A007489 -id:A007489 - cp's OEIS frontend

A220655 For n with a unique factorial base representation n = duu! + ... + d22! + d11! (each di in range 0..i, cf. A007623), a(n) = (du+1)u! + ... + (d2+1)2! + (d1+1)1!; a(n) = n + A007489(A084558(n)).

2, 5, 6, 7, 8, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99

Offset: 1

Antti Karttunen, Dec 17 2012

nonn

Used for computing A107346.

Term a(n) can be obtained by adding one to each digit of factorial base representation of n (A007623(n)) and then reinterpreting it as a kind of pseudo-factorial base representation, ignoring the fact that now some of the digits might be over the maximum allowed in that position. Please see the example section. - Antti Karttunen, Nov 29 2013

			1 has a factorial base representation A007623(1) = '1', as 1 = 1*1!. Incrementing the digit 1 with 1, we get 2*1! = 2, thus a(1) = 2. (Note that although '2' is not a valid factorial base representation, it doesn't matter here.)
2 has a factorial base representation '10', as 2 = 1*2! + 0*1!. Incrementing the digits by one, we get 2*2! + 1*1! = 5, thus a(2) = 5.
3 has a factorial base representation '11', as 3 = 1*2! + 1*1!. Incrementing the digits by one, we get 2*2! + 2*1! = 6, thus a(3) = 6.

Antti Karttunen, Table of n, a(n) for n = 1..5039

Complement: A220695.
One less than A220656.
Cf. A000142, A003422, A007489, A007623, A084558, A212598, A153880, A231720.

Block[{nn = 66, m = 1}, While[Factorial@ m < nn, m++]; m = MixedRadix[Reverse@ Range[2, m]]; Array[FromDigits[1 + IntegerDigits[#, m], m] &, nn]] (* Michael De Vlieger, Jan 20 2020 *)

;; Standalone iterative implementation (Nov 29 2013):
(define (A220655 n) (let loop ((n n) (z 0) (i 2) (f 1)) (cond ((zero? n) z) (else (loop (quotient n i) (+ (* f (+ 1 (remainder n i))) z) (+ 1 i) (* f i))))))
;; Alternative implementation:
(define (A220655 n) (+ n (A007489 (A084558 n))))

a(n) = A220656(n)-1 = A003422(A084558(n)+1) + A000142(A084558(n)) + A212598(n) - 1. [The original definition]
a(n) = n + A007489(A084558(n)). [The above formula reduces to this, which proves that the original Dec 17 2012 description and the new main description produce the same sequence. Essentially, we are adding to n a factorial base repunit '1...111' with as many fact.base digits as n has.] - Antti Karttunen, Nov 29 2013
For n >= 1, A231720(n) = a(A153880(n)).

Name changed by Antti Karttunen, Nov 29 2013

A283984 Sums of distinct nonzero terms of A007489: a(n) = Sum_{k>=0} A030308(n,k)*A007489(1+k).

Original entry on oeis.org

0, 1, 3, 4, 9, 10, 12, 13, 33, 34, 36, 37, 42, 43, 45, 46, 153, 154, 156, 157, 162, 163, 165, 166, 186, 187, 189, 190, 195, 196, 198, 199, 873, 874, 876, 877, 882, 883, 885, 886, 906, 907, 909, 910, 915, 916, 918, 919, 1026, 1027, 1029, 1030, 1035, 1036, 1038, 1039, 1059, 1060, 1062, 1063, 1068, 1069, 1071, 1072, 5913

Offset: 0

Antti Karttunen, Mar 19 2017

nonn

Indexing starts from zero, with a(0) = 0.

Cf. A000142, A007489, A030308, A059590, A070939, A276075, A276091, A283477, A283985.

A007489(n) = sum(k=1, n, k!);
A030308(n,k) = bittest(n,k);
A283984(n) = sum(i=0,(#binary(n)-1),A030308(n,i)*A007489(1+i));

(define (A283984 n) (A276075 (A283477 n)))

a(n) = Sum_{i=0..A070939(n)} A030308(n,i)*A007489(1+i).
a(n) = A276075(A283477(n)).
Other identities. For all n >= 0:
a(2^n) = A007489(n+1).

A244311 Primes (with d digits, say) that generate another prime when acted on by the "standard" superpermutation of length A007489(d) of d elements (cf. comment).

