A227153 -id:A227153 - cp's OEIS frontend

A257687 Discard the most significant digit from factorial base representation of n, then convert back to decimal: a(n) = n - A257686(n).

0, 0, 0, 1, 0, 1, 0, 1, 2, 3, 4, 5, 0, 1, 2, 3, 4, 5, 0, 1, 2, 3, 4, 5, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 0

Offset: 0

Antti Karttunen, May 04 2015

A060130(n) gives the number of steps needed to reach zero, when starting iterating as a(k), a(a(k)), etc., from the starting value k = n.

			Factorial base representation (A007623) of 1 is "1", discarding the most significant digit leaves nothing, taken to be zero, thus a(1) = 0.
Factorial base representation of 2 is "10", discarding the most significant digit leaves "0", thus a(2) = 0.
Factorial base representation of 3 is "11", discarding the most significant digit leaves "1", thus a(3) = 1.
Factorial base representation of 4 is "20", discarding the most significant digit leaves "0", thus a(4) = 0.

Antti Karttunen, Table of n, a(n) for n = 0..10080

Cf. A007623, A257686.
Can be used (together with A099563) to define simple recurrences for sequences like A034968, A060130, A227153, A246359, A257511, A257679, A257680.
Cf. also A257684.

f[n_] := Block[{m = p = 1}, While[p*(m + 1) <= n, p = p*m; m++]; Mod[n, p]]; Array[f, 101, 0] (* Robert G. Wilson v, Jul 21 2015 *)

from sympy import factorial as f
def a007623(n, p=2): return n if nIndranil Ghosh, Jun 21 2017

Scheme
```
(define (A257687 n) (- n (A257686 n)))
```

a(n) = n - A257686(n).

A278236 Filter-sequence for factorial base (digit values): least number with the same prime signature as A276076(n).

Original entry on oeis.org

1, 2, 2, 6, 4, 12, 2, 6, 6, 30, 12, 60, 4, 12, 12, 60, 36, 180, 8, 24, 24, 120, 72, 360, 2, 6, 6, 30, 12, 60, 6, 30, 30, 210, 60, 420, 12, 60, 60, 420, 180, 1260, 24, 120, 120, 840, 360, 2520, 4, 12, 12, 60, 36, 180, 12, 60, 60, 420, 180, 1260, 36, 180, 180, 1260, 900, 6300, 72, 360, 360, 2520, 1800, 12600, 8, 24, 24, 120, 72, 360, 24, 120, 120, 840, 360, 2520

Offset: 0

Antti Karttunen, Nov 16 2016

This sequence can be used for filtering certain factorial base related sequences, because it matches only with any such sequence b that can be computed as b(n) = f(A276076(n)), where f(n) is any function that depends only on the prime signature of n (some of these are listed under the index entry for "sequences computed from exponents in ...").
Matching in this context means that the sequence a matches with the sequence b iff for all i, j: a(i) = a(j) => b(i) = b(j). In other words, iff the sequence b partitions the natural numbers to the same or coarser equivalence classes (as/than the sequence a) by the distinct values it obtains.
Any such sequence should match where the result is computed from the nonzero digits (that may also be > 9) in the factorial base representation of n, but does not depend on their order. Some of these are listed on the last line of the Crossrefs section.
Note that as A275735 is present in that list it means that the sequences matching to its filter-sequence A278235 form a subset of the sequences matching to this sequence. Also, for A275735 there is a stronger condition that for any i, j: a(i) = a(j) <=> A275735(i) = A275735(j), which if true, would imply that there is an injective function f such that f(A275735(n)) = A278236(n), and indeed, this function seems to be A181821.

