A227157 -id:A227157 - cp's OEIS frontend

A003422 Left factorials: !n = Sum_{k=0..n-1} k!.

0, 1, 2, 4, 10, 34, 154, 874, 5914, 46234, 409114, 4037914, 43954714, 522956314, 6749977114, 93928268314, 1401602636314, 22324392524314, 378011820620314, 6780385526348314, 128425485935180314, 2561327494111820314, 53652269665821260314, 1177652997443428940314

Offset: 0

N. J. A. Sloane, R. K. Guy

Number of {12, 12*, 1*2, 21*}- and {12, 12*, 21, 21*}-avoiding signed permutations in the hyperoctahedral group.
a(n) is the number of permutations on [n] that avoid the patterns 2n1 and n12. An occurrence of a 2n1 pattern is a (scattered) subsequence a-n-b with a > b. - David Callan, Nov 29 2007
Also, numbers left over after the following sieving process: At step 1, keep all numbers of the set N = {0, 1, 2, ...}. In step 2, keep only every second number after a(2) = 2: N' = {0, 1, 2, 4, 6, 8, 10, ...}. In step 3, keep every third of the numbers following a(3) = 4, N" = {0, 1, 2, 4, 10, 16, 22, ...}. In step 4, keep every fourth of the numbers beyond a(4) = 10: {0, 1, 2, 4, 10, 34, 58, ...}, and so on. - M. F. Hasler, Oct 28 2010
If s(n) is a second-order recurrence defined as s(0) = x, s(1) = y, s(n) = n*(s(n - 1) - s(n - 2)), n > 1, then s(n) = n*y - n*a(n - 1)*x. - Gary Detlefs, May 27 2012
a(n) is the number of lists of {1, ..., n} with (1st element) = (smallest element) and (k-th element) <> (k-th smallest element) for k > 1, where a list means an ordered subset. a(4) = 10 because we have the lists: [1], [2], [3], [4], [1, 3, 2], [1, 4, 2], [1, 4, 3], [2, 4, 3], [1, 3, 4, 2], [1, 4, 2, 3]. Cf. A000262. - Geoffrey Critzer, Oct 04 2012
Consider a tree graph with 1 vertex. Add an edge to it with another vertex. Now add 2 edges with vertices to this vertex, and then 3 edges to each open vertex of the tree (not the first one!), and the next stage is to add 4 edges, and so on. The total number of vertices at each stage give this sequence (see example). - Jon Perry, Jan 27 2013
Additive version of the superfactorials A000178. - Jon Perry, Feb 09 2013
Repunits in the factorial number system (see links). - Jon Perry, Feb 17 2013
Whether n|a(n) only for 1 and 2 remains an open problem. A published 2004 proof was retracted in 2011. This is sometimes known as Kurepa's conjecture. - Robert G. Wilson v, Jun 15 2013, corrected by Jeppe Stig Nielsen, Nov 07 2015.
!n is not always squarefree for n > 3. Miodrag Zivkovic found that 54503^2 divides !26541. - Arkadiusz Wesolowski, Nov 20 2013
a(n) gives the position of A007489(n) in A227157. - Antti Karttunen, Nov 29 2013
Matches the total domination number of the Bruhat graph from n = 2 to at least n = 5. - Eric W. Weisstein, Jan 11 2019
For the connection with Kurepa trees, see A. Petojevic, The {K_i(z)}{i=1..oo} functions, Rocky Mtn. J. Math., 36 (2006), 1637-1650. - _Aleksandar Petojevic, Jun 29 2018
This sequence converges in the p-adic topology, for every prime number p. - Harry Richman, Aug 13 2024

			!5 = 0! + 1! + 2! + 3! + 4! = 1 + 1 + 2 + 6 + 24 = 34.
x + 2*x^2 + 4*x^3 + 10*x^4 + 34*x^5 + 154*x^6 + 874*x^7 + 5914*x^8 + 46234*x^9 + ...
From _Arkadiusz Wesolowski_, Aug 06 2012: (Start)
Illustration of initial terms:
.
. o        o         o            o                         o
.          o         o            o                         o
.                   o o          o o                       o o
.                              ooo ooo                   ooo ooo
.                                             oooo oooo oooo oooo oooo oooo
.
. 1        2         4            10                        34
.
(End)
The tree graph. The total number of vertices at each stage is 1, 2, 4, 10, ...
    0 0
    |/
    0-0
   /
0-0
   \
    0-0
    |\
    0 0
- _Jon Perry_, Jan 27 2013