Original entry on oeis.org

13, 19, 31, 37, 79, 109, 113, 139, 193, 317, 331, 911, 991, 1453, 1481, 1669, 1831, 1901, 7127, 7561, 7589, 7687, 9343, 9413, 9811, 10223, 11821, 12889, 13627, 13633, 16979, 17551, 32297, 33529, 34157, 35747, 37409, 39521, 39829, 70957, 71339, 75653, 79633, 90289, 94793, 97583, 99877

Offset: 1

Abhiram R Devesh, Jun 25 2014

For primes with more than four digits, the sequence is based on the current information about the conjectured minimal length.
Since 2013 it is known that the superpermutations with minimal length are not unique for n > 4, and several ones are known for n = 5, cf. Wikipedia. Accordingly, the sequence is ill-defined if the choice of the superpermutation is not made precise. It also turns out that the 6-digit terms in the b-file correspond to the palindromic superpermutation of length A007489(d) obtained by the standard algorithm described on Wikipedia or Johnston's blog. For n = 5 this is of minimal length but only third in lexicographic order, for n >= 6 it is of non-minimal length. See A332088 for the analog sequence using the lexico-first superpermutation of minimal length and including the single-digit terms. - M. F. Hasler, Jul 28 2020
"Acted on" in the definition means that the digits of the prime are 'selected' according to those of the superpermutation. This sequence uses the palindromic superpermutations generated through the standard recursive algorithm, so the corresponding primes (with A007489(d) digits) are palindromic primes (A002385). - M. F. Hasler, Jul 29 2020

			The super-permutation of 3 objects abc with minimal length is abcabacba.
p = 109 is in this sequence as under the super-permutation with minimal length, the number 109101901 is also prime.

Abhiram R Devesh, Table of n, a(n) for n = 1..74
Nathaniel Johnston, Super Permutation
Wikipedia, Superpermutation

Cf. A007489, A002385, A180632, A332088.

my(s); #SSP=vector(6,n,s=if(n--,my(t);concat([if(#Set(s)A332088 for those of minimal length
is_A244311(n)=ispseudoprime(fromdigits(vecextract(n=digits(n), SSP[#n])))
(A244311_upto(N)=select(is_A244311, primes([1,N])))(10^5) \\ (End)

Definition corrected and keyword 'hard' removed; data and b-file double-checked by M. F. Hasler, Jul 29 2020

A082648 Consider f(m) = Sum_{k=1..m} k! (A007489) when m is very large; a(n) = n-th digit from end.

Original entry on oeis.org

3, 1, 3, 0, 4, 9, 0, 2, 4, 0, 2, 9, 8, 2, 5, 6, 3, 3, 2, 4, 4, 6, 5, 5, 2, 5, 0, 9, 3, 0, 5, 0, 1, 3, 9, 5, 3, 2, 3, 4, 0, 8, 4, 9, 9, 7, 0, 1, 1, 2, 6, 8, 3, 7, 4, 8, 6, 8, 7, 4, 9, 7, 4, 7, 4, 2, 2, 9, 0, 0, 4, 3, 3, 0, 5, 6, 5, 8, 6, 5

Offset: 1

Alexander Adamchuk, May 15 2003

Apart from the first term, the same as A025016. - R. J. Mathar, Sep 17 2008

Since A007845 gives the smallest factorial having at least n trailing zeros, the first n digits read from the right are determined for m >= A007845(n) - 1. - Martin Renner, Feb 14 2021

			Sum_{k=1..30} k! = 274410818470142134209703780940313.
The last 7 digits in reverse order give us the first 7 terms of this sequence: 3,1,3,0,4,9,0.
From _Jon E. Schoenfield_, Feb 16 2021: (Start)
The table below shows the 12 least-significant digits of Sum_{k=1..m} k! converging to the first 12 terms of this sequence (in reverse order) as m increases:
.
    m  Sum_{k=1..m} k!  # corresponding digits
   --  ---------------  ----------------------
    0                0                       0
    4               33                       1
    9           409113                       2
   14      93928268313                       3
   19  ...485935180313                       4
   24  ...567844940313                       6
   29  ...395300940313                       7
   34  ...323620940313                       8
   39  ...232420940313                       9
   44  ...080420940313                      10
   49  ...920420940313                      12
  ...
   oo  ...920420940313
(End)

Cf. A007489, A003422, A007845, A045748.

Take[Reverse[IntegerDigits[Sum[n!, {n, 1, 500}]]], 100] (* generates first 100 terms *)

A122990 Numbers m such that (1/99)*Sum_{k=1..m} k! = A007489(m)/99 is prime.

Original entry on oeis.org

8, 10, 11, 16, 22, 30, 34, 40, 42, 47, 49, 68, 74, 79, 168, 202, 245, 280, 463, 534, 803, 936, 958, 1299, 2455, 2546, 7391

Offset: 1

Alexander Adamchuk, Oct 28 2006

A007489(n) = Sum_{k=1..m} k! = (!(n+1) - 1) = A003422(n+1) - 1 = {0, 1, 3, 9, 33, 153, 873, 5913, 46233, 409113, 4037913, ...}. A007489(n) is divisible by 99 for n=8 and n>9. Corresponding primes of the form (!(n+1) - 1)/99 are {467, 40787, 443987, 225498914387, 11895484822660898387, 2771826449193354891007108898387, 3072603482270933019578343003268898387, ...}.