Cf. A007623, A046523, A181821, A276076.
Similar sequences: A278222 (base-2 related), A069877 (base-10), A278226 (primorial base), A278225, A278234, A278235 (other variants for factorial base),
Differs from A278226 for the first time at n=24, where a(24)=2, while A278226(24)=16.
Sequences that partition N into same or coarser equivalence classes: A275735 (<=>), A034968, A060130, A227153, A227154, A246359, A257079, A257511, A257679, A257694, A257695, A257696, A264990, A275729, A275806, A275948, A275964, A278235.

a[n_] := Module[{k = n, m = 2, r, s = {}}, While[{k, r} = QuotientRemainder[k, m]; k != 0|| r != 0, AppendTo[s, r]; m++]; s = ReverseSort[s]; Times @@ (Prime[Range[Length[s]]] ^ s)]; Array[a, 100, 0] (* Amiram Eldar, Feb 07 2024 *)

(define (A278236 n) (A046523 (A276076 n)))

a(n) = A046523(A276076(n)).

a(n) = A181821(A275735(n)). [Empirical formula found with the help of equivalence class matching. Not yet proved.]

A208575 Product of digits of n in factorial base.

Original entry on oeis.org

0, 1, 0, 1, 0, 2, 0, 0, 0, 1, 0, 2, 0, 0, 0, 2, 0, 4, 0, 0, 0, 3, 0, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 0, 0, 2, 0, 4, 0, 0, 0, 3, 0, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 4, 0, 0, 0, 4, 0, 8, 0, 0, 0, 6, 0, 12, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 6, 0, 0, 0, 6, 0, 12, 0, 0, 0, 9, 0, 18

Offset: 0

Charles R Greathouse IV, Feb 28 2012

Antti Karttunen, Table of n, a(n) for n = 0..10080
M. R. Diamond and D. D. Reidpath, A counterexample to conjectures by Sloane and Erdős concerning the persistence of numbers, Journal of Recreational Mathematics 29:2 (1998), pp. 89-92.
Index entries for sequences related to factorial base representation

Cf. A007623, A007954, A031346, A208576, A208277, A227153, A227154, A286590.

(* For the definition of the factorial base version of IntegerDigits, see A007623 *) Table[Times@@factBaseIntDs[n], {n, 0, 99}] (* Alonso del Arte, Feb 28 2012 *)

a(n)=my(k=1,s=1);while(n,s*=n%k++;n\=k);s

from functools import reduce
from operator import mul
def A(n, p=2):
    return n if nIndranil Ghosh, Jun 19 2017

A277012 Factorial base representation of n is rewritten as a base-2 number with each nonzero digit k replaced by a run of k 1's (followed by one extra zero if not the rightmost run of 1's) and with each 0 kept as 0.

Original entry on oeis.org

0, 1, 2, 5, 6, 13, 4, 9, 10, 21, 22, 45, 12, 25, 26, 53, 54, 109, 28, 57, 58, 117, 118, 237, 8, 17, 18, 37, 38, 77, 20, 41, 42, 85, 86, 173, 44, 89, 90, 181, 182, 365, 92, 185, 186, 373, 374, 749, 24, 49, 50, 101, 102, 205, 52, 105, 106, 213, 214, 429, 108, 217, 218, 437, 438, 877, 220, 441, 442, 885, 886, 1773, 56, 113, 114, 229, 230, 461, 116, 233

Offset: 0

Antti Karttunen, Sep 25 2016

			9 = "111" in factorial base (3! + 2! + 1! = 9) is converted to three 1-bits with separating zeros between, in binary as "10101" = A007088(21), thus a(9) = 21.
91 = "3301" in factorial base (91 = 3*4! + 3*3! + 1!) is converted to binary number "1110111001" = A007088(953), thus a(91) = 953. Between the rightmost 1-runs the other zero comes from the factorial base representation, while the other zero is an extra separating zero inserted after each run of 1-bits apart from the rightmost 1-run. The single zero between the two leftmost 1-runs is similarly used to separate the two "unary representations" of 3's.

