cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A050278 Pandigital numbers: numbers containing the digits 0-9. Version 1: each digit appears exactly once.

Original entry on oeis.org

1023456789, 1023456798, 1023456879, 1023456897, 1023456978, 1023456987, 1023457689, 1023457698, 1023457869, 1023457896, 1023457968, 1023457986, 1023458679, 1023458697, 1023458769, 1023458796, 1023458967, 1023458976, 1023459678, 1023459687, 1023459768
Offset: 1

Views

Author

Eric W. Weisstein, Dec 11 1999

Keywords

Comments

This is a finite sequence with 9*9! = 3265920 terms: a(9*9!) = 9876543210.
A171102 is the infinite version, where each digit must appear at least once.
More precisely, this is exactly the subset of the first 9*9! terms of A171102. - M. F. Hasler, Jan 05 2020
Subsequence of A134336 and of A178403; A178401(a(n)) = 1. - Reinhard Zumkeller, May 27 2010
Smallest prime factors: A178775(n) = A020639(a(n)). - Reinhard Zumkeller, Jun 11 2010
A178788(a(n)) = 1. - Reinhard Zumkeller, Jun 30 2010
All these numbers are composite because the sum of the digits, 45, is divisible by 9. - T. D. Noe, Nov 09 2011
This is the 10th row of the array T(k,n) = n-th number in which the number of distinct base-10 digits is k. A031969 is the 4th row. A220063 is the 5th row. A220076 is the 6th row. A218019 is the 7th row. A219743 is the 8th row. - Jonathan Vos Post, Dec 05 2012
From Hieronymus Fischer, Feb 13 2013: (Start)
The sum of all terms is 9!*49444444440 = 17942399998387200.
General formula for the sum of all terms of the finite sequence of the corresponding base-p pandigital numbers with p places: sum = ((p^2 - p - 1)*(p^p - 1) + p - 1)*(p-2)!/2.
General formula for the sum of all terms (interpreted as decimal permutational numbers with exactly d+1 different digits from the range 0..d < 10): sum = (d+1)!*((10d - 1)*10^d - d + 1)/18, d > 1.
(End)

Links

Crossrefs

Cf. A171102, A050288, A050289.
Cf. A199630, A199631, A114260, A199632, A199633.
Cf. A031969, A050278, A220063, A220076, A218019, A219743.

Programs

  • Mathematica
    Select[ FromDigits@# & /@ Permutations[ Range[0, 9]], # > 10^9 &, 20] (* Robert G. Wilson v, May 30 2010, Jan 17 2012 *)
  • PARI
    A050278(n)={ my(b=vector(9,k,1+(n+9!-1)%(k+1)!\k!), t=b[9]-1, d=vector(9,i,i+(i>t)-1)); for(i=1,8, t=10*t+d[b[9-i]]; d=vecextract(d,Str("^"b[9-i]))); t*10+d[1]} \\ M. F. Hasler, Jan 15 2012
  • PARI
    is_A050278(n)={ 9<#vecsort(Vecsmall(Str(n)),,8) & n<1e10 } /* assuming that n is a nonnegative integer */ /* M. F. Hasler, Jan 10 2012 */
  • PARI
    a(n)=my(d=numtoperm(10,n+9!-1));sum(i=1,#d,(d[i]-1)*10^(#d-i)) \\ David A. Corneth, Jun 01 2014
  • Python
    from itertools import permutations
A050278_list = [int(''.join(d)) for d in permutations('0123456789',10) if d[0] != '0'] # Chai Wah Wu, May 25 2015

Formula

A050278 = 9*A171571. - M. F. Hasler, Jan 12 2012
A050278(n) = A171102(n) for n <= 9*9!.

Extensions

Edited by N. J. A. Sloane, Sep 25 2010 to clarify that this is a finite sequence

A219743 Number for which the number of distinct base 10 digits is 8.

