A060117 -id:A060117 - cp's OEIS frontend

A275734 Prime-factorization representations of "factorial base slope polynomials": a(0) = 1; for n >= 1, a(n) = A275732(n) * a(A257684(n)).

1, 2, 3, 6, 2, 4, 5, 10, 15, 30, 10, 20, 3, 6, 9, 18, 6, 12, 2, 4, 6, 12, 4, 8, 7, 14, 21, 42, 14, 28, 35, 70, 105, 210, 70, 140, 21, 42, 63, 126, 42, 84, 14, 28, 42, 84, 28, 56, 5, 10, 15, 30, 10, 20, 25, 50, 75, 150, 50, 100, 15, 30, 45, 90, 30, 60, 10, 20, 30, 60, 20, 40, 3, 6, 9, 18, 6, 12, 15, 30, 45, 90, 30, 60, 9, 18, 27

Offset: 0

Antti Karttunen, Aug 08 2016

These are prime-factorization representations of single-variable polynomials where the coefficient of term x^(k-1) (encoded as the exponent of prime(k) in the factorization of n) is equal to the number of nonzero digits that occur on the slope (k-1) levels below the "maximal slope" in the factorial base representation of n. See A275811 for the definition of the "digit slopes" in this context.

			For n=23 ("321" in factorial base representation, A007623), all three nonzero digits are maximal for their positions (they all occur on "maximal slope"), thus a(23) = prime(1)^3 = 2^3 = 8.
For n=29 ("1021"), there are three nonzero digits, where both 2 and the rightmost 1 are on the "maximal slope", while the most significant 1 is on the "sub-sub-sub-maximal", thus a(29) = prime(1)^2 * prime(4)^1 = 2*7 = 28.
For n=37 ("1201"), there are three nonzero digits, where the rightmost 1 is on the maximal slope, 2 is on the sub-maximal, and the most significant 1 is on the "sub-sub-sub-maximal", thus a(37) = prime(1) * prime(2) * prime(4) = 2*3*7 = 42.
For n=55 ("2101"), the least significant 1 is on the maximal slope, and the digits "21" at the beginning are together on the sub-sub-maximal slope (as they are both two less than the maximal digit values 4 and 3 allowed in those positions), thus a(55) = prime(1)^1 * prime(3)^2 = 2*25 = 50.

Antti Karttunen, Table of n, a(n) for n = 0..40320
Index entries for sequences related to factorial base representation

Cf. A001221, A001222, A002110, A007489, A007814, A048675, A051903, A056169, A056170, A060130, A060502, A225901.
Cf. A257684, A275732.
Cf. A275811.
Cf. A260736, A275728, A275808, A275812, A275946, A275947, A275962.
Cf. A275804 (indices of squarefree terms), A275805 (of terms not squarefree).
Cf. also A275725, A275733, A275735, A276076 for other such prime factorization encodings of A060117/A060118-related polynomials.

from operator import mul
from sympy import prime, factorial as f
def a007623(n, p=2): return n if n0 else '0' for i in x)[::-1]
    return 0 if n==1 else sum(int(y[i])*f(i + 1) for i in range(len(y)))
def a(n): return 1 if n==0 else a275732(n)*a(a257684(n))
print([a(n) for n in range(101)]) # Indranil Ghosh, Jun 19 2017

a(0) = 1; for n >= 1, a(n) = A275732(n) * a(A257684(n)).
Other identities and observations. For all n >= 0:
a(n) = A275735(A225901(n)).
a(A007489(n)) = A002110(n).
A001221(a(n)) = A060502(n).
A001222(a(n)) = A060130(n).
A007814(a(n)) = A260736(n).
A051903(a(n)) = A275811(n).
A048675(a(n)) = A275728(n).
A248663(a(n)) = A275808(n).
A056169(a(n)) = A275946(n).
A056170(a(n)) = A275947(n).
A275812(a(n)) = A275962(n).

A275735 Prime-factorization representations of "factorial base level polynomials": a(0) = 1; for n >= 1, a(n) = 2^A257511(n) * A003961(a(A257684(n))).