Cf. A000035, A000079, A000120, A000225, A007088, A007623, A034968, A060130, A069010, A156552, A227153, A227349.
Cf. A277008 (terms sorted into ascending order).
Cf. A277011 (a left inverse).
Differs from analogous A277022 for the first time at n=24, where a(24) = 8, while A277022(24) = 60.

(define (A277012 n) (let loop ((n n) (z 0) (i 2) (j 0)) (if (zero? n) z (let ((d (remainder n i))) (loop (quotient n i) (+ z (* (A000225 d) (A000079 j))) (+ 1 i) (+ 1 j d))))))

a(n) = A156552(A276076(n)).
Other identities. For all n >= 0:
A277011(a(n)) = n.
A005940(1+a(n)) = A276076(n).
A000035(a(n)) = A000035(n). [Preserves the parity of n.]
A000120(a(n)) = A034968(n).
A069010(a(n)) = A060130(n).
A227349(a(n)) = A227153(n).

A227154 Product of digits+1 of n in factorial base.

Original entry on oeis.org

1, 2, 2, 4, 3, 6, 2, 4, 4, 8, 6, 12, 3, 6, 6, 12, 9, 18, 4, 8, 8, 16, 12, 24, 2, 4, 4, 8, 6, 12, 4, 8, 8, 16, 12, 24, 6, 12, 12, 24, 18, 36, 8, 16, 16, 32, 24, 48, 3, 6, 6, 12, 9, 18, 6, 12, 12, 24, 18, 36, 9, 18, 18, 36, 27, 54, 12, 24, 24, 48, 36, 72, 4, 8, 8

Offset: 0

Antti Karttunen, Jul 04 2013

			5 has factorial base representation A007623(5) = "21" (as 2*2 + 1*1 = 5). Adding one to these digits and multiplying, we get 3*2 = 6, thus a(5)=6.

Antti Karttunen, Table of n, a(n) for n = 0..5040 (corrected by Ray Chandler, Jan 19 2019)
Tyler Ball, Joanne Beckford, Paul Dalenberg, Tom Edgar, and Tina Rajabi, Some Combinatorics of Factorial Base Representations, J. Int. Seq., Vol. 23 (2020), Article 20.3.3.
Index entries for sequences related to factorial base representation.

Cf. A007623, A208575, A227153.

a[n_] := Module[{k = n, m = 2, r, p = 1}, While[{k, r} = QuotientRemainder[k, m]; k != 0|| r != 0, p *= (r+1); m++]; p]; Array[a, 100, 0] (* Amiram Eldar, Feb 13 2024 *)

a(n)=my(b=2,t=1); while(n, t *= n%b + 1; n \= b; b++); t \\ Charles R Greathouse IV, Jun 06 2017

a(0)=1 added by Tom Edgar, Jun 05 2017

A257694 a(0) = 1; for n >= 1, a(n) = A060130(n) * a(A257684(n)).

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 2, 2, 3, 2, 3, 1, 2, 2, 3, 4, 6, 1, 2, 2, 3, 4, 6, 1, 2, 2, 3, 2, 3, 2, 3, 3, 4, 3, 4, 2, 3, 3, 4, 6, 8, 2, 3, 3, 4, 6, 8, 1, 2, 2, 3, 4, 6, 2, 3, 3, 4, 6, 8, 4, 6, 6, 8, 9, 12, 4, 6, 6, 8, 9, 12, 1, 2, 2, 3, 4, 6, 2, 3, 3, 4, 6, 8, 4, 6, 6, 8, 9, 12, 8, 12, 12, 16, 18, 24, 1, 2, 2, 3, 4, 6, 2, 3, 3, 4, 6, 8, 4, 6, 6, 8, 9, 12, 8, 12, 12, 16, 18, 24, 1

Offset: 0

Antti Karttunen, May 05 2015

nonn

Antti Karttunen, Table of n, a(n) for n = 0..10080
Index entries for sequences related to factorial base representation

Cf. A034968, A060130, A257684, A257695, A257696, A227153.

a(0) = 1; for n >= 1, a(n) = A060130(n) * a(A257684(n)).
Other identities:
For all n >= 1, a(A033312(n)) = A000142(n-1).

