A030299 -id:A030299 - cp's OEIS frontend

A178477 Permutations of 1234567: Numbers having each of the decimal digits 1,...,7 exactly once, and no other digit.

1234567, 1234576, 1234657, 1234675, 1234756, 1234765, 1235467, 1235476, 1235647, 1235674, 1235746, 1235764, 1236457, 1236475, 1236547, 1236574, 1236745, 1236754, 1237456, 1237465, 1237546, 1237564, 1237645, 1237654

Offset: 1

M. F. Hasler, Oct 09 2010

It would be nice to have a simple explicit formula for the n-th term.
Contains A000142(7) = 5040 terms. - R. J. Mathar, Apr 08 2011
An efficient procedure for generating the n-th term of this sequence can be found at A178475. - Nathaniel Johnston, May 19 2011

Nathaniel Johnston, Table of n, a(n) for n = 1..5040 (full sequence)

Cf. A030298, A030299, A055089, A060117, A178475, A178476.

FromDigits/@Take[Permutations[Range[7]],50] (* Harvey P. Dale, Nov 11 2012 *)

is_A178477(x)= { vecsort(Vec(Str(x)))==Vec("1234567") }

A178478 Permutations of 12345678: Numbers having each of the decimal digits 1..8 exactly once, and no other digit.

Original entry on oeis.org

12345678, 12345687, 12345768, 12345786, 12345867, 12345876, 12346578, 12346587, 12346758, 12346785, 12346857, 12346875, 12347568, 12347586, 12347658, 12347685, 12347856, 12347865, 12348567, 12348576, 12348657, 12348675, 12348756, 12348765

Offset: 1

M. F. Hasler, Oct 09 2010

It would be nice to have a simple explicit formula for the n-th term.

An efficient procedure for generating the n-th term of this sequence can be found at A178475. - Nathaniel Johnston, May 19 2011

Nathaniel Johnston, Table of n, a(n) for n = 1..10000

Cf. A030298, A030299, A055089, A060117, A178475-A178477, A191820.

Take[FromDigits/@Permutations[Range[8]],40] (* Harvey P. Dale, Oct 29 2014 *)

is_A178478(x)= { vecsort(Vec(Str(x)))==Vec("12345678") }

A178478(n)={my(b=vector(7,k,1+(n-1)%(k+1)!\k!),t=b[7], d=vector(7,i,i+(i>=t)));for(i=1,6,t=10*t+d[b[7-i]]; d=vecextract(d,Str("^"b[7-i]))); t*10+d[1]} \\ - M. F. Hasler (following N. Johnston's comment), Jan 10 2012

A220696 The positions of those permutations in A030298 where the first element is one (fixed).

Original entry on oeis.org

1, 2, 4, 5, 10, 11, 12, 13, 14, 15, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 154, 155, 156, 157, 158, 159, 160, 161, 162, 163, 164, 165, 166, 167, 168, 169, 170, 171, 172, 173, 174, 175, 176, 177, 178, 179

Offset: 1

Antti Karttunen, Dec 17 2012

nonn

Correspondingly gives the positions of those terms in A030299 whose first digit is 1, as long as the decimal encoding system employed is valid.

A. Karttunen, Table of n, a(n) for n = 1..5914

Complement: A220656. Cf. A072795.

a(1)=1; and for n>1, a(n)=A220695(n-1)+1.

A305701 Nonnegative integers whose decimal digits span an initial interval of {0,...,9}.

Original entry on oeis.org

0, 10, 100, 101, 102, 110, 120, 201, 210, 1000, 1001, 1002, 1010, 1011, 1012, 1020, 1021, 1022, 1023, 1032, 1100, 1101, 1102, 1110, 1120, 1200, 1201, 1202, 1203, 1210, 1220, 1230, 1302, 1320, 2001, 2010, 2011, 2012, 2013, 2021, 2031, 2100, 2101, 2102, 2103

Offset: 1

Gus Wiseman, Jun 08 2018

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000670, A030299, A050289, A144688, A240763, A305712, A305714, A305715.

filter:= proc(n) local L;
  L:= convert(convert(n,base,10),set);
  L = {$0..max(L)}
end proc:
select(filter, [$0..3000]); # Robert Israel, Jun 10 2018

Select[Range[0,10000],Union[IntegerDigits[#]]==Range[0,Max[IntegerDigits[#]]]&]

isok(n) = if (n==0, return (1)); my(d=Set(digits(n))); (vecmin(d) == 0) && (vecmax(d) == #d - 1); \\ Michel Marcus, Jul 05 2018

A305714 Number of finite sequences of positive integers of length n that are polydivisible and strictly pandigital.

Original entry on oeis.org

1, 1, 1, 2, 0, 0, 2, 0, 1, 1, 1

Offset: 0

Gus Wiseman, Jun 08 2018

A sequence q of length k is strictly pandigital if it is a permutation of {1,2,...,k}. It is polydivisible if Sum_{i = 1...m} 10^(m - i) * q_i is a multiple of m for all 1 <= m <= k.