Original entry on oeis.org

10234567, 10234568, 10234569, 10234576, 10234578, 10234579, 10234586, 10234587, 10234589, 10234596, 10234597, 10234598, 10234657, 10234658, 10234659, 10234675, 10234678, 10234679, 10234685, 10234687, 10234689, 10234695, 10234697, 10234698
Offset: 1

Views

Author

Jonathan Vos Post, Dec 05 2012

Keywords

Links

Crossrefs

Cf. A010785 (1 digits), A031955 (2 digits), A031962 (3 digits), A031969 (4 digits), A031987 (5 digits), A220076 (6 digits), A218019 (7 digits), A116670 (9 digits), A171102 (10 digits).

Programs

  • Mathematica
    Select[Range[10^7, 10^7 + 1000000], Length[Union[IntegerDigits[#]]] == 8 &] (* T. D. Noe, Dec 05 2012 *)

Extensions

Corrected and extended by T. D. Noe, Dec 05 2012

A337127 Table with 10 columns read by rows: T(n, k) is the number of n-digit positive integers with exactly k distinct base 10 digits (0 < k <= 10).

Original entry on oeis.org

9, 0, 0, 0, 0, 0, 0, 0, 0, 0, 9, 81, 0, 0, 0, 0, 0, 0, 0, 0, 9, 243, 648, 0, 0, 0, 0, 0, 0, 0, 9, 567, 3888, 4536, 0, 0, 0, 0, 0, 0, 9, 1215, 16200, 45360, 27216, 0, 0, 0, 0, 0, 9, 2511, 58320, 294840, 408240, 136080, 0, 0, 0, 0, 9, 5103, 195048, 1587600, 3810240, 2857680, 544320, 0, 0, 0
Offset: 1

Views

Author

Stefano Spezia, Aug 17 2020

Keywords

Examples

			The table T(n, k) begins:
9     0      0       0       0       0  0  0  0  0
9    81      0       0       0       0  0  0  0  0
9   243    648       0       0       0  0  0  0  0
9   567   3888    4536       0       0  0  0  0  0
9  1215  16200   45360   27216       0  0  0  0  0
9  2511  58320  294840  408240  136080  0  0  0  0
...

Links

Crossrefs

Cf. A010785, A031955, A031962, A031969, A031987, A116670, A171102, A218019, A219743, A220076.
Cf. A010734, A048993, A052268 (row sums), A073531 (diagonal), A180599 (k = 1), A335843 (k = 2), A337313 (k = 3).

Programs

  • Mathematica
    T[n_,k_]:=9Pochhammer[11-k,k-1]/k!*n!*Coefficient[Series[(Exp[x]-1)^k,{x,0,n}],x,n]; Table[T[n,k],{n,7},{k,10}]//Flatten

Formula

T(n, k) = 9*Pochhammer(11-k, k-1)*n! * [x^n] (exp(x) - 1)^k/k!.
T(n, k) = 9*Pochhammer(11-k, k-1) * [x^n] x^k/Product_{j=1..k} (1-j*x).
T(n, k) = 9*Pochhammer(11-k, k-1)*S2(n, k) where S2(n, k) = A048993(n, k) are the Stirling numbers of the 2nd kind.

A220111 Main diagonal of array T(k,n) = n-th number in which the number of distinct base 10 digits is k.

Original entry on oeis.org

0, 12, 104, 1026, 10238, 102354, 1023468, 10234587, 1023457869
Offset: 1

Views

Author

Jonathan Vos Post, Dec 05 2012

Keywords

Examples

			T(1,1) = 0, which is first number in which the number of distinct base 10 digits is 1. That row has only one other element, 1.
T(2,2) = 12, which is first number not beginning with a zero in which the number of distinct base 10 digits is 2. The row begins: 10, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 23,....
0,1,2,3,4,5,6,...
10,12,13,14,15,16,17,...
102,103,104,105,106,107,108,...
1023,1024,1025,1026,1027,1028,1029,...
10234,10235,10236,10237,10238,10239,10243,...
102345,102346,102347,102348,102349,102354,102356,...
1023456,1023457,1023458,1023459,1023465,1023467,1023468,...

Crossrefs

Cf. A043537, A031969 (case 4), A220076 (case 6), A218019 (case 7).
Cf. A219743 (case 8), A050278 (pandigital numbers).
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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