Hisanori Mishima, Factorizations of many number sequences.
Eric Weisstein's World of Mathematics, Left Factorial.

Cf. A007489, A003422 (Left factorial).

f=0;Do[f=f+n!;If[PrimeQ[f/99],Print[{n,f/99}]],{n,1,534}]
Position[Accumulate[Range[1000]!]/99,?PrimeQ]//Flatten (* The program generates the first 23 terms of the sequence. *) (* _Harvey P. Dale, Sep 21 2023 *)

from math import factorial
from sympy import isprime, prime
def afind(limit, startk=8):
    if startk <= 8 <= limit: print(8, end=", ")
    f, s, startk = 1, 0, max(startk, 10)
    for i in range(1, startk):
        f *= i
        s += f
    for k in range(startk, limit+1):
        f *= k
        s += f
        if isprime(s//99):
            print(k, end=", ")
afind(535) # Michael S. Branicky, Jan 17 2022

a(21)-a(26) from Michael S. Branicky, Jan 17 2022

a(27) from Michael S. Branicky, Apr 05 2023

A001563 a(n) = n*n! = (n+1)! - n!.

Original entry on oeis.org

0, 1, 4, 18, 96, 600, 4320, 35280, 322560, 3265920, 36288000, 439084800, 5748019200, 80951270400, 1220496076800, 19615115520000, 334764638208000, 6046686277632000, 115242726703104000, 2311256907767808000, 48658040163532800000, 1072909785605898240000

Offset: 0

N. J. A. Sloane

A similar sequence, with the initial 0 replaced by 1, namely A094258, is defined by the recurrence a(2) = 1, a(n) = a(n-1)*(n-1)^2/(n-2). - Andrey Ryshevich (ryshevich(AT)notes.idlab.net), May 21 2002
Denominators in power series expansion of E_1(x) + gamma + log(x), x > 0. - Michael Somos, Dec 11 2002
If all the permutations of any length k are arranged in lexicographic order, the n-th term in this sequence (n <= k) gives the index of the permutation that rotates the last n elements one position to the right. E.g., there are 24 permutations of 4 items. In lexicographic order they are (0,1,2,3), (0,1,3,2), (0,2,1,3), ... (3,2,0,1), (3,2,1,0). Permutation 0 is (0,1,2,3), which rotates the last 1 element, i.e., it makes no change. Permutation 1 is (0,1,3,2), which rotates the last 2 elements. Permutation 4 is (0,3,1,2), which rotates the last 3 elements. Permutation 18 is (3,0,1,2), which rotates the last 4 elements. The same numbers work for permutations of any length. - Henry H. Rich (glasss(AT)bellsouth.net), Sep 27 2003
Stirling transform of a(n+1)=[4,18,96,600,...] is A083140(n+1)=[4,22,154,...]. - Michael Somos, Mar 04 2004
From Michael Somos, Apr 27 2012: (Start)
Stirling transform of a(n)=[1,4,18,96,...] is A069321(n)=[1,5,31,233,...].
Partial sums of a(n)=[0,1,4,18,...] is A033312(n+1)=[0,1,5,23,...].
Binomial transform of A000166(n+1)=[0,1,2,9,...] is a(n)=[0,1,4,18,...].
Binomial transform of A000255(n+1)=[1,3,11,53,...] is a(n+1)=[1,4,18,96,...].
Binomial transform of a(n)=[0,1,4,18,...] is A093964(n)=[0,1,6,33,...].
Partial sums of A001564(n)=[1,3,4,14,...] is a(n+1)=[1,4,18,96,...].
(End)
Number of small descents in all permutations of [n+1]. A small descent in a permutation (x_1,x_2,...,x_n) is a position i such that x_i - x_(i+1) =1. Example: a(2)=4 because there are 4 small descents in the permutations 123, 13\2, 2\13, 231, 312, 3\2\1 of {1,2,3} (shown by \). a(n)=Sum_{k=0..n-1}k*A123513(n,k). - Emeric Deutsch, Oct 02 2006
Equivalently, in the notation of David, Kendall and Barton, p. 263, this is the total number of consecutive ascending pairs in all permutations on n+1 letters (cf. A010027). - N. J. A. Sloane, Apr 12 2014
a(n-1) is the number of permutations of n in which n is not fixed; equivalently, the number of permutations of the positive integers in which n is the largest element that is not fixed. - Franklin T. Adams-Watters, Nov 29 2006
Number of factors in a determinant when writing down all multiplication permutations. - Mats Granvik, Sep 12 2008
a(n) is also the sum of the positions of the left-to-right maxima in all permutations of [n]. Example: a(3)=18 because the positions of the left-to-right maxima in the permutations 123,132,213,231,312 and 321 of [3] are 123, 12, 13, 12, 1 and 1, respectively and 1+2+3+1+2+1+3+1+2+1+1=18. - Emeric Deutsch, Sep 21 2008
Equals eigensequence of triangle A002024 ("n appears n times"). - Gary W. Adamson, Dec 29 2008
Preface the series with another 1: (1, 1, 4, 18, ...); then the next term = dot product of the latter with "n occurs n times". Example: 96 = (1, 1, 4, 8) dot (4, 4, 4, 4) = (4 + 4 + 16 + 72). - Gary W. Adamson, Apr 17 2009
Row lengths of the triangle in A030298. - Reinhard Zumkeller, Mar 29 2012
a(n) is also the number of minimum (n-)distinguishing labelings of the star graph S_{n+1} on n+1 nodes. - Eric W. Weisstein, Oct 14 2014
When the numbers denote finite permutations (as row numbers of A055089) these are the circular shifts to the right, i.e., a(n) is the permutation with the cycle notation (0 1 ... n-1 n). Compare array A051683 for circular shifts to the right in a broader sense. Compare sequence A007489 for circular shifts to the left. - Tilman Piesk, Apr 29 2017
a(n-1) is the number of permutations on n elements with no cycles of length n. - Dennis P. Walsh, Oct 02 2017
The number of pandigital numbers in base n+1, such that each digit appears exactly once. For example, there are a(9) = 9*9! = 3265920 pandigital numbers in base 10 (A050278). - Amiram Eldar, Apr 13 2020