A227191 a(n) = n minus (product of nonzero digits in factorial base representation of n).

Original entry on oeis.org

0, 1, 2, 2, 3, 5, 6, 7, 8, 8, 9, 10, 11, 12, 13, 12, 13, 15, 16, 17, 18, 16, 17, 23, 24, 25, 26, 26, 27, 29, 30, 31, 32, 32, 33, 34, 35, 36, 37, 36, 37, 39, 40, 41, 42, 40, 41, 46, 47, 48, 49, 48, 49, 52, 53, 54, 55, 54, 55, 56, 57, 58, 59, 56, 57, 60, 61, 62

Offset: 1

Antti Karttunen, Jul 04 2013

			22 has factorial expansion A007623(22) = "320", and multiplying the nonzero digits, we get 3*2 = 6, and 22-6 = 16, thus a(22)=16.

Antti Karttunen, Table of n, a(n) for n = 1..5040
Index entries for sequences related to factorial base representation.

Cf. A007623, A227153, A219651, A227190.

a[n_] := Module[{k = n, m = 2, r, p = 1}, While[{k, r} = QuotientRemainder[k, m]; k != 0|| r != 0, If[r > 0, p *= r]; m++]; n - p]; Array[a, 100] (* Amiram Eldar, Feb 14 2024 *)

Scheme
```
(define (A227191 n) (- n (A227153 n)))
```

a(n) = n - A227153(n).

A328581 Product of nonzero digits in primorial base expansion of n.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 4, 4, 3, 3, 3, 3, 6, 6, 4, 4, 4, 4, 8, 8, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 4, 4, 3, 3, 3, 3, 6, 6, 4, 4, 4, 4, 8, 8, 2, 2, 2, 2, 4, 4, 2, 2, 2, 2, 4, 4, 4, 4, 4, 4, 8, 8, 6, 6, 6, 6, 12, 12, 8, 8, 8, 8, 16, 16, 3, 3, 3, 3, 6, 6, 3, 3, 3, 3, 6, 6, 6, 6, 6, 6

Offset: 0

Antti Karttunen, Oct 21 2019

a(0) = 1 as an empty product.

Antti Karttunen, Table of n, a(n) for n = 0..32768
Index entries for sequences related to primorial base.

Cf. A005361, A049345, A276086, A324655, A328582.
Cf. A276156 (positions of 1's).
Cf. also A227153 (an analogous sequence).

a[n_] := Module[{k = n, p = 2, s = 1, r}, While[{k, r} = QuotientRemainder[k, p]; k != 0 || r != 0, If[r > 0, s *= r]; p = NextPrime[p]]; s]; Array[a, 100, 0] (* Amiram Eldar, Mar 06 2024 *)

A328581(n) = { my(m=1, p=2); while(n, if(n%p, m *= (n%p)); n = n\p; p = nextprime(1+p)); (m); };

a(n) = A005361(A276086(n)).

A286606 a(n) = n mod product of nonzero digits of n in factorial base.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 2, 0, 4, 5, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 2, 0, 4, 5, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 2, 3, 0, 1, 2, 3, 0, 1, 0, 1, 2, 3, 10, 11, 0, 1, 2, 0, 4, 5, 0, 1, 2, 0, 4, 5, 0, 1, 2, 3, 4, 5, 0, 1, 2, 3, 4, 5, 0, 1, 2, 3, 4, 5, 2, 3, 0, 1, 2, 3, 4, 5, 6, 7, 0, 1, 6, 7, 8, 9, 22, 23, 0

Offset: 1

Antti Karttunen, Jun 18 2017

Antti Karttunen, Table of n, a(n) for n = 1..10080
Index entries for sequences related to factorial base representation.