Original entry on oeis.org

1, 2, 2, 4, 3, 6, 2, 4, 4, 8, 6, 12, 3, 6, 6, 12, 9, 18, 5, 10, 10, 20, 15, 30, 2, 4, 4, 8, 6, 12, 4, 8, 8, 16, 12, 24, 6, 12, 12, 24, 18, 36, 10, 20, 20, 40, 30, 60, 3, 6, 6, 12, 9, 18, 6, 12, 12, 24, 18, 36, 9, 18, 18, 36, 27, 54, 15, 30, 30, 60, 45, 90, 5, 10, 10, 20, 15, 30, 10, 20, 20, 40, 30, 60, 15, 30, 30, 60, 45, 90, 25, 50, 50, 100, 75

Offset: 0

Antti Karttunen, Aug 09 2016

These are prime-factorization representations of single-variable polynomials where the coefficient of term x^(k-1) (encoded as the exponent of prime(k) in the factorization of n) is equal to the number of times a nonzero digit k occurs in the factorial base representation of n. See the examples.

			For n = 0 whose factorial base representation (A007623) is also 0, there are no nonzero digits at all, thus there cannot be any prime present in the encoding, and a(0) = 1.
For n = 1 there is just one 1, thus a(1) = prime(1) = 2.
For n = 2 ("10"), there is just one 1-digit, thus a(2) = prime(1) = 2.
For n = 3 ("11") there are two 1-digits, thus a(3) = prime(1)^2 = 4.
For n = 18 ("300") there is just one 3, thus a(18) = prime(3) = 5.
For n = 19 ("301") there is one 1 and one 3, thus a(19) = prime(1)*prime(3) = 2*5 = 10.
For n = 141 ("10311") there are three 1's and one 3, thus a(141) = prime(1)^3 * prime(3) = 2^3 * 5^1 = 40.

Antti Karttunen, Table of n, a(n) for n = 0..40320
Index entries for sequences related to factorial base representation

Cf. A000079, A001221, A001222, A003961, A007623, A008683, A181819, A225901, A257511, A257684, A265349, A265350, A264990, A275729, A275806, A351954.
Cf. also A275725, A275733, A275734 for other such prime factorization encodings of A060117/A060118-related polynomials, and also A276076.
Differs from A227154 for the first time at n=18, where a(18) = 5, while A227154(18) = 4.

A276076(n) = { my(i=0,m=1,f=1,nextf); while((n>0),i=i+1; nextf = (i+1)*f; if((n%nextf),m*=(prime(i)^((n%nextf)/f));n-=(n%nextf));f=nextf); m; };
A181819(n) = factorback(apply(e->prime(e),(factor(n)[,2])));
A275735(n) = A181819(A276076(n)); \\ Antti Karttunen, Apr 03 2022

from sympy import prime
from operator import mul
import collections
def a007623(n, p=2): return n if nIndranil Ghosh, Jun 19 2017

a(0) = 1; for n >= 1, a(n) = 2^A257511(n) * A003961(a(A257684(n))).
Other identities and observations. For all n >= 0:
a(n) = A275734(A225901(n)).
A001221(a(n)) = A275806(n).
A001222(a(n)) = A060130(n).
A048675(a(n)) = A275729(n).
A051903(a(n)) = A264990(n).
A008683(a(A265349(n))) = -1 or +1 for all n >= 0.
A008683(a(A265350(n))) = 0 for all n >= 1.
From Antti Karttunen, Apr 03 2022: (Start)
A342001(a(n)) = A351954(n).
a(n) = A181819(A276076(n)). (End)

A060112 Sums of nonconsecutive factorial numbers.

Original entry on oeis.org

0, 1, 2, 6, 7, 24, 25, 26, 120, 121, 122, 126, 127, 720, 721, 722, 726, 727, 744, 745, 746, 5040, 5041, 5042, 5046, 5047, 5064, 5065, 5066, 5160, 5161, 5162, 5166, 5167, 40320, 40321, 40322, 40326, 40327, 40344, 40345, 40346, 40440, 40441, 40442

Offset: 1

Antti Karttunen, Mar 01 2001

Zeckendorf (Fibonacci) expansion of n (A003714) reinterpreted as a factorial expansion.
Also positions in A055089, A060117 and A060118 of the permutations that are composed of disjoint adjacent transpositions only. (That these positions are same can be seen by comparing algorithms PermRevLexUnrankAMSD, PermUnrank3R, PermUnrank3L in the respective sequences). Thus also positions of the fixed terms in A065181-A065184. See comment at A065163.
Written as disjoint cycles the permutations are: (), (1 2), (2 3), (3 4), (1 2)(3 4), (4 5), (1 2)(4 5), (2 3)(4 5), etc. Apart from the first one (the identity), these are the only kind of permutations used in campanology when moving from one "change" to next.