Richard K. Guy, Unsolved Problems Number Theory, Section B44.
D. Kurepa, On the left factorial function !n. Math. Balkanica 1 1971 147-153.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..100
P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), Article 00.1.5.
Bernd C. Kellner, Some remarks on Kurepa's left factorial, arXiv:math/0410477 [math.NT], 2004.
D. Kurepa, On the left factorial function !N, Math. Balkanica 1 (1971), 147-153. (Annotated scanned copy).
T. Mansour and J. West, Avoiding 2-letter signed patterns, arXiv:math/0207204 [math.CO], 2002.
Romeo Meštrović, Variations of Kurepa's left factorial hypothesis, arXiv preprint arXiv:1312.7037 [math.NT], 2013.
Romeo Meštrović, The Kurepa-Vandermonde matrices arising from Kurepa's left factorial hypothesis, Filomat 29:10 (2015), 2207-2215; DOI 10.2298/FIL1510207M.
Hisanori Mishima, Factorizations of many number sequences.
Hisanori Mishima, Factorizations of many number sequences
F. J. Papp, Letter to N. J. A. Sloane, Nov 1974.
Jon Perry, Sum of Factorials. [Broken link?]
Aleksandar Petojevic, On Kurepa's hypothesis for the left factorial, Filomat, Vol. 12, No. 1, 1998.
Alexsandar Petojevic, The Function vM_m(s; a; z) and Some Well-Known Sequences, Journal of Integer Sequences, Vol. 5 (2002), Article 02.1.7.
Aleksandar Petojevic, The {K_i(z)}_{i=1..oo} functions, Rocky Mtn. J. Math., 36 (2006), 1637-1650.
Aleksandar Petojević, Marjana Gorjanac Ranitović, Dragan Rastovac, and Milinko Mandić, The Golden Ratio, Factorials, and the Lambert W Function, Journal of Integer Sequences, Vol. 27 (2024), Article 24.5.7.
Eric Weisstein's World of Mathematics, Factorial Sums.
Eric Weisstein's World of Mathematics, Left Factorial.
Eric Weisstein's World of Mathematics, Repunit.
Wikipedia, Factorial number system.
Miodrag Zivkovic, The number of primes sum_{i=1..n} (-1)^(n-i)*i! is finite, Math. Comp. 68 (1999), pp. 403-409.
Index entries for sequences related to factorial numbers.

Equals A007489(n-1)+1 for n>=1. Cf. A000142, A014144, A005165.
Twice A014288. See also A049782, A100612.
Cf. A102639, A102411, A102412, A101752, A094216, A094638, A008276, A000166, A000110, A000204, A000045, A000108, A135722, A227157, A000178, A137338, A232845, A357145.

a003422 n = a003422_list !! n
a003422_list = scanl (+) 0 a000142_list
-- Reinhard Zumkeller, Dec 27 2011

A003422 := proc(n) local k; add(k!,k=0..n-1); end proc:
# Alternative, using the Charlier polynomials A137338:
C := proc(n, x) option remember; if n > 0 then (x-n)*C(n-1, x) - n*C(n-2, x)
elif n = 0 then 1 else 0 fi end: A003422 := n -> (-1)^(n+1)*C(n-1, -1):
seq(A003422(n), n=0..22); # Peter Luschny, Nov 28 2018
# third Maple program:
a:= proc(n) option remember; `if`(n=0, 0, a(n-1)+(n-1)!) end:
seq(a(n), n=0..23);  # Alois P. Heinz, Feb 24 2022