			E_1(x) + gamma + log(x) = x/1 - x^2/4 + x^3/18 - x^4/96 + ..., x > 0. - _Michael Somos_, Dec 11 2002
G.f. = x + 4*x^2 + 18*x^3 + 96*x^4 + 600*x^5 + 4320*x^6 + 35280*x^7 + 322560*x^8 + ...

A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, id. 218.
J. M. Borwein and P. B. Borwein, Pi and the AGM, Wiley, 1987, p. 336.
F. N. David, M. G. Kendall, and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 263.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Hemisphere Publishing Corp., 1987, chapter 37, equation 37:6:1 at page 354.

T. D. Noe, Table of n, a(n) for n = 0..100
J. D. H. Dickson, Discussion of two double series arising from the number of terms in determinants of certain forms, Proc. London Math. Soc., 10 (1879), 120-122. [Annotated scanned copy]
Loïc Foissy, The antipode of of [sic] a Com-PreLie Hopf algebra, arXiv:2406.01120 [math.CO], 2024. See p. 9.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 30.
Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets.
L. B. W. Jolley, Summation of Series, Dover, 1961.
I. Kortchemski, Asymptotic behavior of permutation records, arXiv: 0804.0446v2 [math.CO], 18 May 2008.
C. Lanczos, Applied Analysis (Annotated scans of selected pages).
Rezsö L. Lovas and István Mezö, On an exotic topology of the integers, arXiv:1008.0713 [math.GN], 2010. See p. 4.
Daniel J. Mundfrom, A problem in permutations: the game of `Mousetrap', European J. Combin. 15 (1994), no. 6, 555-560.
Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv preprint arXiv:1406.3081 [math.CO], 2014.
J. Ser, Les Calculs Formels des Séries de Factorielles, Gauthier-Villars, Paris, 1933 [Local copy].
J. Ser, Les Calculs Formels des Séries de Factorielles (Annotated scans of some selected pages).
A. van Heemert, Cyclic permutations with sequences and related problems, J. Reine Angew. Math., 198 (1957), 56-72.
Dennis P. Walsh, The number of permutations with no k-cycles.
Eric Weisstein's World of Mathematics, Distinguishing Number.
Eric Weisstein's World of Mathematics, Exponential Integral.

Cf. A000142, A002024, A047920, A047922, A053495, A055089, A123513, A143946, A229837, A239069.
Cf. A163931 (E(x,m,n)), A002775 (n^2*n!), A091363 (n^3*n!), A091364 (n^4*n!).
Cf. sequences with formula (n + k)*n! listed in A282466.
Row sums of A185105, A322383, A322384, A094485.