			Zeckendorf Expansions of first few natural numbers and the corresponding values when interpreted as factorial expansions: 0 = 0 = 0, 1 = 1 = 1, 2 = 10 = 2, 3 = 100 = 6, 4 = 101 = 7, 5 = 1000 = 24, 6 = 1001 = 25, 7 = 1010 = 26, 8 = 10000 = 120, etc.,

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Arthur T. White, Ringing the Changes, Math. Proc. Camb. Phil. Soc., September 1983, Vol. 94, part 2, pp. 203-215.
Index entries for sequences related to bell ringing

Subset of A059590. Cf. also A001611, A064640.

For PermRevLexRank, see A056019, for fibbinary see A048679 and A003714.

CampanoPerm := proc(n) local z,p,i; p := []; z := fibbinary(n); i := 1; while(z > 0) do if(1 = (z mod 2)) then p := permul(p,[[i,i+1]]); fi; i := i+1; z := floor(z/2); od; RETURN(convert(p,'permlist',i)); end;

With[{b = MixedRadix[Range[12, 2, -1]]}, FromDigits[#, b] & /@ Select[Tuples[{0, 1}, 8], SequenceCount[#, {1, 1}] == 0 &]] (* Michael De Vlieger, Jun 26 2017 *)

fill(lim,k,val)=if(k>#f, return); my(t=val+f[k]); if(t<=lim, listput(v,t); fill(lim,k+2,t)); fill(lim,k+1,val)
list(lim)=my(k,t=1); local(f=List(),v=List([0])); while((t*=k++)<=lim, listput(f,t)); f=Vecrev(f); fill(lim,1,0); Set(v) \\ Charles R Greathouse IV, Jun 25 2017

first(n) = my(res = [0, 1], k = 1, t = 1, p = 1); while(#res < n, k++; t++; p *= t; res = concat(res, vector(fibonacci(k), i, res[i]+p))); vector(n, i, res[i]) \\ David A. Corneth, Jun 26 2017

a(n) = PermRevLexRank(CampanoPerm(n))

a(A001611(n)) = (n-1)! for n > 2. - David A. Corneth, Jun 25 2017

A227130 Numbers k for which there is an even number of nonzero digits when k is written in the factorial base (A007623).

Original entry on oeis.org

0, 3, 5, 7, 8, 10, 13, 14, 16, 19, 20, 22, 25, 26, 28, 30, 33, 35, 36, 39, 41, 42, 45, 47, 49, 50, 52, 54, 57, 59, 60, 63, 65, 66, 69, 71, 73, 74, 76, 78, 81, 83, 84, 87, 89, 90, 93, 95, 97, 98, 100, 102, 105, 107, 108, 111, 113, 114, 117, 119, 121, 122, 124

Offset: 1

Antti Karttunen, Jul 02 2013

This sequence offers one possible analog to A001969 (evil numbers) in the factorial base system. A227148 gives another kind of analog.
In each range [0,n!-1] exactly half of the integers are found in this sequence, and the other half of them are found in the complement, A227132.
The sequence gives the positions of even permutations in the tables A060117 and A060118.

Antti Karttunen, Table of n, a(n) for n = 1..2520
Wikipedia, Parity of permutation.

Complement: A227132. Cf. also A001969, A060130, A227148.

q[n_] := Module[{k = n, m = 2, c = 0, r}, While[{k, r} = QuotientRemainder[k, m]; k != 0 || r != 0, If[r != 0, c++]; m++]; EvenQ[c]]; Select[Range[0, 150], q] (* Amiram Eldar, Jan 23 2024 *)

A227132 Numbers k for which there is an odd number of nonzero digits when k is written in the factorial base (A007623).