Cf. A227153, A286604.

a[n_] := Module[{k = n, m = 2, r, p = 1}, While[{k, r} = QuotientRemainder[k, m]; k != 0|| r != 0, If[r > 0, p *= r]; m++]; Mod[n, p]]; Array[a, 100] (* Amiram Eldar, Feb 21 2024 *)

from operator import mul
from functools import reduce
def a007623(n, p=2): return n if nIndranil Ghosh, Jun 21 2017

(define (A286606 n) (modulo n (A227153 n)))

a(n) = n mod A227153(n).

A301652 Triangle read by rows: row n gives the digits of n in factorial base in reversed order.

Original entry on oeis.org

0, 1, 0, 1, 1, 1, 0, 2, 1, 2, 0, 0, 1, 1, 0, 1, 0, 1, 1, 1, 1, 1, 0, 2, 1, 1, 2, 1, 0, 0, 2, 1, 0, 2, 0, 1, 2, 1, 1, 2, 0, 2, 2, 1, 2, 2, 0, 0, 3, 1, 0, 3, 0, 1, 3, 1, 1, 3, 0, 2, 3, 1, 2, 3, 0, 0, 0, 1, 1, 0, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 2, 0, 1, 1, 2, 0, 1, 0, 0, 1, 1

Offset: 0

Seiichi Manyama, Mar 25 2018

Row n gives exponents for successive primes 2, 3, 5, 7, 11, etc., in the prime factorization of A276076(n). - Antti Karttunen, Mar 11 2024

			   n | 1  2  6
  ---+---------
   0 | 0;
   1 | 1;
   2 | 0, 1;
   3 | 1, 1;
   4 | 0, 2;
   5 | 1, 2;
   6 | 0, 0, 1;
   7 | 1, 0, 1;
   8 | 0, 1, 1;
   9 | 1, 1, 1;
  10 | 0, 2, 1;
  11 | 1, 2, 1;
  12 | 0, 0, 2;
  13 | 1, 0, 2;
  14 | 0, 1, 2;
  15 | 1, 1, 2;
  16 | 0, 2, 2;
  17 | 1, 2, 2;
  18 | 0, 0, 3;
  19 | 1, 0, 3;

Seiichi Manyama, Rows n = 0..2000, flattened
Wikipedia, Factorial number system.
Index entries for sequences related to factorial base representation.

Triangle A108731 with rows reversed.

Cf. A007623, A034968 (row sums), A208575 (row products), A227153 (products of nonzero terms on row n), A276076, A301593.

row[n_] := Module[{k = n, m = 2, r, s = {}}, While[{k, r} = QuotientRemainder[k, m]; k != 0 || r != 0, AppendTo[s, r]; m++]; s]; row[0] = {0}; Array[row, 31, 0] // Flatten (* Amiram Eldar, Mar 11 2024 *)

terms=25; print([0]+[x for sublist in [[floor(n/factorial(i))%(i+1) for i in [k for k in [1..n] if factorial(k)<=n]] for n in [1..terms]] for x in sublist]) # Tom Edgar, Aug 15 2018

T(n,k) = floor(n/k!) mod k+1. - Tom Edgar, Aug 15 2018

cp's OEIS Frontend

A257687 Discard the most significant digit from factorial base representation of n, then convert back to decimal: a(n) = n - A257686(n).

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A278236 Filter-sequence for factorial base (digit values): least number with the same prime signature as A276076(n).

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A208575 Product of digits of n in factorial base.

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A277012 Factorial base representation of n is rewritten as a base-2 number with each nonzero digit k replaced by a run of k 1's (followed by one extra zero if not the rightmost run of 1's) and with each 0 kept as 0.

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A227154 Product of digits+1 of n in factorial base.

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A257694 a(0) = 1; for n >= 1, a(n) = A060130(n) * a(A257684(n)).

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A227191 a(n) = n minus (product of nonzero digits in factorial base representation of n).

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A328581 Product of nonzero digits in primorial base expansion of n.

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