			Sequence of sets of n-digit numbers that are weakly polydivisible and strictly pandigital is (with A = 10):
  {0}
  {1}
  {12}
  {123,321}
  {}
  {}
  {123654,321654}
  {}
  {38165472}
  {381654729}
  {381654729A}

Cf. A000670, A010784, A030299, A050289, A143671, A144688, A156069, A156071, A158242, A163574, A240763, A305701, A305712, A305715.

A181073 Permutations of lengths 1,2,3,... arranged lexicographically that are relatively prime to 30.

Original entry on oeis.org

1, 1243, 1423, 2143, 2341, 2413, 2431, 3241, 3421, 4123, 4213, 4231, 4321, 1234567, 1234657, 1235467, 1235647, 1236457, 1236547, 1243567, 1243657, 1245367, 1245637, 1245673, 1245763, 1246357, 1246537, 1246573, 1246753

Offset: 1

Marco Ripà, Jan 23 2011

Numbers whose digits are permutations of (1,2,3,4,...,n) for some n >= 1, for which the first 3 primes (2,3,5) do not appear in their factorization. This constitutes the smallest subset of A030299 for which it's possible to synthesize a compact formula to express the generic term: it contains every prime number already in A030299.

In fact it corresponds to the subset of the terms of A030299 constructed from the concatenation of k:=1+3*i (for i >= 0) elements belonging to (1,2,3,4,...,n) that are congruent in modulo 10 to (1,3,7,9).

Marco Ripà, Divisibilità per 3 degli elementi di particolari sequenze numeriche
Marco Ripà, On prime factors in old and new sequences of integers

Cf. A030299, A030298.

f[n_] := Select[ FromDigits@ # & /@ Permutations@ Range@ n, Mod[#, 2] != 0 && Mod[#, 3] != 0 && Mod[#, 5] != 0 &]; Take[ Flatten@ Array[f, 7], 35]

A305712 Polydivisible nonnegative integers whose decimal digits span an initial interval of {0,...,9}.

Original entry on oeis.org

0, 10, 102, 120, 201, 1020, 1200, 2012, 10200, 12000, 12320, 20120, 32120, 102000, 120000, 123204, 321204, 1024023, 1200003, 1232042, 1444023, 2220001, 3212041, 10240232, 12000032, 12320424, 14440232, 32125240, 50165432

Offset: 0

Gus Wiseman, Jun 08 2018

A number with decimal digit sequence {q_1, ..., q_k} is polydivisible if Sum_{i = 1...m} 10^(m - i) * q_i is a multiple of m for all 1 <= m <= k.

Matt Parker, Things to make and do in the fourth dimension, 2015, pages 7-9.

Wikipedia, Polydivisible number

Cf. A000670, A030299, A050289, A144688, A156069, A156071, A240763, A305701, A305714, A305715.

polyQ[q_]:=And@@Table[Divisible[FromDigits[Take[q,k]],k],{k,Length[q]}];
normseqs[n_]:=Join@@Permutations/@Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1];
Sort[FromDigits/@Join@@Table[Select[normseqs[n]-1,First[#]>0&&polyQ[#]&],{n,8}]]

A305715 Irregular triangle whose rows are all finite sequences of positive integers that are polydivisible and strictly pandigital.

Original entry on oeis.org

1, 1, 2, 1, 2, 3, 3, 2, 1, 1, 2, 3, 6, 5, 4, 3, 2, 1, 6, 5, 4, 3, 8, 1, 6, 5, 4, 7, 2, 3, 8, 1, 6, 5, 4, 7, 2, 9, 3, 8, 1, 6, 5, 4, 7, 2, 9, 10

Offset: 1

Gus Wiseman, Jun 08 2018

A positive integer sequence q of length k is strictly pandigital if it is a permutation of {1,2,...,k}. It is polydivisible if Sum_{i = 1...m} 10^(m - i) * q_i is a multiple of m for all 1 <= m <= k.

			Triangle is:
  {1}
  {1,2}
  {1,2,3}
  {3,2,1}
  {1,2,3,6,5,4}
  {3,2,1,6,5,4}
  {3,8,1,6,5,4,7,2}
  {3,8,1,6,5,4,7,2,9}
  {3,8,1,6,5,4,7,2,9,10}

Matt Parker, Things to make and do in the fourth dimension, 2015, pages 7-9.

Wikipedia, Polydivisible number

Cf. A000670, A010784, A030299, A050289, A143671, A144688, A156069, A156071, A158242, A163574, A240763, A305701, A305712, A305714 (row lengths).

polyQ[q_]:=And@@Table[Divisible[FromDigits[Take[q,k]],k],{k,Length[q]}];
Flatten[Table[Select[Permutations[Range[n]],polyQ],{n,8}]]

A352991 Concatenation of all the distinct permutations of the first 1, 2, 3, ... (strictly) positive integers, arranged in ascending numerical order.