List([0..20], n-> n*Factorial(n) ); # G. C. Greubel, Dec 30 2019

a001563 n = a001563_list !! n
a001563_list = zipWith (-) (tail a000142_list) a000142_list
-- Reinhard Zumkeller, Aug 05 2013

[Factorial(n+1)-Factorial(n): n in [0..20]]; // Vincenzo Librandi, Aug 08 2014

Maple
```
A001563 := n->n*n!;
```

Table[n!n,{n,0,25}] (* Harvey P. Dale, Oct 03 2011 *)

{a(n) = if( n<0, 0, n * n!)} /* Michael Somos, Dec 11 2002 */

[n*factorial(n) for n in (0..20)] # G. C. Greubel, Dec 30 2019

From Michael Somos, Dec 11 2002: (Start)
E.g.f.: x / (1 - x)^2.
a(n) = -A021009(n, 1), n >= 0. (End)
The coefficient of y^(n-1) in expansion of (y+n!)^n, n >= 1, gives the sequence 1, 4, 18, 96, 600, 4320, 35280, ... - Artur Jasinski, Oct 22 2007
Integral representation as n-th moment of a function on a positive half-axis: a(n) = Integral_{x=0..oo} x^n*(x*(x-1)*exp(-x)) dx, for n>=0. This representation may not be unique. - Karol A. Penson, Sep 27 2001
a(0)=0, a(n) = n*a(n-1) + n!. - Benoit Cloitre, Feb 16 2003
a(0) = 0, a(n) = (n - 1) * (1 + Sum_{i=1..n-1} a(i)) for i > 0. - Gerald McGarvey, Jun 11 2004
Arises in the denominators of the following identities: Sum_{n>=1} 1/(n*(n+1)*(n+2)) = 1/4, Sum_{n>=1} 1/(n*(n+1)*(n+2)*(n+3)) = 1/18, Sum_{n>=1} 1/(n*(n+1)*(n+2)*(n+3)*(n+4)) = 1/96, etc. The general expression is Sum_{n>=k} 1/C(n, k) = k/(k-1). - Dick Boland, Jun 06 2005 [And the general expression implies that Sum_{n>=1} 1/(n*(n+1)*...*(n+k-1)) = (Sum_{n>=k} 1/C(n, k))/k! = 1/((k-1)*(k-1)!) = 1/a(k-1), k >= 2. - Jianing Song, May 07 2023]
a(n) = Sum_{m=2..n+1} |Stirling1(n+1, m)|, n >= 1 and a(0):=0, where Stirling1(n, m) = A048994(n, m), n >= m = 0.
a(n) = 1/(Sum_{k>=0} k!/(n+k+1)!), n > 0. - Vladeta Jovovic, Sep 13 2006
a(n) = Sum_{k=1..n(n+1)/2} k*A143946(n,k). - Emeric Deutsch, Sep 21 2008
The reciprocals of a(n) are the lead coefficients in the factored form of the polynomials obtained by summing the binomial coefficients with a fixed lower term up to n as the upper term, divided by the term index, for n >= 1: Sum_{k = i..n} C(k, i)/k = (1/a(n))*n*(n-1)*..*(n-i+1). The first few such polynomials are Sum_{k = 1..n} C(k, 1)/k = (1/1)*n, Sum_{k = 2..n} C(k, 2)/k = (1/4)*n*(n-1), Sum_{k = 3..n} C(k, 3)/k = (1/18)*n*(n-1)*(n-2), Sum_{k = 4..n} C(k, 4)/k = (1/96)*n*(n-1)*(n-2)*(n-3), etc. - Peter Breznay (breznayp(AT)uwgb.edu), Sep 28 2008
If we define f(n,i,x) = Sum_{k=i..n} Sum_{j=i..k} binomial(k,j)*Stirling1(n,k)* Stirling2(j,i)*x^(k-j) then a(n) = (-1)^(n-1)*f(n,1,-2), (n >= 1). - Milan Janjic, Mar 01 2009
Sum_{n>=1} (-1)^(n+1)/a(n) =  0.796599599... [Jolley eq. 289]
G.f.: 2*x*Q(0), where Q(k) = 1 - 1/(k+2 - x*(k+2)^2*(k+3)/(x*(k+2)*(k+3)-1/Q(k+1))); (continued fraction). - Sergei N. Gladkovskii, Apr 19 2013
G.f.: W(0)*(1-sqrt(x)) - 1, where W(k) = 1 + sqrt(x)/( 1 - sqrt(x)*(k+2)/(sqrt(x)*(k+2) + 1/W(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Aug 18 2013
G.f.: T(0)/x - 1/x, where T(k) = 1 - x^2*(k+1)^2/( x^2*(k+1)^2 - (1-x-2*x*k)*(1-3*x-2*x*k)/T(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Oct 17 2013
G.f.: Q(0)*(1-x)/x - 1/x, where Q(k) = 1 - x*(k+1)/( x*(k+1) - 1/(1 - x*(k+1)/( x*(k+1) - 1/Q(k+1) ))); (continued fraction). - Sergei N. Gladkovskii, Oct 22 2013
D-finite with recurrence: a(n) +(-n-2)*a(n-1) +(n-1)*a(n-2)=0. - R. J. Mathar, Jan 14 2020
a(n) = (-1)^(n+1)*(n+1)*Sum_{k=1..n} A094485(n,k)*Bernoulli(k). The inverse of the Worpitzky representation of the Bernoulli numbers. - Peter Luschny, May 28 2020
From Amiram Eldar, Aug 04 2020: (Start)
Sum_{n>=1} 1/a(n) = Ei(1) - gamma = A229837.
Sum_{n>=1} (-1)^(n+1)/a(n) = gamma - Ei(-1) = A239069. (End)
a(n) = Gamma(n)*A000290(n) for n > 0. - Jacob Szlachetka, Jan 01 2022