Original entry on oeis.org

1, 2, 4, 6, 9, 11, 12, 15, 17, 18, 21, 23, 24, 27, 29, 31, 32, 34, 37, 38, 40, 43, 44, 46, 48, 51, 53, 55, 56, 58, 61, 62, 64, 67, 68, 70, 72, 75, 77, 79, 80, 82, 85, 86, 88, 91, 92, 94, 96, 99, 101, 103, 104, 106, 109, 110, 112, 115, 116, 118, 120, 123, 125

Offset: 1

Antti Karttunen, Jul 02 2013

This sequence offers one possible analog to A000069 (odious numbers) in the factorial base system. A227149 gives another kind of analog.
In each range [0,n!-1] exactly half of the integers are found in this sequence, and the other half of them are found in the complement, A227130.
The sequence gives the positions of odd permutations in the tables A060117 and A060118.

Antti Karttunen, Table of n, a(n) for n = 1..2520
Wikipedia, Parity of permutation.

Complement: A227130.
Cf. A007623, A060117, A060118.
Cf. also A000069, A060130, A227149.

q[n_] := Module[{k = n, m = 2, c = 0, r}, While[{k, r} = QuotientRemainder[k, m]; k != 0 || r != 0, If[r != 0, c++]; m++]; OddQ[c]]; Select[Range[150], q] (* Amiram Eldar, Jan 24 2024 *)

A275733 a(0) = 1; for n >= 1, a(n) = A275732(n) * A003961(a(A257684(n))).

Original entry on oeis.org

1, 2, 3, 6, 3, 6, 5, 10, 15, 30, 15, 30, 5, 10, 15, 30, 15, 30, 5, 10, 15, 30, 15, 30, 7, 14, 21, 42, 21, 42, 35, 70, 105, 210, 105, 210, 35, 70, 105, 210, 105, 210, 35, 70, 105, 210, 105, 210, 7, 14, 21, 42, 21, 42, 35, 70, 105, 210, 105, 210, 35, 70, 105, 210, 105, 210, 35, 70, 105, 210, 105, 210, 7, 14, 21, 42, 21, 42

Offset: 0

Antti Karttunen, Aug 08 2016

a(n) = product of primes whose indices are positions of nonzero-digits in factorial base representation of n (see A007623). Here positions are one-based, so that the least significant digit is the position 1, the next least significant the position 2, etc.

			For n=19, A007623(19) = 301, thus a(19) = prime(3)*prime(1) = 5*2 = 10.
For n=52, A007623(52) = 2020, thus a(52) = prime(2)*prime(4) = 3*7 = 21.

Antti Karttunen, Table of n, a(n) for n = 0..40320
Indranil Ghosh, Python program for computing this sequence
Index entries for sequences related to factorial base representation

Cf. A001221, A002110, A003961, A005117, A007489, A007623, A048675, A060130, A061395, A084558, A257684, A275732.
Subsequence of A005117.
Cf. A275727.
Cf. also A275725, A275734, A275735 for other such prime factorization encodings of A060117/A060118-related polynomials.

a(0) = 1; for n >= 1, a(n) = A275732(n) * A003961(a(A257684(n))).
Other identities and observations. For all n >= 0:
a(A007489(n)) = A002110(n).
A001221(a(n)) = A001222(a(n)) = A060130(n).
A048675(a(n)) = A275727(n).
A061395(a(n)) = A084558(n).

A276005 Numbers with hit-free factorial base representations; positions of zeros in A276004 & A276007.

Original entry on oeis.org

0, 1, 2, 4, 5, 6, 7, 12, 14, 16, 18, 19, 20, 22, 23, 24, 25, 26, 28, 29, 48, 49, 54, 55, 60, 66, 67, 72, 74, 76, 78, 84, 86, 88, 90, 92, 94, 96, 97, 98, 100, 101, 102, 103, 108, 110, 112, 114, 115, 116, 118, 119, 120, 121, 122, 124, 125, 126, 127, 132, 134, 136, 138, 139, 140, 142, 143, 240, 241, 242, 244, 245, 264, 265, 266, 268, 269, 288, 289, 312, 314, 316