Table[Sum[i!, {i, 0, n - 1}], {n, 0, 20}] (* Stefan Steinerberger, Mar 31 2006 *)
Join[{0}, Accumulate[Range[0, 25]!]] (* Harvey P. Dale, Nov 19 2011 *)
a[0] = 0; a[1] = 1; a[n_] := a[n] = n*a[n - 1] - (n - 1)*a[n - 2]; Array[a, 23, 0] (* Robert G. Wilson v, Jun 15 2013 *)
a[n_] := (-1)^n*n!*Subfactorial[-n-1]-Subfactorial[-1]; Table[a[n] // FullSimplify, {n, 0, 22}] (* Jean-François Alcover, Jan 09 2014 *)
RecurrenceTable[{a[n] == n a[n - 1] - (n - 1) a[n - 2], a[0] == 0, a[1] == 1}, a, {n, 0, 10}] (* Eric W. Weisstein, Jan 11 2019 *)
Range[0, 20]! CoefficientList[Series[(ExpIntegralEi[1] - ExpIntegralEi[1 - x]) Exp[x - 1], {x, 0, 20}], x] (* Eric W. Weisstein, Jan 11 2019 *)
Table[(-1)^n n! Subfactorial[-n - 1] - Subfactorial[-1], {n, 0, 20}] // FullSimplify (* Eric W. Weisstein, Jan 11 2019 *)
Table[(I Pi + ExpIntegralEi[1] + (-1)^n n! Gamma[-n, -1])/E, {n, 0, 20}] // FullSimplify (* Eric W. Weisstein, Jan 11 2019 *)

makelist(sum(k!,k,0,n-1), n, 0, 20); /* Stefano Spezia, Jan 11 2019 */

a003422(n)=sum(k=0,n-1,k!) \\ Charles R Greathouse IV, Jun 15 2011

from itertools import count, islice
def A003422_gen(): # generator of terms
    yield from (0,1)
    c, f = 1, 1
    for n in count(1):
        yield (c:= c + (f:= f*n))
A003422_list = list(islice(A003422_gen(),20)) # Chai Wah Wu, Jun 22 2022

def a(n):
    if n == 0: return 0
    s = f = 1
    for k in range(1, n):
        f *= k
        s += f
    return round(s)
print([a(n) for n in range(24)])  # Peter Luschny, Mar 05 2024

D-finite with recurrence: a(n) = n*a(n - 1) - (n - 1)*a(n - 2). - Henry Bottomley, Feb 28 2001
Sequence is given by 1 + 1*(1 + 2*(1 + 3*(1 + 4*(1 + ..., terminating in n*(1)...). - Jon Perry, Jun 01 2004
a(n) = Sum_{k=0..n-1} P(n, k) / C(n, k). - Ross La Haye, Sep 20 2004
E.g.f.: (Ei(1) - Ei(1 - x))*exp(-1 + x) where Ei(x) is the exponential integral. - Djurdje Cvijovic and Aleksandar Petojevic, Apr 11 2000
a(n) = Integral_{x = 0..oo} [(x^n - 1)/(x - 1)]*exp(-x) dx. - Gerald McGarvey, Oct 12 2007
A007489(n) = !(n + 1) - 1 = a(n + 1) - 1. - Artur Jasinski, Nov 08 2007. Typos corrected by Antti Karttunen, Nov 29 2013
Starting (1, 2, 4, 10, 34, 154, ...), = row sums of triangle A135722. - Gary W. Adamson, Nov 25 2007
a(n) = a(n - 1) + (n - 1)! for n >= 2. - Jaroslav Krizek, Jun 16 2009
E.g.f. A(x) satisfies the differential equation A'(x) = A(x) + 1/(1 - x). - Vladimir Kruchinin, Jan 19 2011
a(n + 1) = p(-1) where p(x) is the unique degree-n polynomial such that p(k) = A182386(k) for k = 0, 1, ..., n. - Michael Somos, Apr 27 2012
From Sergei N. Gladkovskii, May 09 2013 to Oct 22 2013: (Start)
Continued fractions:
G.f.: x/(1-x)*Q(0) where Q(k) = 1 + (2*k + 1)*x/( 1 - 2*x*(k+1)/(2*x*(k+1) + 1/Q(k+1))).
G.f.: G(0)*x/(1-x)/2 where G(k) = 1 + 1/(1 - x*(k+1)/(x*(k+1) + 1/G(k+1))).
G.f.: 2*x/(1-x)/G(0) where G(k) = 1 + 1/(1 - 1/(1 - 1/(2*x*(k+1)) + 1/G(k+1))).
G.f.: W(0)*x/(1+sqrt(x))/(1-x) where W(k) = 1 + sqrt(x)/(1 - sqrt(x)*(k+1)/(sqrt(x)*(k+1) + 1/W(k+1))).
G.f.: B(x)*(1+x)/(1-x) where B(x) is the g.f. of A153229.
G.f.: x/(1-x) + x^2/(1-x)/Q(0) where Q(k) = 1 - 2*x*(2*k+1) - x^2*(2*k+1)*(2*k+2)/(1 - 2*x*(2*k+2) - x^2*(2*k+2)*(2*k+3)/Q(k+1)).
G.f.: x*(1+x)*B(x) where B(x) is the g.f. of A136580. (End)
a(n) = (-1)^(n+1)*C(n-1, -1) where C(n, x) are the Charlier polynomials (with parameter a=1) as given in A137338. (Evaluation at x = 1 gives A232845.) - Peter Luschny, Nov 28 2018
a(n) = (a(n-3)*(n-2)^2*(n-3)! + a(n-1)^2)/a(n-2) (empirical). - Gary Detlefs, Feb 25 2022
a(n) = signum(n)/b(1,n) with b(i,n) = i - [iMohammed Bouras, Sep 07 2022
Sum_{n>=1} 1/a(n) = A357145. - Amiram Eldar, Oct 01 2022

