604 |  6053444 |
10 || 1 | 12 | 123 | 1234 | 12345 | 123456 | 1234567 | 12345678 |
11 || 1 | 13 | 146 | 1610 | 17715 | 194871 | 2143588 | 23579476 |

Davis Smith, Upper bounds for A350510(n, k) and various conjectured patterns.

Cf. A023811, A035239, A049363, A056744, A057544, A061845, A102305.

The first n - 1 terms of rows: 2: A047778, 3: A048435, 4: A048436, 5: A048437, 6: A048438, 7: A048439, 8: A048440, 9: A048441, 10: A007908, 11: A048442, 12: A048443, 13: A048444, 14: A048445, 15: A048446, 16: A048447.

T[n_,k_]:=(m=0;While[!ContainsAll[Subsequences@IntegerDigits[++m,n],IntegerDigits[Range@k,n]]];m);Flatten@Table[T[1+i,j+1-i],{j,9},{i,j}] (* Giorgos Kalogeropoulos, Jan 09 2022 *)

A350510_rows(n,k,N=0)= my(L=List(concat(apply(z->fromdigits([1..z],n),[1..n-1]),if(n>2,fromdigits(concat([1,0],[2..n-1]),n),[]))),T1(x)=digits(x,n),T2(x)=fromdigits(x,n),A(x)=my(S=T1(x));setbinop((y,z)->T2(S[y..z]),[1..#S]),N=if(N,N,L[#L]),A1=A(N));while(#Lsetsearch(A1,z),[1..#L+1])),A1=A(N++));listput(L,N));Vec(L)

For k < n, A(n,k) = A(n,k - 1)*n + k = Sum_{i=1..k} i*(n^(k - i)).
A(n,n) = A049363(n).
A(n,2) = A057544(n).
For n > 3, A(n,3) = A102305(n).
A(n,n - 1) = A023811(n).

A123668 Smallest pandigital palindrome in base n, with a(1) = 1.

Original entry on oeis.org

1, 5, 100, 4833, 434176, 64896625, 14555276100, 4566338422401, 1907710008707584, 1023456789876543201, 685593403921020830500, 560806213771094855054689, 550049712286417194431060352

Offset: 1

Franklin T. Adams-Watters, Nov 15 2006

G. C. Greubel, Table of n, a(n) for n = 1..214

Cf. A049363, A068792.

Join[{1, 5}, Table[1 + n^(2 n - 2) + (n - 1) n^(n - 1) + Sum[ i*(n^(2 n - 2 - i) + n^i), {i, 2, n - 2}], {n, 3, 50}]] (* G. C. Greubel, Oct 26 2017 *)

for(n=1,50, print1(if(n==1,1, if(n==2,5, 1 + n^(2*n - 2) + (n - 1)* n^(n - 1) + sum(i=2,n-2, i*(n^(2*n - 2 - i) + n^i)))), ", ")) \\ G. C. Greubel, Oct 26 2017

def A123668(n): return n*((n**(n-1)-1)//(n-1))**2 + (n-1)*(n**(2*n-3)-1) if n>2 else 4*n-3 # Chai Wah Wu, Mar 18 2024

For n>2, a(n) = n*A068792(n) + (n-1)(n^(2n-3) - 1).

For n>2, a(n) = 1 + n^(2n-2) + (n-1)n^(n-1) + Sum_{i=2..(n-2)} i*(n^(2n-2-i)+n^i).

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

A217535 Least number having in its decimal representation each digit n times.

Original entry on oeis.org

1023456789, 10012233445566778899, 100011222333444555666777888999, 1000011122223333444455556666777788889999, 10000011112222233333444445555566666777778888899999, 100000011111222222333333444444555555666666777777888888999999

Offset: 1

M. F. Hasler, Oct 05 2012

Alois P. Heinz, Table of n, a(n) for n = 1..100

Cf. A050278, A051018, A049363, A074205, A154566, A204047, A020666, A020667, A061604, A180489.

A subsequence of A171102.

a:= n-> parse(cat(1,0$n,1$(n-1),seq(i$n, i=2..9))):
seq(a(n), n=1..10);  # Alois P. Heinz, Jun 25 2017

A217535(n)=sum(d=1,9,10^(n-(d==1))\9*d*10^(n*(9-d)))+10^(10*n-1)

a(n) ~ 10^(10n-1). See PARI code for an exact formula.

A235708 Largest base in which n is pandigital.

Original entry on oeis.org

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

Offset: 1

Joonas Pohjonen, May 02 2014

The first occurrence of k in this sequence is given by A049363(k).

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

Cf. A049363.

import Data.List (unfoldr, nub); import Data.Tuple (swap)
a235708 n = f n where
   f 1 = 1
   f b = if isPandigital b n then b else f (b - 1) where
         isPandigital b = (== b) . length . nub . unfoldr
           (\x -> if x == 0 then Nothing else Just $ swap $ divMod x b)
-- Reinhard Zumkeller, May 12 2014

A239437 Least number that is pandigital in some base >= n but not pandigital in bases 3 through n-1.

Original entry on oeis.org

11, 78, 698, 12280, 179685, 5518135, 1037845296, 1037845296, 46935813565, 2860727439460, 285947759601954, 1018897102759406, 672654273047783383

Offset: 3

Charles R Greathouse IV, Mar 18 2014

Gives an upper bound on the testing needed to check membership in A239348.

Charles R Greathouse IV, GP script for computing terms

Subsequence of A154314. Cf. A239348.

PARI
```
See Greathouse link.
```

from itertools import permutations
from gmpy2 import digits
def A239437(n): # requires 3 <= n <= 62
    m = n
    while True:
        s = ''.join([digits(i,m) for i in range(m)])
        for d in permutations(s,m):
            if d[0] != '0':
                c = mpz(''.join(d),m)
                for b in range(3,n):
                    if len(set(digits(c,b))) == b:
                        break
                else:
                    return int(c)
        m += 1 # Chai Wah Wu, May 31 2015

Trivially a(n) >= A049363(n) > n^(n-1).

a(15) from Giovanni Resta, Mar 19 2014

cp's OEIS Frontend

A318725 a(n) is the smallest number k such that k! is pandigital in base n.

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A318779 Smallest n-th power that is pandigital in base n.

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A318780 a(n) is the smallest positive integer k such that k^n is pandigital in base n.

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A350510 Square array read by descending antidiagonals: A(n,k) is the least number m such that the base-n expansion of m contains the base-n expansions of 1..k as substrings.

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Mathematica

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A123668 Smallest pandigital palindrome in base n, with a(1) = 1.

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Mathematica

PARI

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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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A217535 Least number having in its decimal representation each digit n times.

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