A208575 -id:A208575 - cp's OEIS frontend

A286590 Numbers that are divisible by the product of their factorial base digits (A208575).

1, 3, 9, 21, 33, 45, 81, 153, 165, 201, 393, 405, 441, 645, 873, 885, 921, 1113, 1125, 1161, 1365, 2313, 2565, 3765, 4005, 5913, 5925, 5961, 6153, 6165, 6201, 6405, 7353, 7641, 8805, 9045, 15993, 16281, 17433, 26085, 26325, 36393, 36645, 46233, 46245, 46281, 46473, 46485, 46521, 46725, 47673

Offset: 1

Antti Karttunen, Jun 18 2017

After the initial 1, all terms are multiples of three.

Amiram Eldar, Table of n, a(n) for n = 1..500 (terms 1..120 from Antti Karttunen)
Index entries for sequences related to factorial base representation

Cf. A007489 (a subsequence), A208575, A118363.

Cf. A007602 (for base-10 analog).

max = 8; Select[Range[max!], FreeQ[(d = IntegerDigits[#, MixedRadix[Range[max, 2, -1]]]), 0] && Divisible[#, Times @@ d] &] (* Amiram Eldar, Feb 16 2021 *)

A227157 Numbers k whose factorial base representation A007623(k) does not contain any nonleading zeros.

Original entry on oeis.org

1, 3, 5, 9, 11, 15, 17, 21, 23, 33, 35, 39, 41, 45, 47, 57, 59, 63, 65, 69, 71, 81, 83, 87, 89, 93, 95, 105, 107, 111, 113, 117, 119, 153, 155, 159, 161, 165, 167, 177, 179, 183, 185, 189, 191, 201, 203, 207, 209, 213, 215, 225, 227, 231, 233, 237, 239, 273

Offset: 1

Antti Karttunen, Jul 04 2013

a(A003422(n)) = A007489(n).
a(A007489(n)) = (n+1)!-1 thus A007489(n) gives the number of terms less than (n+1)! in this sequence.
Equivalently, there are n! terms in the sequence with their magnitude in range n!..(n+1)!.
Also numbers k such that A304036(k) = 1 for k > 0. - Seiichi Manyama, May 06 2018

Seiichi Manyama, Table of n, a(n) for n = 1..10000 (terms 1..5913 from Antti Karttunen)

The sequence gives all n for which A208575(n) is not zero. Complement of A227187. Subsets: A071156 (apart from zero), A231716, A231720.

Cf. also A003422, A007489, A007623, A153880, A227130, A227132, A304036.

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++]; c == 0]; Select[Range[300], q] (* Amiram Eldar, Jan 23 2024 *)

A227153 Product of nonzero digits of n in factorial base.

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

Offset: 0

Antti Karttunen, Jul 04 2013

a(0) = 1 as an empty product always gives 1.

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

A227157 gives the positions where equal with A208575.

Cf. A007623, A227154, A227191.

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++]; p]; Array[a, 100, 0] (* Amiram Eldar, Feb 07 2024 *)

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

def a(n, k=2): return max(n % k, 1) * a(n // k, k + 1) if n else 1 # David Radcliffe, May 22 2025

For all n, a(A227157(n)) = A208575(A227157(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

A208576 Multiplicative persistence of n in factorial base.

Original entry on oeis.org

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

Offset: 0

Charles R Greathouse IV, Feb 28 2012

Diamond and Reidpath prove that a(2n) = 1 for n > 0, a(n) = 2 if n is contains an even digit but no 0's in its factorial base representation. If a(n) > 2 then 3 | n.

Further modular properties can be easily proved. For example, a(n) > 2 implies that n is 33, 45, 81, or 93 mod 120.

Antti Karttunen, Table of n, a(n) for n = 0..65537
M. R. Diamond and D. D. Reidpath, A counterexample to conjectures by Sloane and Erdos 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, A031346, A208575, A208277.

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

a(0) = a(1) = 0; for n > 1, a(n) = 1 + a(A208575(n)). - Antti Karttunen, Nov 14 2018

A227187 Numbers n whose factorial base representation A007623(n) contains at least one nonleading zero. (Zero is also included as a(0)).