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

Original entry on oeis.org

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

A006939 Chernoff sequence: a(n) = Product_{k=1..n} prime(k)^(n-k+1).

Original entry on oeis.org

1, 2, 12, 360, 75600, 174636000, 5244319080000, 2677277333530800000, 25968760179275365452000000, 5793445238736255798985527240000000, 37481813439427687898244906452608585200000000, 7517370874372838151564668004911177464757864076000000000, 55784440720968513813368002533861454979548176771615744085560000000000

Offset: 0

N. J. A. Sloane

Product of first n primorials: a(n) = Product_{i=1..n} A002110(i).
Superprimorials, from primorials by analogy with superfactorials.
Smallest number k with n distinct exponents in its prime factorization, i.e., A071625(k) = n.
Subsequence of A130091. - Reinhard Zumkeller, May 06 2007
Hankel transform of A171448. - Paul Barry, Dec 09 2009
This might be a good place to explain the name "Chernoff sequence" since his name does not appear in the References or Links as of Mar 22 2014. - Jonathan Sondow, Mar 22 2014
Pickover (1992) named this sequence after Paul Chernoff of California, who contributed this sequence to his book. He was possibly referring to American mathematician Paul Robert Chernoff (1942 - 2017), a professor at the University of California. - Amiram Eldar, Jul 27 2020

			a(4) = 360 because 2^3 * 3^2 * 5 = 1 * 2 * 6 * 30 = 360.
a(5) = 75600 because 2^4 * 3^3 * 5^2 * 7 = 1 * 2 * 6 * 30 * 210 = 75600.

Clifford A. Pickover, Mazes for the Mind, St. Martin's Press, NY, 1992, p. 351.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
James K. Strayer, Elementary number theory, Waveland Press, Inc., Long Grove, IL, 1994. See p. 37.

T. D. Noe, Table of n, a(n) for n=0..25

Cf. A000178 (product of first n factorials), A007489 (sum of first n factorials), A060389 (sum of first n primorials).
Cf. A002110, A051357.
A000142 counts divisors of superprimorials.
A000325 counts uniform divisors of superprimorials.
A008302 counts divisors of superprimorials by bigomega.
A022915 counts permutations of prime indices of superprimorials.
A076954 is a sister-sequence.
A118914 has row a(n) equal to {1..n}.
A124010 has row a(n) equal to {n..1}.
A130091 lists numbers with distinct prime multiplicities.
A317829 counts factorizations of superprimorials.
A336417 counts perfect-power divisors of superprimorials.
A336426 gives non-products of superprimorials.
Cf. A001221, A001222, A005117, A022559, A071625, A181796, A181819, A336419, A336420, A336496.

a006939 n = a006939_list !! n
a006939_list = scanl1 (*) a002110_list -- Reinhard Zumkeller, Jul 21 2012

[1] cat [(&*[NthPrime(k)^(n-k+1): k in [1..n]]): n in [1..15]]; // G. C. Greubel, Oct 14 2018

a := []; printlevel := -1; for k from 0 to 20 do a := [op(a),product(ithprime(i)^(k-i+1),i=1..k)] od; print(a);

Rest[FoldList[Times,1,FoldList[Times,1,Prime[Range[15]]]]] (* Harvey P. Dale, Jul 07 2011 *)
Table[Times@@Table[Prime[i]^(n - i + 1), {i, n}], {n, 12}] (* Alonso del Arte, Sep 30 2011 *)

a(n)=prod(k=1,n,prime(k)^(n-k+1)) \\ Charles R Greathouse IV, Jul 25 2011

from math import prod
from sympy import prime
def A006939(n): return prod(prime(k)**(n-k+1) for k in range(1,n+1)) # Chai Wah Wu, Aug 12 2025

a(n) = m(1)*m(2)*m(3)*...*m(n), where m(n) = n-th primorial number. - N. J. A. Sloane, Feb 20 2005
a(0) = 1, a(n) = a(n - 1)p(n)#, where p(n)# is the n-th primorial A002110(n) (the product of the first n primes). - Alonso del Arte, Sep 30 2011
log a(n) = n^2(log n + log log n - 3/2 + o(1))/2. - Charles R Greathouse IV, Mar 14 2011
A181796(a(n)) = A000110(n+1). It would be interesting to have a bijective proof of this theorem, which is stated at A181796 without proof. See also A336420. - Gus Wiseman, Aug 03 2020

Corrected and extended by Labos Elemer, May 30 2001

A034968 Minimal number of factorials that add to n.