A255411 Shift factorial base representation of n one digit left (with 0 added to right), increment all nonzero digits by one, then convert back to decimal; Numbers with no digit 1 in their factorial base representation.

Original entry on oeis.org

0, 4, 12, 16, 18, 22, 48, 52, 60, 64, 66, 70, 72, 76, 84, 88, 90, 94, 96, 100, 108, 112, 114, 118, 240, 244, 252, 256, 258, 262, 288, 292, 300, 304, 306, 310, 312, 316, 324, 328, 330, 334, 336, 340, 348, 352, 354, 358, 360, 364, 372, 376, 378, 382, 408, 412, 420, 424, 426, 430, 432, 436, 444

Offset: 0

Antti Karttunen, Apr 16 2015

Nonnegative integers such that the number of ones (A257511) in their factorial base representation (A007623) is zero.
Nonnegative integers such that the least missing nonzero digit (A257079) in their factorial base representation is one.
a(n) can be also directly computed from n by "shifting left" its factorial base representation (that is, by appending one zero to the right, see A153880) and then incrementing all nonzero digits by one, and then converting the resulting (still valid) factorial base number back to decimal. See the examples.
The sequences A227130 and A227132 are closed under a(n), in other words, permutation listed as the a(n)-th entry in tables A060117 & A060118 has the same parity as the n-th entry in those same tables.

			Factorial base representation (A007623) of 1 is "1", shifting it left yields "10", and when we increment all nonzero digits by one, we get "20", which is the factorial base representation of 4 (as 4 = 2*2! + 0*1!), thus a(1) = 4.
F.b.r. of 2 is "10", shifting it left yields "100", and "200" is f.b.r. of 12, thus a(2) = 12.
F.b.r. of 43 is "1301", shifting it left and incrementing all nonzeros by one yields "24020", which is f.b.r of 340, thus a(43) = 340.

Antti Karttunen, Table of n, a(n) for n = 0..40319

Complement: A256450.
Positions of ones in A257079, fixed points of A257080, positions of zeros in A257511, A257081 and A257261.
Cf. A007623, A227157, A153880, A231720.
Cf. A255341, A255342, A255343.
Cf. A257262, A257263.
Cf. also A227130/A227132, A060117/A060118 and also arrays A257503 & A257505.

