A266193 -id:A266193 - cp's OEIS frontend

A225901 Write n in factorial base, then replace each nonzero digit d of radix k with k-d.

0, 1, 4, 5, 2, 3, 18, 19, 22, 23, 20, 21, 12, 13, 16, 17, 14, 15, 6, 7, 10, 11, 8, 9, 96, 97, 100, 101, 98, 99, 114, 115, 118, 119, 116, 117, 108, 109, 112, 113, 110, 111, 102, 103, 106, 107, 104, 105, 72, 73, 76, 77, 74, 75, 90, 91, 94, 95, 92, 93, 84, 85, 88, 89, 86, 87, 78, 79, 82, 83, 80, 81, 48, 49, 52, 53, 50, 51, 66, 67, 70, 71, 68

Offset: 0

Paul Tek, May 20 2013

Analogous to A004488 or A048647 for the factorial base.
A self-inverse permutation of the natural numbers.
From Antti Karttunen, Aug 16-29 2016: (Start)
Consider the following way to view a factorial base representation of nonnegative integer n. For each nonzero digit d_i present in the factorial base representation of n (where i is the radix = 2.. = one more than 1-based position from the right), we place a pebble to the level (height) d_i at the corresponding column i of the triangular diagram like below, while for any zeros the corresponding columns are left empty:
.
Level
  6        o
          ─ ─
  5        . .
          ─ ─ ─
  4        . . .
          ─ ─ ─ ─
  3        . . . .
          ─ ─ ─ ─ ─
  2        . . o . .
          ─ ─ ─ ─ ─ ─
  1        . o . . o o
          ─ ─ ─ ─ ─ ─ ─
  Radix:   7 6 5 4 3 2
  Digits:  6 1 2 0 1 1 = A007623(4491)
Instead of levels, we can observe on which "slope" each pebble (nonzero digit) is located at. Formally, the slope of nonzero digit d_i with radix i is (i - d_i). Thus in above example, both the most significant digit (6) and the least significant 1 are on slope 1 (called "maximal slope", because it contains digits that are maximal allowed in those positions), while the second 1 from the right is on slope 2 ("submaximal slope").
This involution (A225901) sends each nonzero digit at level k to the slope k (and vice versa) by flipping such a diagram by the shallow diagonal axis that originates from the bottom right corner. Thus, from above diagram we obtain:
Slope (= digit's radix - digit's value)
   1
   2 .
   3 .  .╲
   4 .  .╲o╲
   5 .  .╲.╲.╲
   6 .  .╲.╲o╲.╲
     .  .╲.╲.╲.╲o╲
        o╲.╲.╲.╲.╲o╲
        -----------------
        1  5  3 0  2  1  = A007623(1397)
and indeed, a(4491) = 1397 and a(1397) = 4491.
Thus this permutation maps between polynomial encodings A275734 & A275735 and all the respective sequences obtained from them, where the former set of sequences are concerned with the "slopes" and the latter set with the "levels" of the factorial base representation. See the Crossrefs section.
Sequences A231716 and A275956 are closed with respect to this sequence, in other words, for all n, a(A231716(n)) is a term of A231716 and a(A275956(n)) is a term of A275956.
(End)

			a(1000) = a(1*6! + 2*5! + 1*4! + 2*3! + 2*2!) = (7-1)*6! + (6-2)*5! + (5-1)*4! + (4-2)*3! + (3-2)*2! = 4910.
a(1397) = a(1*6! + 5*5! + 3*4! + 0*3! + 2*2! + 1*1!) = (7-1)*6! + (6-5)*5! + (5-3)*4! + (3-2)*2! + (2-1)*1! = 4491.

Cf. A000142, A007623, A004488, A048647, A001563, A007489, A257684, A257687, A276091, A275736, A276149.
Cf. A275959 (fixed points), A231716, A275956.
Cf. A153880 & A255411.
Cf. also A275734 & A275735, A275952 & A275954.
This involution maps between the following sequences related to "levels" and "slopes" (see comments): A275806 <--> A060502, A257511 <--> A260736, A264990 <--> A275811, A275729 <--> A275728, A275948 <--> A275946, A275949 <--> A275947, A275964 <--> A275962, A059590 <--> A276091.
Related permutations: A275957, A275958, A275835, A275836, A275837, A275838, A276011, A276012.