Original entry on oeis.org

0, 1, 1, 2, 2, 3, 1, 2, 2, 3, 3, 4, 2, 3, 3, 4, 4, 5, 3, 4, 4, 5, 5, 6, 1, 2, 2, 3, 3, 4, 2, 3, 3, 4, 4, 5, 3, 4, 4, 5, 5, 6, 4, 5, 5, 6, 6, 7, 2, 3, 3, 4, 4, 5, 3, 4, 4, 5, 5, 6, 4, 5, 5, 6, 6, 7, 5, 6, 6, 7, 7, 8, 3, 4, 4, 5, 5, 6, 4, 5, 5, 6, 6, 7, 5, 6, 6, 7, 7, 8, 6, 7, 7, 8, 8, 9, 4, 5, 5, 6, 6, 7, 5, 6, 6, 7

Offset: 0

Erich Friedman

nonn

Equivalently, sum of digits when n is written in factorial base (A007623).
Equivalently, a(0)...a(n!-1) give the total number of inversions of the permutations of n elements in lexicographic order (the factorial numbers in rising base are the inversion tables of the permutations and their sum of digits give the total number of inversions, see example and the Fxtbook link). - Joerg Arndt, Jun 17 2011
Also minimum number of adjacent transpositions needed to produce each permutation in the list A055089, or number of swappings needed to bubble sort each such permutation. (See A055091 for the minimum number of any transpositions.)

			a(205) = a(1!*1 + 3!*2 + 4!*3 + 5!*1) = 1+2+3+1 = 7. [corrected by Shin-Fu Tsai, Mar 23 2021]
From _Joerg Arndt_, Jun 17 2011: (Start)
   n:    permutation   inv. table a(n)  cycles
   0:    [ 0 1 2 3 ]   [ 0 0 0 ]   0    (0) (1) (2) (3)
   1:    [ 0 1 3 2 ]   [ 0 0 1 ]   1    (0) (1) (2, 3)
   2:    [ 0 2 1 3 ]   [ 0 1 0 ]   1    (0) (1, 2) (3)
   3:    [ 0 2 3 1 ]   [ 0 1 1 ]   2    (0) (1, 2, 3)
   4:    [ 0 3 1 2 ]   [ 0 2 0 ]   2    (0) (1, 3, 2)
   5:    [ 0 3 2 1 ]   [ 0 2 1 ]   3    (0) (1, 3) (2)
   6:    [ 1 0 2 3 ]   [ 1 0 0 ]   1    (0, 1) (2) (3)
   7:    [ 1 0 3 2 ]   [ 1 0 1 ]   2    (0, 1) (2, 3)
   8:    [ 1 2 0 3 ]   [ 1 1 0 ]   2    (0, 1, 2) (3)
   9:    [ 1 2 3 0 ]   [ 1 1 1 ]   3    (0, 1, 2, 3)
  10:    [ 1 3 0 2 ]   [ 1 2 0 ]   3    (0, 1, 3, 2)
  11:    [ 1 3 2 0 ]   [ 1 2 1 ]   4    (0, 1, 3) (2)
  12:    [ 2 0 1 3 ]   [ 2 0 0 ]   2    (0, 2, 1) (3)
  13:    [ 2 0 3 1 ]   [ 2 0 1 ]   3    (0, 2, 3, 1)
  14:    [ 2 1 0 3 ]   [ 2 1 0 ]   3    (0, 2) (1) (3)
  15:    [ 2 1 3 0 ]   [ 2 1 1 ]   4    (0, 2, 3) (1)
  16:    [ 2 3 0 1 ]   [ 2 2 0 ]   4    (0, 2) (1, 3)
  17:    [ 2 3 1 0 ]   [ 2 2 1 ]   5    (0, 2, 1, 3)
  18:    [ 3 0 1 2 ]   [ 3 0 0 ]   3    (0, 3, 2, 1)
  19:    [ 3 0 2 1 ]   [ 3 0 1 ]   4    (0, 3, 1) (2)
  20:    [ 3 1 0 2 ]   [ 3 1 0 ]   4    (0, 3, 2) (1)
  21:    [ 3 1 2 0 ]   [ 3 1 1 ]   5    (0, 3) (1) (2)
  22:    [ 3 2 0 1 ]   [ 3 2 0 ]   5    (0, 3, 1, 2)
  23:    [ 3 2 1 0 ]   [ 3 2 1 ]   6    (0, 3) (1, 2)
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..10000
F. T. Adams-Watters and F. Ruskey, Generating Functions for the Digital Sum and Other Digit Counting Sequences, JIS 12 (2009) 09.5.6.
Joerg Arndt, Matters Computational (The Fxtbook), fig.10-1.B on p.234.
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.
FindStat - Combinatorial Statistic Finder, The number of inversions of a permutation
Index entries for sequences related to factorial base representation