factBaseIntDs[n_] := Module[{m, i, len, dList, currDigit}, i = 1; While[n > i!, i++]; m = n; len = i; dList = Table[0, {len}]; Do[currDigit = 0; While[m >= j!, m = m - j!; currDigit++]; dList[[len - j + 1]] = currDigit, {j, i, 1, -1}]; If[dList[[1]] == 0, dList = Drop[dList, 1]]; dList]; s = Table[FromDigits[factBaseIntDs[n]], {n, 500}]; {0}~Join~Flatten@ Position[s, x_ /; DigitCount[x][[1]] == 0](* Michael De Vlieger, Apr 27 2015, after Alonso del Arte at A007623 *)
Select[Range[0, 444], ! MemberQ[IntegerDigits[#, MixedRadix[Reverse@ Range@ 12]], 1] &] (* Michael De Vlieger, May 30 2016, Version 10.2 *)
r = MixedRadix[Reverse@Range[2, 12]]; Table[FromDigits[Map[If[# == 0, 0, # + 1] &, IntegerDigits[n, r]]~Join~{0}, r], {n, 0, 60}] (* Michael De Vlieger, Aug 14 2016, Version 10.2 *)

from sympy import factorial as f
def a007623(n, p=2): return n if n0 else '0' for i in x)[::-1]
    return 0 if n==0 else sum(int(y[i])*f(i + 1) for i in range(len(y)))
print([a(n) for n in range(101)]) # Indranil Ghosh, Jun 20 2017

A339013 Class number m containing n in a partitioning of the natural numbers into classes B_m by William J. Keith.

Original entry on oeis.org

2, 3, 2, 4, 2, 4, 2, 3, 2, 5, 2, 5, 2, 3, 2, 5, 2, 5, 2, 3, 2, 5, 2, 5, 2, 3, 2, 4, 2, 4, 2, 3, 2, 6, 2, 6, 2, 3, 2, 6, 2, 6, 2, 3, 2, 6, 2, 6, 2, 3, 2, 4, 2, 4, 2, 3, 2, 6, 2, 6, 2, 3, 2, 6, 2, 6, 2, 3, 2, 6, 2, 6, 2, 3, 2, 4, 2, 4, 2, 3, 2, 6, 2, 6, 2, 3, 2

Offset: 1

Kevin Ryde, Nov 19 2020

a(n)=m when n is in class B_m. Keith's residues formula in lemma 1 is equivalent to requiring that n-1 in factorial base representation ends in m-2 nonzero digits, so m = A339012(n-1) + 2.
a(n)=m iff n mod m! is among certain residue classes determined by m. The residues for A339012 are rows of A227157 and here add +1 to each residue (mod m!).  For example 3 or 5 (mod 24) in A339012 becomes here 4 or 6 (mod 24).
The frequency of appearance of the term k = 2, 3, ... in this sequence is 1/(k*(k-1)). - Amiram Eldar, Feb 15 2021

Kevin Ryde, Table of n, a(n) for n = 1..10080
William J. Keith, Sequences of Density zeta(K) - 1, INTEGERS, Vol. 10 (2010), Article #A19, pp. 233-241. Also arXiv preprint, arXiv:0905.3765 [math.NT], 2009 and author's copy.

Cf. A005408 (class B_2), A016933 (class B_3).

Cf. A161189 (class number in partition A_k), A339012.

a[n_] := Module[{k = n - 1, m = 2, r}, While[{k, r} = QuotientRemainder[k, m]; r != 0, m++]; m]; Array[a, 30] (* Amiram Eldar, Feb 15 2021 after Kevin Ryde's PARI code *)

a(n) = n--; my(b=2,r); while([n,r]=divrem(n,b);r!=0, b++); b;

a(n) = A339012(n-1) + 2.
a(n) = m iff n == 1 + Sum_{j=1..m-2} d[j]*j! (mod m!) with d[j] in ranges 1 <= d[j] <= j. [Keith, section 2.1 lemma 1]
a(n)=2 iff n mod 2 = 1. [Keith section 4 residues].
a(n)=3 iff n mod 6 = 2.
a(n)=4 iff n mod 24 = 4 or 6.
a(n)=5 iff n mod 120 = any of 10, 12, 16, 18, 22, 24.

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)).

A325616 Triangle read by rows where T(n,k) is the number of length-k integer partitions of n into factorial numbers.