b = MixedRadix[Reverse@ Range[2, 12]]; Table[FromDigits[Map[Boole[# > 0] &, #] (Reverse@ Range[2, Length@ # + 1] - #), b] &@ IntegerDigits[n, b], {n, 0, 82}] (* Version 10.2, or *)
f[n_] := Block[{a = {{0, n}}}, Do[AppendTo[a, {First@ #, Last@ #} &@ QuotientRemainder[a[[-1, -1]], Times @@ Range[# - i]]], {i, 0, #}] &@ NestWhile[# + 1 &, 0, Times @@ Range[# + 1] <= n &]; Most@ Rest[a][[All, 1]] /. {} -> {0}]; g[w_List] := Total[Times @@@ Transpose@ {Map[Times @@ # &, Range@ Range[0, Length@ w]], Reverse@ Append[w, 0]}]; Table[g[Map[Boole[# > 0] &, #] (Reverse@ Range[2, Length@ # + 1] - #)] &@ f@ n, {n, 0, 82}] (* Michael De Vlieger, Aug 29 2016 *)

a(n)=my(s=0,d,k=2);while(n,d=n%k;n=n\k;if(d,s=s+(k-d)*(k-1)!);k=k+1);return(s)

from sympy import factorial as f
def a(n):
    s=0
    k=2
    while(n):
        d=n%k
        n=(n//k)
        if d: s=s+(k - d)*f(k - 1)
        k+=1
    return s
print([a(n) for n in range(101)]) # Indranil Ghosh, Jun 19 2017

(define (A225901 n) (let loop ((n n) (z 0) (m 2) (f 1)) (cond ((zero? n) z) (else (loop (quotient n m) (if (zero? (modulo n m)) z (+ z (* f (- m (modulo n m))))) (+ 1 m) (* f m))))))
;; One implementing the first recurrence, with memoization-macro definec:
(definec (A225901 n) (if (zero? n) n (+ (A276091 (A275736 n)) (A153880 (A225901 (A257684 n))))))
;; Antti Karttunen, Aug 29 2016

From Antti Karttunen, Aug 29 2016: (Start)
a(0) = 0; for n >= 1, a(n) = A276091(A275736(n)) + A153880(a(A257684(n))).
or, for n >= 1, a(n) = A276149(n) + a(A257687(n)).
(End)
Other identities. For n >= 0:
a(n!) = A001563(n).
a(n!-1) = A007489(n-1).
From Antti Karttunen, Aug 16 2016: (Start)
A275734(a(n)) = A275735(n) and vice versa, A275735(a(n)) = A275734(n).
A060130(a(n)) = A060130(n). [The flip preserves the number of nonzero digits.]
A153880(n) = a(A255411(a(n))) and A255411(n) = a(A153880(a(n))). [This involution conjugates between the two fundamental factorial base shifts.]
a(n) = A257684(a(A153880(n))) = A266193(a(A255411(n))). [Follows from above.]
A276011(n) = A273662(a(A273670(n))).
A276012(n) = A273663(a(A256450(n))).
(End)

A153880 Shift factorial base representation left by one digit.

Original entry on oeis.org

0, 2, 6, 8, 12, 14, 24, 26, 30, 32, 36, 38, 48, 50, 54, 56, 60, 62, 72, 74, 78, 80, 84, 86, 120, 122, 126, 128, 132, 134, 144, 146, 150, 152, 156, 158, 168, 170, 174, 176, 180, 182, 192, 194, 198, 200, 204, 206, 240, 242, 246, 248, 252, 254, 264, 266, 270, 272

Offset: 0

Antti Karttunen, Jan 03 2009

Equally, append 0 to the end of the factorial base representation of n (= A007623(n)), then convert back to decimal.

Involution A225901 maps each term of this sequence to a unique term of A255411, and vice versa.

			Factorial base representation of 5 is A007623(5) = "21". Shifting this once left (that is, appending 0 to the end) yields "210", which is factorial base representation for 14. Thus a(5) = 14.

Antti Karttunen, Table of n, a(n) for n = 0..40319
Index entries for sequences related to factorial base representation

Indices of zeros in A260736.
Cf. A153883 (terms divided by 2).
Cf. A266193 (a left inverse).
Cf. A273670 (complement).
Cf. also A007623, A225901, A255411.

Table[Function[b, FromDigits[IntegerDigits[n, b]~Join~{0}, b]]@ MixedRadix[Reverse@ Range@ 12], {n, 0, 57}] (* Michael De Vlieger, May 30 2016, Version 10.2 *)

from sympy import factorial as f
def a007623(n, p=2): return n if nIndranil Ghosh, Jun 20 2017

(define (A153880 n) (let loop ((n n) (z 0) (i 2) (f 2)) (cond ((zero? n) z) (else (loop (floor->exact (/ n i)) (+ (* f (modulo n i)) z) (+ 1 i) (* f (+ i 1)))))))

Other identities. For all n >= 0:

A266193(a(n)) = n.