Cf. A368342 (partial sums), A001809 (sums of n! terms).
Cf. A000142, A007489, A007623, A033312, A055091, A139365.
Cf. A000217, A001222, A056019, A060130, A099563, A225901, A257684, A257687, A257694, A276076, A276146.
Cf. A227148 (positions of even terms), A227149 (of odd terms).
Cf. also A219650, A219651, A219666, A230423.
Differs from analogous A276150 for the first time at n=24.
Positions of records are A200748.

[seq(convert(fac_base(j),`+`),j=0..119)]; # fac_base and PermRevLexUnrank given in A055089. Perm2InversionVector in A064039
Or alternatively: [seq(convert(Perm2InversionVector(PermRevLexUnrank(j)),`+`),j=0..119)];
# third Maple program:
b:= proc(n, i) local q;
      `if`(n=0, 0, b(irem(n, i!, 'q'), i-1)+q)
    end:
a:= proc(n) local k;
      for k while k!Alois P. Heinz, Nov 15 2012

a[n_] := Module[{s=0, i=2, k=n}, While[k > 0, k = Floor[n/i!]; s = s + (i-1)*k; i++]; n-s]; Table[a[n], {n, 0, 105}] (* Jean-François Alcover, Nov 06 2013, after Benoit Cloitre *)

a(n)=local(k,r);k=2;r=0;while(n>0,r+=n%k;n\=k;k++);r \\ Franklin T. Adams-Watters, May 13 2009

def a(n):
    k=2
    r=0
    while n>0:
        r+=n%k
        n=n//k
        k+=1
    return r
print([a(n) for n in range(201)]) # Indranil Ghosh, Jun 19 2017, after PARI program

def A034968(n, p=2): return n if nA034968(n//p, p+1) + n%p
print([A034968(n) for n in range(106)]) # Michael S. Branicky, Oct 27 2024

(define (A034968 n) (let loop ((n n) (i 2) (s 0)) (cond ((zero? n) s) (else (loop (quotient n i) (+ 1 i) (+ s (remainder n i)))))))
;; Antti Karttunen, Aug 29 2016

a(n) = n - Sum_{i>=2} (i-1)*floor(n/i!). - Benoit Cloitre, Aug 26 2003
G.f.: 1/(1-x)*Sum_{k>0} (Sum_{i=1..k} i*x^(i*k!))/(Sum_{i=0..k} x^(i*k!)). - Franklin T. Adams-Watters, May 13 2009
From Antti Karttunen, Aug 29 2016: (Start)
a(0) = 0; for n >= 1, a(n) = A099563(n) + a(A257687(n)).
a(0) = 0; for n >= 1, a(n) = A060130(n) + a(A257684(n)).
Other identities. For all n >= 0:
a(n) = A001222(A276076(n)).
a(n) = A276146(A225901(n)).
a(A000142(n)) = 1, a(A007489(n)) = n, a(A033312(n+1)) = A000217(n).
a(A056019(n)) = a(n).
A219651(n) = n - a(n).
(End)

Additional comments from Antti Karttunen, Aug 23 2001

cp's OEIS Frontend

A014145 Partial sums of A007489.

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A220655 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) = (du+1)*u! + ... + (d2+1)*2! + (d1+1)*1!; a(n) = n + A007489(A084558(n)).

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A283984 Sums of distinct nonzero terms of A007489: a(n) = Sum_{k>=0} A030308(n,k)*A007489(1+k).

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A244311 Primes (with d digits, say) that generate another prime when acted on by the "standard" superpermutation of length A007489(d) of d elements (cf. comment).

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PARI

Extensions

A082648 Consider f(m) = Sum_{k=1..m} k! (A007489) when m is very large; a(n) = n-th digit from end.

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Mathematica

A122990 Numbers m such that (1/99)*Sum_{k=1..m} k! = A007489(m)/99 is prime.

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Mathematica

Python

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A001563 a(n) = n*n! = (n+1)! - n!.

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GAP

Haskell

Magma

Maple

A220655 For n with a unique factorial base representation n = duu! + ... + d22! + d11! (each di in range 0..i, cf. A007623), a(n) = (du+1)u! + ... + (d2+1)2! + (d1+1)1!; a(n) = n + A007489(A084558(n)).