Original entry on oeis.org

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

Offset: 0

Gus Wiseman, May 12 2019

			Triangle begins:
  1
  0 1
  0 1 1
  0 0 1 1
  0 0 1 1 1
  0 0 0 1 1 1
  0 1 0 1 1 1 1
  0 0 1 0 1 1 1 1
  0 0 1 1 1 1 1 1 1
  0 0 0 1 1 1 1 1 1 1
  0 0 0 1 1 2 1 1 1 1 1
  0 0 0 0 1 1 2 1 1 1 1 1
  0 0 1 0 1 1 2 2 1 1 1 1 1
  0 0 0 1 0 1 1 2 2 1 1 1 1 1
  0 0 0 1 1 1 1 2 2 2 1 1 1 1 1
  0 0 0 0 1 1 1 1 2 2 2 1 1 1 1 1
  0 0 0 0 1 1 2 1 2 2 2 2 1 1 1 1 1
  0 0 0 0 0 1 1 2 1 2 2 2 2 1 1 1 1 1
  0 0 0 1 0 1 1 2 2 2 2 2 2 2 1 1 1 1 1
  0 0 0 0 1 0 1 1 2 2 2 2 2 2 2 1 1 1 1 1
  0 0 0 0 1 1 1 1 2 2 3 2 2 2 2 2 1 1 1 1 1
Row n = 12 counts the following partitions:
  (66)
  (6222)
  (62211)
  (222222) (621111)
  (2222211) (6111111)
  (22221111)
  (222111111)
  (2211111111)
  (21111111111)
  (111111111111)

Row sums are A064986.
Cf. A008284.
Factorial numbers: A000142, A007489, A076934, A108731, A115944, A227157, A284605, A322583, A325509, A325617.
Reciprocal factorial sum: A325618, A325619, A325620, A325622.

Table[SeriesCoefficient[Product[1/(1-y*x^(i!)),{i,1,n}],{x,0,n},{y,0,k}],{n,0,15},{k,0,n}]

T(n,k) is the coefficient of x^n * y^k in the expansion of Product_{i > 0} 1/(1 - y * x^(i!)).

A231716 Numbers with restricted residue set factorial base representation: numbers n which can be formed as a sum n = duu! + ... + d22! + d1*1!, where each di is in range 1..i and gcd(di,i+1)=1.

Original entry on oeis.org

1, 3, 5, 9, 11, 21, 23, 33, 35, 45, 47, 57, 59, 69, 71, 81, 83, 93, 95, 105, 107, 117, 119, 153, 155, 165, 167, 177, 179, 189, 191, 201, 203, 213, 215, 225, 227, 237, 239, 633, 635, 645, 647, 657, 659, 669, 671, 681, 683, 693, 695, 705, 707, 717, 719, 873, 875

Offset: 1

Antti Karttunen, Nov 12 2013

A001088(n+1) gives the number of terms x in sequence for which A084558(x)=n.

Because totatives (the reduced residue set) of each natural number k form a multiplicative group of integers modulo same k, it means that taking e.g. inverses of each digit modulo same k or multiplying them (again modulo k) by some member of that set keeps the set closed, and thus applying these operations to each digit modulo i+1 (2 for the least significant digit in factorial base, 3 for the next, and so on) yield only digits allowed in this sequence, and thus they induce various permutations of this sequence. These can be further "normalized" to be permutations of natural numbers with a suitable ranking function, which is to be submitted later.

			This can be viewed as an irregular table, where row n has A001088(n+1) elements, starts from position A231721(n) and ends at position A231722(n+1):
1;
3, 5;
9, 11, 21, 23;
33, 35, 45, 47, 57, 59, 69, 71, 81, 83, 93, 95, 105, 107, 117, 119;
...

Antti Karttunen, Rows 1..7, flattened
Wikipedia, Totative

Positions of ones in A231715.
The first term of each row: A007489(n) = a(A231721(n)).
The last term of each row: A033312(n+1) = a(A231722(n+1)).
Subset of A227157.
Cf. also A007623, A000010, A001088, A084558.