A273667 Permutation of nonnegative integers: a(0) = 0, a(A153880(n)) = A255411(a(n)), a(A273670(n)) = A256450(a(n)).

Original entry on oeis.org

0, 1, 4, 2, 6, 3, 18, 8, 12, 5, 24, 10, 48, 15, 16, 7, 30, 13, 56, 20, 21, 9, 36, 17, 96, 67, 60, 26, 27, 11, 72, 42, 22, 23, 120, 81, 240, 73, 66, 32, 33, 14, 87, 49, 28, 29, 144, 101, 360, 270, 88, 89, 80, 38, 90, 39, 52, 19, 107, 57, 288, 34, 76, 35, 168, 125, 416, 303, 109, 110, 99, 44, 420, 111, 108, 45, 61, 25, 112, 131, 64, 68, 327, 40

Offset: 0

Antti Karttunen, May 30 2016

Antti Karttunen, Table of n, a(n) for n = 0..5039
Indranil Ghosh, Python program for computing this sequence
Index entries for sequences related to factorial base representation
Index entries for sequences that are permutations of the natural numbers

Inverse: A273668.
Cf. A153880, A225901, A255411, A256450, A257680, A266193, A273663, A273670.
Similar or related permutations: A255566, A273665.

a(0) = 0; for n >= 1: if A257680(A225901(n)) = 0 [when n is one of the terms of A153880] then a(n) = A255411(a(A266193(n))), otherwise [when n is one of the terms of A273670], a(n) = A256450(a(A273663(n))).
As a composition of other permutations:
a(n) = A255566(A273665(n)).

A275847 Permutation of natural numbers: a(0) = 0, a(A153880(n)) = A255411(a(n)), a(A273670(n)) = A256450(n).

Original entry on oeis.org

0, 1, 4, 2, 3, 5, 18, 6, 12, 7, 8, 9, 16, 10, 22, 11, 13, 14, 15, 17, 19, 20, 21, 23, 96, 24, 48, 25, 26, 27, 72, 28, 52, 29, 30, 31, 60, 32, 64, 33, 34, 35, 36, 37, 38, 39, 40, 41, 90, 42, 66, 43, 44, 45, 114, 46, 70, 47, 49, 50, 76, 51, 84, 53, 54, 55, 56, 57, 58, 59, 61, 62, 88, 63, 94, 65, 67, 68, 100, 69, 108, 71, 73, 74, 112, 75, 118, 77, 78

Offset: 0

Antti Karttunen, Aug 13 2016

Inverse: A275848.
Cf. A153880, A225901, A255411, A256450, A257680, A266193, A273663, A273670.
Similar permutations: A273667 (a more recursed variant), A275845, A275846.

a(0) = 0; for n >= 1: if A257680(A225901(n)) = 0 [when n is one of the terms of A153880] then a(n) = A255411(a(A266193(n))), otherwise [when n is one of the terms of A273670], a(n) = A256450(A273663(n)).

A276949 Index of row where n is located in array A276953 (equally: column in A276955).

Original entry on oeis.org

0, 1, 2, 1, 1, 1, 3, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 2, 1, 1, 1, 3, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 2, 1, 1, 1, 3, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5

Offset: 0

Antti Karttunen, Sep 22 2016

This is the smallest difference that occurs between any nonzero digit's radix (which is one more than its one-based position from the right) and that digit's value in the factorial base representation of n. See A225901 and the example.

a(0) = 0 by convention, as there are no nonzero digits present, and neither does 0 occur in arrays A276953 & A276955.

			For n=8, its factorial base representation (A007623) is "110", where the radix for each digit position 1, 2, 3 (from the right) is 2, 3, 4 (one larger than the position). For the 1 in the middle position the difference is 3-1 = 2, while for the 1 at the left we obtain 4-1 = 3. Of these two differences 2 is smaller, thus a(8)=2.

Antti Karttunen, Table of n, a(n) for n = 0..5040
Index entries for sequences related to factorial base representation

Cf. A007623, A257679, A266193, A276950.
Cf. A276951 (for the other index).
Cf. arrays A276953 & A276955.
Cf. also A225901, A273667, A275847.

a(0) = 0, and for n >= 1: if A276950(n) = 1, then a(n) = 1, otherwise a(n) = 1 + a(A266193(n)).
Other identities. For all n >= 0:
a(n) = A257679(A225901(n)) = A257679(A275847(n)) = A257679(A273667(n)).

