A178475 -id:A178475 - cp's OEIS frontend

A178485 (A178475(n)-6)/9.

1371, 1372, 1381, 1383, 1392, 1393, 1471, 1472, 1491, 1494, 1502, 1504, 1581, 1583, 1591, 1594, 1613, 1614, 1692, 1693, 1702, 1704, 1713, 1714, 2371, 2372, 2381, 2383, 2392, 2393, 2571, 2572, 2601, 2605, 2612, 2615, 2681, 2683, 2701, 2705, 2723, 2725

Offset: 1

M. F. Hasler, May 28 2010

There are 5!=120 terms in this finite sequence. Its origin is the fact that numbers whose decimal expansion is a permutation of 12345 are all of the form 9k+6.

Nathaniel Johnston, Table of n, a(n) for n = 1..120 (full sequence)

Cf. A030298, A030299, A055089, A060117, A178486, A191819, A191820.

v=vector(5,i,10^(i-1))~; vecsort(vector(5!,i,numtoperm(5,i)*v))
is_A178475(x)= { vecsort(Vec(Str(x)))==Vec("12345") }
forstep( m=12345,54321,9, is_A178475(m) & print1(m","))

a(n) + a(5!+1-n) = 7406.
a(n) == 1, 2, 3, 4 or 5 (mod 10).
a(n+6)-a(n) is an element of { 100, 110, 111, 200, 220, 222, 679 }.
a(n+6)-a(n) = 679 iff (n-1)%24 > 17, where % denotes the remainder upon division.
a(n+6)-a(n) = 200, 220 or 222 iff (n-1)%30 > 23, i.e. n==25,...,30 (mod 30).

A209280 First difference of A050289 = numbers whose digits are a permutation of (1,...,9).

Original entry on oeis.org

9, 81, 18, 81, 9, 702, 9, 171, 27, 72, 18, 693, 18, 72, 27, 171, 9, 702, 9, 81, 18, 81, 9, 5913, 9, 81, 18, 81, 9, 1602, 9, 261, 36, 63, 27, 594, 18, 162, 36, 162, 18, 603, 9, 171, 27, 72, 18, 5814, 9, 171, 27, 72, 18, 603, 9, 261, 36, 63, 27, 1584, 27, 63, 36, 261, 9

Offset: 1

M. F. Hasler, Jan 12 2013

This sequence is the natural extension of A107346 (and others, see below) from 5!-1 to 9!-1 terms, which is the natural (since maximal) length, given that OEIS sequence data are stored as decimal numbers. On the other hand, it is quite different from A219664 in many aspects, not only for the reason that the other sequence is infinite and therefore differs from this one in all terms beyond n = 9!-1.
The sequence is finite, with 9!-1 terms, and symmetric: a(n)=a(9!-n).
All terms are multiples of 9, cf. formula.
The subsequence of the first n!-1 terms (n=2,...,9) yields the first differences of the sequence of numbers whose digits are a permutation of (1,...,n):
The first 8!-1 terms yield the first differences of A178478: numbers whose digits are a permutation of 12345678.
The first 7!-1 terms yield the first differences of A178477: numbers whose digits are a permutation of 1234567.
The first 6!-1 terms yield the first differences of A178476: numbers whose digits are a permutation of 123456.
The first 5!-1 terms yield A107346, the first differences of A178475: numbers whose digits are a permutation of 12345.

			The same initial terms are obtained for the permutations of any set of the form {1,...,m}, e.g., {1,2,3} or {1,...,9}: In the first case we have P = (123,132,213,231,312,321) and P(4)-P(3) = 231 - 213 = 18 = a(3), and in the latter case P(4)-P(3) = 123456897 - 123456879 = 18, again. - _M. F. Hasler_, Jan 12 2013

Cf. A030299, A219664.

Take[Differences[Sort[FromDigits/@Permutations[Range[9]]]],70] (* Harvey P. Dale, Mar 31 2018 *)

A209280_list(N=5)={my(v=vector(N,i,10^(N-i))~); v=vecsort(vector(N!,k,numtoperm(N,k)*v)); vecextract(v,"^1")-vecextract(v,"^-1")} \\ return the N!-1 first terms as a vector

A209280(n)={if(a209280=='a209280 || #a209280A209280_list(A090529(n+1)));a209280[n]}

a(n) = A219664(n) = 9*A217626(n) (for n < 9!). - M. F. Hasler, Jan 12 2013

a(n) = a(m!-n) for any m < 10 such that n < m!.

A178476 Permutations of 123456: Numbers having each of the decimal digits 1,...,6 exactly once, and no other digit.

Original entry on oeis.org

123456, 123465, 123546, 123564, 123645, 123654, 124356, 124365, 124536, 124563, 124635, 124653, 125346, 125364, 125436, 125463, 125634, 125643, 126345, 126354, 126435, 126453, 126534, 126543, 132456, 132465, 132546, 132564, 132645, 132654

Offset: 1

M. F. Hasler, May 28 2010

This finite sequence contains 6!=720 terms.
This is a subsequence of A030299, consisting of elements A030299(154)..A030299(873).
If individual digits are be split up into separate terms, we get a subsequence of A030298.
It would be interesting to conceive simple and/or efficient functions which yield (a) the n-th term of this sequence: f(n)=a(n), (b) for a given term, the subsequent one: f(a(n)) = a(1 + (n mod 6!)).
The expression a(n+6) - a(n) takes only 18 different values for n = 1..6!-6.
An efficient procedure for generating the n-th term of this sequence can be found at A178475. - Nathaniel Johnston, May 19 2011
From Hieronymus Fischer, Feb 13 2013: (Start)
The sum of all terms as decimal numbers is 279999720.
General formula for the sum of all terms (interpreted as decimal permutational numbers with exactly d different digits from the range 1..d < 10): sum = (d+1)!*(10^d-1)/18.
If the terms are interpreted as base-7 numbers the sum is 49412160.
General formula for the sum of all terms of the corresponding sequence of base-p permutational numbers (numbers with exactly p-1 different digits excluding the zero digit): sum = (p-2)!*(p^p-p)/2. (End)

Nathaniel Johnston, Table of n, a(n) for n = 1..720 (full sequence)

Cf. A030298, A030299, A055089, A060117.

Take[FromDigits/@Permutations[Range[6]],40] (* Harvey P. Dale, Jun 05 2012 *)

v=vector(6,i,10^(i-1))~; A178476=vecsort(vector(6!,i,numtoperm(6,i)*v));
is_A178476(x)= { vecsort(Vec(Str(x)))==Vec("123456") }
forstep( m=123456,654321,9, is_A178476(m) & print1(m","))

a(n) + a(6! + 1 - n) = 777777.
floor( a(n) / 10^5 ) = ceiling( n / 5! ).
a(n) = A030299(n+153).
a(n) == 3 (mod 9).
a(n) = 3 + 9*A178486(n).

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

Original entry on oeis.org

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

cp's OEIS Frontend

A178485 (A178475(n)-6)/9.

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A209280 First difference of A050289 = numbers whose digits are a permutation of (1,...,9).

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A178476 Permutations of 123456: Numbers having each of the decimal digits 1,...,6 exactly once, and no other digit.

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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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