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%I #109 Apr 04 2024 10:00:52
%S 1023456789,1023456798,1023456879,1023456897,1023456978,1023456987,
%T 1023457689,1023457698,1023457869,1023457896,1023457968,1023457986,
%U 1023458679,1023458697,1023458769,1023458796,1023458967,1023458976,1023459678,1023459687,1023459768
%N Pandigital numbers: numbers containing the digits 0-9. Version 1: each digit appears exactly once.
%C This is a finite sequence with 9*9! = 3265920 terms: a(9*9!) = 9876543210.
%C A171102 is the infinite version, where each digit must appear at least once.
%C More precisely, this is exactly the subset of the first 9*9! terms of A171102. - _M. F. Hasler_, Jan 05 2020
%C Subsequence of A134336 and of A178403; A178401(a(n)) = 1. - _Reinhard Zumkeller_, May 27 2010
%C Smallest prime factors: A178775(n) = A020639(a(n)). - _Reinhard Zumkeller_, Jun 11 2010
%C A178788(a(n)) = 1. - _Reinhard Zumkeller_, Jun 30 2010
%C All these numbers are composite because the sum of the digits, 45, is divisible by 9. - _T. D. Noe_, Nov 09 2011
%C 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
%C From _Hieronymus Fischer_, Feb 13 2013: (Start)
%C The sum of all terms is 9!*49444444440 = 17942399998387200.
%C 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.
%C 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.
%C (End)
%H Robert G. Wilson v, <a href="/A050278/b050278.txt">Table of n, a(n) for n = 1..1000</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PandigitalNumber.html">Pandigital Number</a>
%H Chai Wah Wu, <a href="https://arxiv.org/abs/2403.20304">Pandigital and penholodigital numbers</a>, arXiv:2403.20304 [math.GM], 2024. See p. 1.
%F A050278 = 9*A171571. - _M. F. Hasler_, Jan 12 2012
%F A050278(n) = A171102(n) for n <= 9*9!.
%t Select[ FromDigits@# & /@ Permutations[ Range[0, 9]], # > 10^9 &, 20] (* _Robert G. Wilson v_, May 30 2010, Jan 17 2012 *)
%o (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
%o (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 */
%o (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
%o (Python)
%o from itertools import permutations
%o A050278_list = [int(''.join(d)) for d in permutations('0123456789',10) if d[0] != '0'] # _Chai Wah Wu_, May 25 2015
%Y Cf. A171102, A050288, A050289.
%Y Cf. A199630, A199631, A114260, A199632, A199633.
%Y Cf. A031969, A050278, A220063, A220076, A218019, A219743.
%K nonn,base,fini
%O 1,1
%A _Eric W. Weisstein_, Dec 11 1999
%E Edited by _N. J. A. Sloane_, Sep 25 2010 to clarify that this is a finite sequence
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