A275837 Permutation of nonnegative integers: a(n) = A225901(A273667(n)).

Original entry on oeis.org

0, 1, 2, 4, 18, 5, 6, 22, 12, 3, 96, 20, 72, 17, 14, 19, 114, 13, 94, 10, 11, 23, 108, 15, 24, 79, 84, 100, 101, 21, 48, 102, 8, 9, 600, 71, 480, 49, 78, 118, 119, 16, 65, 73, 98, 99, 696, 27, 360, 594, 62, 63, 70, 112, 54, 113, 74, 7, 45, 95, 552, 116, 50, 117, 672, 603, 454, 569, 37, 40, 29, 106, 444, 41, 36, 107, 85, 97, 38, 621, 86, 82, 545, 110, 528, 59, 56, 111

Offset: 0

Antti Karttunen, Aug 13 2016

Inverse: A275838.
Cf. A225901, A273667.
Cf. A275839 (fixed points).
Cf. also A275835, A275836.

(define (A275837 n) (A225901 (A273667 n)))

a(n) = A225901(A273667(n)).
Other identities. For all n >= 0:
a(n) = A266193(a(A153880(n))). [Restriction to A153880 induces the same permutation.]
A275841(n) = A273663(a(A273670(n))). [While restriction to A273670 induces another permutation.]

A275838 Permutation of nonnegative integers: a(n) = A273668(A225901(n)).

Original entry on oeis.org

0, 1, 2, 9, 3, 5, 6, 57, 32, 33, 19, 20, 8, 17, 14, 23, 41, 13, 4, 15, 11, 29, 7, 21, 24, 133, 272, 47, 293, 70, 150, 263, 152, 357, 109, 110, 74, 68, 78, 163, 69, 73, 135, 545, 249, 58, 230, 220, 30, 37, 62, 225, 100, 161, 54, 127, 86, 285, 209, 85, 182, 172, 50, 51, 197, 42, 104, 231, 105, 431, 52, 35, 12, 43, 56, 99, 213, 95, 38, 25, 134, 153, 81, 341, 26, 76

Offset: 0

Antti Karttunen, Aug 13 2016

Inverse: A275837.
Cf. A225901, A273668, A153880, A266193, A273670, A275842.
Cf. A275839 (fixed points).
Cf. also A275835, A275836.

(define (A275838 n) (A273668 (A225901 n)))

a(n) = A273668(A225901(n)).
Other identities. For all n >= 0:
a(n) = A266193(a(A153880(n))). [Restriction to A153880 induces the same permutation.]
A275842(n) = A273663(a(A273670(n))). [While restriction to A273670 induces another permutation.]

A276951 Index of column where n is located in array A276953 (equally: row in A276955).

Original entry on oeis.org

0, 1, 1, 2, 3, 4, 1, 5, 2, 6, 7, 8, 3, 9, 4, 10, 11, 12, 13, 14, 15, 16, 17, 18, 1, 19, 5, 20, 21, 22, 2, 23, 6, 24, 25, 26, 7, 27, 8, 28, 29, 30, 31, 32, 33, 34, 35, 36, 3, 37, 9, 38, 39, 40, 4, 41, 10, 42, 43, 44, 11, 45, 12, 46, 47, 48, 49, 50, 51, 52, 53, 54, 13, 55, 14, 56, 57, 58, 15, 59, 16, 60, 61, 62, 17, 63, 18, 64

Offset: 0

Antti Karttunen, Sep 22 2016

a(0) = 0 by convention, because 0 is not present in arrays A276953 and A276955.

Antti Karttunen, Table of n, a(n) for n = 0..5040
Index entries for sequences related to factorial base representation

Cf. A260736, A266193, A273670, A276950, A276952, A273663.
Cf. arrays A276953 & A276955. A276949 gives their other index.
Cf. also A257681, A275847.

(definec (A276951 n) (cond ((zero? n) n) ((not (zero? (A276950 n))) (A276952 n)) (else (A276951 (A266193 n)))))

a(0) = 0; for n >= 1, if A260736(n) > 0 [when A276950(n) is not zero, when n is in A273670], then a(n) = A276952(n) = 1 + A273663(n), otherwise a(n) = a(A266193(n)).
Other identities. For all n >= 0:
a(n) = A257681(A275847(n)).

A275845 Permutation of natural numbers: a(0) = 0, a(A153880(n)) = A255411(n), a(A273670(n)) = A256450(a(n)).