A257510 Number of nonleading zeros in factorial base representation of n (A007623).

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Apr 27 2015

Sequence starts from n=1, because 0 is an ambiguous case.

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

Cf. A227157 (numbers n such that a(n) = 0), A227187 (n for which a(n) > 0).
Cf. A007623, A060130, A084558, A230403.
Cf. also A257511.
Cf. also A023416, A080791 (analogous sequences for base-2), A055641 (for base-10).

factBaseIntDs[n_] := Module[{m, i, len, dList, currDigit}, i = 1; While[n > i!, i++]; m = n; len = i; dList = Table[0, {len}]; Do[currDigit = 0; While[m >= j!, m = m - j!; currDigit++]; dList[[len - j + 1]] = currDigit, {j, i, 1, -1}]; If[dList[[1]] == 0, dList = Drop[dList, 1]]; dList]; s = Table[FromDigits[factBaseIntDs[n]], {n, 120}]; Last@ DigitCount[#] & /@ s (* Michael De Vlieger, Apr 27 2015, after Alonso del Arte at A007623 *)

(define (A257510 n) (let loop ((n n) (i 2) (s 0)) (cond ((zero? n) s) (else (loop (floor->exact (/ n i)) (+ 1 i) (+ s (if (zero? (modulo n i)) 1 0)))))))

a(n) = A084558(n) - A060130(n).
Other identities and observations:
For all n >= 0, a(A000142(n+1)) = n. [(n+1)! gives the position where n first appears.]
For all n, a(n) >= A230403(n).

A328574 a(1) = 0, and, for n >= 2, numbers n whose primorial base expansion doesn't contain any nonleading zeros.

Original entry on oeis.org

0, 1, 3, 5, 9, 11, 15, 17, 21, 23, 27, 29, 39, 41, 45, 47, 51, 53, 57, 59, 69, 71, 75, 77, 81, 83, 87, 89, 99, 101, 105, 107, 111, 113, 117, 119, 129, 131, 135, 137, 141, 143, 147, 149, 159, 161, 165, 167, 171, 173, 177, 179, 189, 191, 195, 197, 201, 203, 207, 209, 249, 251, 255, 257, 261, 263, 267, 269, 279, 281, 285

Offset: 1

Antti Karttunen, Oct 20 2019

After the initial zero, numbers n for which A276086(n) produces an even number with no gaps in its prime factorization.
Numbers n such that A276086(n) is in A055932; numbers for which A328475(n) is equal to A328572(n) = A003557(A276086(n)).
The number of positive terms below prime(m)# = A002110(m) is Sum_{k=1..m} A005867(k). - Amiram Eldar, Feb 16 2021

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences related to primorial base

Cf. A002110, A003557, A005867, A055932, A276086, A328475, A328572.
Positions of 1's in A328573, positions of 0's in A329027, cf. also A328840.
Cf. A227157 for analogous sequence.

max = 4; bases = Prime @ Range[max, 1, -1]; nmax = Times @@ bases - 1; Join[{0}, Select[Range[nmax], FreeQ[IntegerDigits[#, MixedRadix[bases]], 0] &]] (* Amiram Eldar, Feb 16 2021 *)

A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
isA055932(n) = { my(f=factor(n)[, 1]~); f==primes(#f); }; \\ From A055932
isA328574(n) = isA055932(A276086(n));

A328475(n) = { my(m=1, p=2, y=1); while(n, if(n%p, m *= p^((n%p)-y), y=0); n = n\p; p = nextprime(1+p)); (m); };
A328572(n) = { my(m=1, p=2); while(n, if(n%p, m *= p^((n%p)-1)); n = n\p; p = nextprime(1+p)); (m); };
isA328574(n) = (A328475(n) == A328572(n));

Primary definition changed, the old definition moved to comment section by Antti Karttunen, Nov 03 2019

cp's OEIS Frontend

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A257510 Number of nonleading zeros in factorial base representation of n (A007623).

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A231716 Numbers with restricted residue set factorial base representation: numbers n which can be formed as a sum n = duu! + ... + d22! + d1*1!, where each di is in range 1..i and gcd(di,i+1)=1.