Original entry on oeis.org

0, 2, 4, 6, 7, 8, 10, 12, 13, 14, 16, 18, 19, 20, 22, 24, 25, 26, 27, 28, 29, 30, 31, 32, 34, 36, 37, 38, 40, 42, 43, 44, 46, 48, 49, 50, 51, 52, 53, 54, 55, 56, 58, 60, 61, 62, 64, 66, 67, 68, 70, 72, 73, 74, 75, 76, 77, 78, 79, 80, 82, 84, 85, 86, 88, 90, 91

Offset: 0

Antti Karttunen, Jul 04 2013

Antti Karttunen, Table of n, a(n) for n = 0..10000
Index entries for sequences related to factorial base representation.

Complement: A227157.
The sequence gives all positions n where A208575 is zero and all terms where A257510 (also A257260) are nonzeros.
Cf. A232745 (a subsequence), A232744.
Cf. also A007623, A132371, A153880, A227130, A227132, A256450 (numbers with at least one 1 in their factorial representation).

q[n_] := Module[{k = n, m = 2, r, s = {}}, While[{k, r} = QuotientRemainder[k, m]; k != 0|| r != 0, AppendTo[s, r]; m++]; MemberQ[s, 0]]; q[0] = True; Select[Range[0, 100], q] (* Amiram Eldar, Feb 07 2024 *)

a(0) = 0, a(1) = 2, and for n > 1, if a(n-1) is odd or A257510(a(n-1)) > 1, then a(n) = a(n-1) + 1, otherwise a(n) = a(n-1) + 2. - Antti Karttunen, Apr 29 2015
Other identities:
For all n >= 2, a(A132371(n)) = A000142(n) = n! [See comments in A227157.]

A208277 Smallest number of multiplicative persistence n in factorial base.

Original entry on oeis.org

0, 2, 5, 633, 443153013

Offset: 0

Charles R Greathouse IV, Feb 28 2012

a(n) exists for all n, unlike (conjecturally) its decimal equivalent A003001. In particular, with k = a(n-1), a(n) <= k * k! + (k-1)! + ... + 2! + 1! < (a(n-1)+1)! for n > 1. Diamond & Reidpath ask if this upper bound can be improved.

a(5) <= 255429978433810461138446192454297813.

			5 = 1*1!+2*2!, and so is 21 in factorial base; the product of its digits is 2*1 = 10_! and the product of its digits in factorial base is 0*1 = 0, so 5 has multiplicative persistence 2. Since it is the smallest, a(2) = 5.
633 = 51111_! -> 21_! -> 10_! -> 0_! is the least chain of length 3 and so a(3) = 633.

M. R. Diamond and D. D. Reidpath, A counterexample to conjectures by Sloane and Erdos concerning the persistence of numbers, Journal of Recreational Mathematics 29:2 (1998), pp. 89-92.

Cf. A003001, A007623, A031346, A208575, A208576.

pr(n)=my(k=1,s=1);while(n,s*=n%k++;n\=k);s
persist(n)=my(t); while(n>1, t++; n=pr(n)); t
a(n)=my(k=0);while(persist(k)!=n, k++); k \\ Charles R Greathouse IV, Jan 21 2013

A231715 For n with a unique factorial base representation n = duu! + ... + d22! + d1*1! (each di in range 0..i, cf. A007623), a(n) = Product_{i=1..u} (gcd(d_i,i+1) mod i+1), where u is given by A084558(n).

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Nov 12 2013

nonn

			For n=13, with factorial base representation '201' (= A007623(13), 2*3! + 0*2! + 1*1! = 13) we have, starting from the least significant digit, (gcd(1,2) mod 2)*(gcd(0,3) mod 3)*(gcd(2,4) mod 4) = (1 mod 2)*(3 mod 3)*(2 mod 4) = 1*0*2 = 0, thus a(13)=0.
For n=17, with factorial base representation '221' (= A007623(17), 2*3! + 2*2! + 1*1! = 17) we have, starting from the least significant digit, (gcd(1,2) mod 2)*(gcd(2,3) mod 3)*(gcd(2,4) mod 4) = (1 mod 2)*(1 mod 3)*(2 mod 4) = 1*1*2 = 2, thus a(17)=2.