Original entry on oeis.org

0, 1, 4, 2, 6, 3, 12, 8, 16, 5, 15, 10, 18, 21, 22, 7, 20, 13, 24, 27, 28, 9, 26, 17, 48, 30, 52, 33, 34, 11, 60, 32, 64, 23, 56, 36, 66, 61, 70, 39, 40, 14, 73, 38, 78, 29, 67, 42, 72, 80, 76, 74, 85, 45, 84, 46, 88, 19, 89, 44, 90, 97, 94, 35, 81, 49, 87, 99, 93, 91, 105, 53, 96, 104, 100, 54, 109, 25, 108, 110, 112, 51, 111, 121, 114, 117, 118, 41

Offset: 0

Antti Karttunen, Aug 13 2016

Inverse: A275846.
Cf. A000142, A052849, A153880, A225901, A255411, A256450, A257680, A266193, A273663, A273670.
Similar permutations: A273667 (a more recursed variant), A275847, A275848.

a(0) = 0; for n >= 1: if A257680(A225901(n)) = 0 [when n is one of the terms of A153880] then a(n) = A255411(A266193(n)), otherwise [when n is one of the terms of A273670], a(n) = A256450(a(A273663(n))).
Other identities:
a(A000142(n)) = A052849(n) for all n >= 2.

A276009 Decrement each nonzero digit by one in factorial base representation of n: a(n) = n - A276008(n).

Original entry on oeis.org

0, 0, 0, 0, 2, 2, 0, 0, 0, 0, 2, 2, 6, 6, 6, 6, 8, 8, 12, 12, 12, 12, 14, 14, 0, 0, 0, 0, 2, 2, 0, 0, 0, 0, 2, 2, 6, 6, 6, 6, 8, 8, 12, 12, 12, 12, 14, 14, 24, 24, 24, 24, 26, 26, 24, 24, 24, 24, 26, 26, 30, 30, 30, 30, 32, 32, 36, 36, 36, 36, 38, 38, 48, 48, 48, 48, 50, 50, 48, 48, 48, 48, 50, 50, 54, 54, 54, 54, 56, 56, 60, 60, 60, 60, 62, 62, 72, 72, 72, 72

Offset: 0

Antti Karttunen, Aug 18 2016

			For n=23 whose factorial base representation is "321", when we subtract one from each digit we get "210", the factorial base representation of 14, thus a(23) = 14.
For n=37 ("1201"), when we subtract one from each digit we get "0100", thus a(37) = 6 as A007623(6) = 100.

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

Cf. A007623, A276008.

Cf. also A257684, A257687, A266123, A266193.

a[n_] := Module[{k = n, m = 2, r, s = {}}, While[{k, r} = QuotientRemainder[k, m]; k != 0|| r != 0, AppendTo[s, r]; m++]; s = Max[# - 1, 0]& /@ s; Total[s*Range[Length[s]]!]]; Array[a, 100, 0] (* Amiram Eldar, Feb 14 2024 *)

from sympy import factorial as f
def a007623(n, p=2): return n if n0 else '0' for i in x)[::-1]
    return sum([int(y[i])*f(i + 1) for i in range(len(y))])
print([a(n) for n in range(201)]) # Indranil Ghosh, Jun 21 2017

(define (A276009 n) (- n (A276008 n)))
;; Standalone version:
(define (A276009 n) (let loop ((n n) (s 0) (f 1) (i 2)) (if (zero? n) s (let ((d (modulo n i))) (if (zero? d) (loop (/ n i) s (* i f) (+ 1 i)) (loop (/ (- n d) i) (+ s (* f (- d 1))) (* i f) (+ 1 i)))))))

a(n) = n - A276008(n).

cp's OEIS Frontend

A225901 Write n in factorial base, then replace each nonzero digit d of radix k with k-d.

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A153880 Shift factorial base representation left by one digit.

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A273667 Permutation of nonnegative integers: a(0) = 0, a(A153880(n)) = A255411(a(n)), a(A273670(n)) = A256450(a(n)).

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A275847 Permutation of natural numbers: a(0) = 0, a(A153880(n)) = A255411(a(n)), a(A273670(n)) = A256450(n).

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A276949 Index of row where n is located in array A276953 (equally: column in A276955).

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A275837 Permutation of nonnegative integers: a(n) = A225901(A273667(n)).

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A275838 Permutation of nonnegative integers: a(n) = A273668(A225901(n)).

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A276951 Index of column where n is located in array A276953 (equally: row in A276955).

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A275845 Permutation of natural numbers: a(0) = 0, a(A153880(n)) = A255411(n), a(A273670(n)) = A256450(a(n)).