Offset: 0

Antti Karttunen, Aug 17 2016

We say there is a "hit" in factorial base representation (A007623) of n when there is any such pair of nonzero digits d_i and d_j in positions i > j so that (i - d_i) = j. Here the rightmost (least significant digit) occurs at position 1. This sequence gives all "hit-free" numbers, meaning that for every nonzero digit d_i (in position i) in their factorial base representation the digit at the position (i - d_i) is 0.
Also numbers n for which A060502(n) = A060128(n), in other words, the numbers n for which the number of slopes in their factorial base representation (A007623) is equal to the number of non-singleton cycles of the permutation listed as n-th permutation in the list A060117 (or A060118).
This can be viewed as a factorial base analog of base-2 related A003714.

			n=14 (factorial base "210") is included because 2 occurs in position 3 and 1 occurs in position 2, thus as (3-2) = 1 <> 2, 2 does not "hit" digit 1.
n=15 ("211") is NOT included because 2 occurring in position 3 hits the rightmost 1 in position 1 (as 3-2 = 1), and moreover, also the middle 1 hits the rightmost 1 as 2-1 = 1.

Antti Karttunen, Table of n, a(n) for n = 0..4140
Index entries for sequences related to factorial base representation

Complement: A276006.
Cf. A000110, A000142, A060128, A060502, A276004, A276007.
Cf. A060112 (a subsequence).
Intersection with A275804 gives A261220.
Cf. also A003714, A060117 and A060118.

Other identities. For all n >= 1:

a(A000110(n)) = n! = A000142(n). [To be proved.]

A065163 Maximal orbit size in the symmetric group partitioned by the upper records version of the Foata transform (i.e., a(n) is the maximum cycle length found in the corresponding permutations A065181-A065184 in range [0, n!-1]).

Original entry on oeis.org

1, 1, 3, 7, 25, 216, 963, 23435, 92225, 2729205, 17313348, 182553725, 4235194171

Offset: 1

Antti Karttunen, Oct 19 2001

Note: the number of fixed terms in each successive range [0, n!-1] is given by A000045(n+1) (Fibonacci numbers) and the corresponding positions by A060112. (Foata transform fixes a permutation only if it is composed of disjoint adjacent transpositions.)

This version of the Foata transform is one of several. This map takes a permutation s in S_n with k cycles to a permutation t in S_n with k upper records, i.e., k indices i for which t(i) > max{t(j): j < i}. - Theodore Zhu, Aug 15 2014

Cf. A065161, A065162.

For the requisite Maple procedures see sequences A057502, A057542, A060117, A060125.

FoataPermutationCycleCounts_Lengths_and_LCM := proc(upto_n) local u,n,a,b,i,f; a := []; b := []; f := 1; for i from 0 to upto_n! -1 do b := [op(b),1+PermRank3R(Foata(PermUnrank3R(i)))]; if((f - 1) = i) then a := [op(a),[CountCycles(b), CycleLengths1(b), CyclesLCM(b)]]; print (a); f := f*(nops(a)+1); fi; od; RETURN(a); end;
lcmlist := proc(a) local z,e; z := 1; for e in a do z := ilcm(z,e); od; RETURN(z); end;
CyclesLCM := b -> lcmlist(map(nops,convert(b,'disjcyc')));

More terms from Theodore Zhu, Aug 15 2014

A261219 Main diagonal of A261216: a(n) = A261216(n,n).

Original entry on oeis.org

0, 0, 0, 5, 0, 3, 0, 0, 14, 16, 22, 20, 0, 19, 8, 20, 0, 7, 0, 13, 0, 7, 10, 16, 0, 0, 0, 5, 0, 3, 54, 54, 60, 65, 66, 69, 84, 90, 78, 95, 84, 81, 114, 108, 114, 107, 102, 111, 0, 0, 74, 76, 100, 98, 30, 30, 78, 83, 102, 105, 0, 19, 26, 45, 100, 119, 0, 13, 74, 87, 28, 41, 0, 97, 50, 98, 0, 49, 0, 97, 26, 117, 22, 47, 36, 108, 60, 113, 36, 63, 0, 25, 50, 33, 10, 59, 0, 73, 0, 49, 52

Offset: 0

Antti Karttunen, Aug 26 2015

nonn

Equally: main diagonal of A261217.