Antti Karttunen, Table of n, a(n) for n = 1..10080

Cf. A231716 (positions of ones), A227157 (the positions of nonzero terms), A007623.

Each a(n) <= A208575(n).

(define (A231715 n) (let loop ((n n) (i 2) (p 1)) (cond ((zero? n) p) (else (loop (floor->exact (/ n i)) (+ i 1) (* p (modulo (gcd (modulo n i) i) i)))))))

A263130 Least number such that the product of its digits in factorial base is n.

Original entry on oeis.org

1, 5, 21, 17, 633, 23, 36153, 65, 93, 635, 443122713, 71, 81474226713, 36155, 645, 113, 6069010670156313, 95, 2318037293294156313, 641, 36165, 443122715, 595774037991797891660313, 119, 4233, 81474226715, 453, 36161, 256727294482662730300616548940313, 647

Offset: 1

Paul Tek, Oct 10 2015

The product of digits in factorial base is given by A208575.
All terms are odd.
Each prime number sets a new record.
a(p) = p*(p!) + Sum_{k=1..p-1} k! for any prime p.
a(n!) = A033312(n+1) for any n>0.
A208576(a(n)) = A208576(n)+1 for any n>1.

			The first terms of the sequence are:
+----+-------------+----------------------------+
| n  | a(n)        | a(n) in factorial base     |
+----+-------------+----------------------------+
|  1 |           1 |                          1 |
|  2 |           5 |                        2_1 |
|  3 |          21 |                      3_1_1 |
|  4 |          17 |                      2_2_1 |
|  5 |         633 |                  5_1_1_1_1 |
|  6 |          23 |                      3_2_1 |
|  7 |       36153 |              7_1_1_1_1_1_1 |
|  8 |          65 |                    2_2_2_1 |
|  9 |          93 |                    3_3_1_1 |
| 10 |         635 |                  5_1_1_2_1 |
| 11 |   443122713 |     11_1_1_1_1_1_1_1_1_1_1 |
| 12 |          71 |                    2_3_2_1 |
| 13 | 81474226713 | 13_1_1_1_1_1_1_1_1_1_1_1_1 |
| 14 |       36155 |              7_1_1_1_1_2_1 |
| 15 |         645 |                  5_1_3_1_1 |
| 16 |         113 |                    4_2_2_1 |
+----+-------------+----------------------------+

Paul Tek, Table of n, a(n) for n = 1..448
Paul Tek, PERL program for this sequence

Cf. A033312, A200748, A208277, A208575, A208576.

f[n_] := Block[{d = Divisors@ n, g, k, m = {1}}, g[x_] := Flatten[Table[#1, {#2}] & @@@ FactorInteger@ x]; Do[k = Max@ Select[d, # <= i &]; If[! IntegerQ@ k, AppendTo[m, 1], d = Divisors[Last[d]/k]; AppendTo[m, k]]; If[d == {1}, Break[]], {i, 2, n}]; Reverse@ m]; Table[FromDigits[#, MixedRadix[Reverse@ Range[2, Length@ #]]] &@ f@ n, {n, 30}] (* Michael De Vlieger, Oct 12 2015, Version 10.2 *)

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

A286590 Numbers that are divisible by the product of their factorial base digits (A208575).

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A227157 Numbers k whose factorial base representation A007623(k) does not contain any nonleading zeros.

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A227153 Product of nonzero digits of n in factorial base.

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

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A208576 Multiplicative persistence of n in factorial base.

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A227187 Numbers n whose factorial base representation A007623(n) contains at least one nonleading zero. (Zero is also included as a(0)).

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A208277 Smallest number of multiplicative persistence n in factorial base.

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A231715 For n with a unique factorial base representation n = du*u! + ... + d2*2! + d1*1! (each di in range 0..i, cf. A007623), a(n) = Product_{i=1..u} (gcd(d_i,i+1) mod i+1), where u is given by A084558(n).

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A231715 For n with a unique factorial base representation n = duu! + ... + d22! + d1*1! (each di in range 0..i, cf. A007623), a(n) = Product_{i=1..u} (gcd(d_i,i+1) mod i+1), where u is given by A084558(n).