Original entry on oeis.org

1, 12, 21, 123, 132, 213, 231, 312, 321, 1234, 1243, 1324, 1342, 1423, 1432, 2134, 2143, 2314, 2341, 2413, 2431, 3124, 3142, 3214, 3241, 3412, 3421, 4123, 4132, 4213, 4231, 4312, 4321, 12345, 12354, 12435, 12453, 12534, 12543, 13245, 13254, 13425

Offset: 1

Marco Ripà, Apr 16 2022

This sequence differs from A030299 starting from a(409114) = 10123456789. All the permutations are listed once and only once (e.g., the concatenation of the permutations of the elements of the set {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11} originates the number 1112345678910 which is a unique element of this sequence and appears only once, since 1_11_23456789 = 11_1_23456789 = 1112345678910).
A001292 is a subsequence of the present sequence. An open problem, published by Kenichiro Kashihara in 1996 (see References, p. 25, #30, Problem 2), is to find how many terms of A001292 (which is a subsequence of A030299) are powers of integers; Kashihara conjectured that there are none (even if, clearly, A001292(1) = 1 should be disregarded in order to keep the conjecture alive). Currently, only the terms up to the prime a(409120) = 10123457689 have been directly checked by the author of this sequence and no nontrivial perfect power has been found. On the other hand, many (maybe infinitely many) terms of the present sequence are nontrivial powers of integers (e.g., A352329(2) to A352329(36) are squares of integers and belong to this sequence).
Although A181129 is a subsequence of the present one, so that A181129(1) = a(19) = 2341, a(14) is the smallest prime in this sequence.
The number of digits of a(n) comes from A058183. There are exactly k! (Cf. A000142) terms having A058183(k) digits. - David A. Corneth, Apr 17 2022

			a(3) = 21, since the number of permutations of {1, 2} is 2! = 2 and the concatenation 1_2 is smaller than 2_1 (while {1} originates only a(1) = 1, so that a(2) = 21).

Kenichiro Kashihara, Comments and Topics on Smarandache Notions and Problems, 25. Erhus University Press, Arizona, 1996. ISBN: 1-879585-55-3.

Index entries for sequences which agree for a long time but are different

Cf. A000142, A001292, A058183, A030299, A181129.

from itertools import count, islice, permutations
def agen(): # generator of terms
    for k in count(1):
        s = (int("".join(map(str, p))) for p in permutations(range(1, k+1)))
        yield from sorted(set(s))
print(list(islice(agen(), 42))) # Michael S. Branicky, Apr 16 2022

A199168 Numbers whose digits are a permutation of (0,...,m) for some m.

Original entry on oeis.org

0, 10, 102, 120, 201, 210, 1023, 1032, 1203, 1230, 1302, 1320, 2013, 2031, 2103, 2130, 2301, 2310, 3012, 3021, 3102, 3120, 3201, 3210, 10234, 10243, 10324, 10342, 10423, 10432, 12034, 12043, 12304, 12340, 12403, 12430, 13024, 13042, 13204, 13240, 13402, 13420

Offset: 1

M. F. Hasler, Jan 08 2013

2013 is the fourth odd term in this sequence: Up to and including the 5 digit terms, odd terms must end in 1 or 3.

Due to the fact that 0 is not allowed as initial digit, this sequence is quite different from A030299, the analog with digits (1,...,m) instead of (0,...,m).

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A187796 (subset of primes), A203569 (also a subset), A030299 (permutations of 1..m) and references therein.
Pandigital numbers A050278 are also a subset.
Cf. A297062.

f:= proc(n) map(L -> add(L[i]*10^(n-i),i=1..n), select(L -> L[1] <> 0, combinat:-permute([$0..n-1]))) end proc:
f(1):= [0]:
seq(op(f(n)),n=1..5); # Robert Israel, Jan 09 2025

n_digit_terms(n)={my(a=[],p=vector(n,i,10^(n-i))~);for(i=(n-1)!,n!-(n>1),a=concat(a,numtoperm(n,i)%n*p));vecsort(a)} \\ - M. F. Hasler, Jan 08 2013

cp's OEIS Frontend

A178477 Permutations of 1234567: Numbers having each of the decimal digits 1,...,7 exactly once, and no other digit.

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A178478 Permutations of 12345678: Numbers having each of the decimal digits 1..8 exactly once, and no other digit.

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A220696 The positions of those permutations in A030298 where the first element is one (fixed).

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A305701 Nonnegative integers whose decimal digits span an initial interval of {0,...,9}.

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A305714 Number of finite sequences of positive integers of length n that are polydivisible and strictly pandigital.

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A181073 Permutations of lengths 1,2,3,... arranged lexicographically that are relatively prime to 30.

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Mathematica

A305712 Polydivisible nonnegative integers whose decimal digits span an initial interval of {0,...,9}.

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Mathematica

A305715 Irregular triangle whose rows are all finite sequences of positive integers that are polydivisible and strictly pandigital.

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Mathematica

A352991 Concatenation of all the distinct permutations of the first 1, 2, 3, ... (strictly) positive integers, arranged in ascending numerical order.

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