For permutation p, which has rank n in permutation list A060117, a(n) gives the rank of the "square" of that permutation (obtained by composing it with itself as: q(i) = p(p(i))) in the same list. Equally, if permutation p has rank n in the order used in list A060118, a(n) gives the rank of the p*p in that same list. Thus zeros (which mark the identity permutation, with rank 0 in both orders) occur at positions where the permutations of A060117 (equally: of A060118) are involutions, listed by A261220.

Antti Karttunen, Table of n, a(n) for n = 0..5040

Main diagonal of A261216 and A261217.
Cf. A261220 (the positions of zeros).
Cf. A060117, A060118.
Cf. also A261099, A089841.
Related permutations: A060119, A060126.

a(n) = A261216(n,n) = A261217(n,n).
By conjugating a similar sequence:
a(n) = A060126(A261099(A060119(n))).

A178475 Permutations of 12345: Numbers having each of the decimal digits 1,...,5 exactly once, and no other digit.

Original entry on oeis.org

12345, 12354, 12435, 12453, 12534, 12543, 13245, 13254, 13425, 13452, 13524, 13542, 14235, 14253, 14325, 14352, 14523, 14532, 15234, 15243, 15324, 15342, 15423, 15432, 21345, 21354, 21435, 21453, 21534, 21543, 23145, 23154, 23415

Offset: 1

M. F. Hasler, May 28 2010

There are 5! = 120 terms in this finite subsequence of A030299.
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 5!)).
From Nathaniel Johnston, May 19 2011: (Start)
Individual terms a(n) can be computed efficiently via the following procedure: Define b(n,k) = 1 + floor(((n-1) mod (k+1)!)/k!) for k = 1, 2, 3, 4. The first digit of a(n) is b(n,4). The second digit of a(n) is the b(n,3)-th number not already used. The third digit of a(n) is the b(n,2)-th number not already used. The fourth digit of a(n) is the b(n,1)-th number not already used, and the final digit of a(n) is the only digit remaining. This procedure generalizes in the obvious way for related sequences such as A178476.
For example, if n = 38 then we compute b(38,1) = 2, b(38,2) = 1, b(38,3) = 3, b(38,4) = 2. Thus a(38) = 24153 (2, followed by the 3rd digit not yet used, followed by the 1st digit not yet used, followed by the 2nd digit not yet used, followed by the last remaining digit).
(End)

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

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

FromDigits/@Permutations[Range[5]] (* Harvey P. Dale, Jan 19 2019 *)

A178475(n)={my(b=vector(4,k,1+(n-1)%(k+1)!\k!),t=b[4],d=vector(4,i,i+(i>=t)));for(i=1,3,t=10*t+d[b[4-i]];d=vecextract(d,Str("^"b[4-i])));t*10+d[1]} \\ - M. F. Hasler (following N. Johnston's comment), Jan 10 2012

v=vector(5,i,10^(i-1))~; A178475=vecsort(vector(5!,i,numtoperm(5,i)*v))
is_A178475(x)={ vecsort(Vecsmall(Str(x)))==Vecsmall("12345") }
forstep( m=12345,54321,9, is_A178475(m) & print1(m","))

a(n) + a(5! + 1 - n) = 66666.
floor( a(n) / 10^4 ) = ceiling( n / 4! ).
a(n) = A030299(n+33).
a(n) == 6 (mod 9).
a(n) = 6 + 9*A178485(n).

cp's OEIS Frontend

A275734 Prime-factorization representations of "factorial base slope polynomials": a(0) = 1; for n >= 1, a(n) = A275732(n) * a(A257684(n)).

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A275735 Prime-factorization representations of "factorial base level polynomials": a(0) = 1; for n >= 1, a(n) = 2^A257511(n) * A003961(a(A257684(n))).

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A060112 Sums of nonconsecutive factorial numbers.

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A227130 Numbers k for which there is an even number of nonzero digits when k is written in the factorial base (A007623).

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A227132 Numbers k for which there is an odd number of nonzero digits when k is written in the factorial base (A007623).

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A275733 a(0) = 1; for n >= 1, a(n) = A275732(n) * A003961(a(A257684(n))).

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A276005 Numbers with hit-free factorial base representations; positions of zeros in A276004 & A276007.

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A065163 Maximal orbit size in the symmetric group partitioned by the upper records version of the Foata transform (i.e., a(n) is the maximum cycle length found in the corresponding permutations A065181-A065184 in range [0, n!-1]).

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