cp's OEIS Frontend

A007489 a(n) = Sum_{k=1..n} k!.

%I A007489 M2818 #192 Jun 12 2025 00:49:07
%S A007489 0,1,3,9,33,153,873,5913,46233,409113,4037913,43954713,522956313,
%T A007489 6749977113,93928268313,1401602636313,22324392524313,378011820620313,
%U A007489 6780385526348313,128425485935180313,2561327494111820313,53652269665821260313,1177652997443428940313
%N A007489 a(n) = Sum_{k=1..n} k!.
%C A007489 Equals row sums of triangle A143122 starting (1, 3, 9, 33, ...). - _Gary W. Adamson_, Jul 26 2008
%C A007489 a(n) for n>=4 is never a perfect square. - _Alexander R. Povolotsky_, Oct 16 2008
%C A007489 Number of cycles that can be written in the form (j,j+1,j+2,...), in all permutations of {1,2,...,n}. Example: a(3)=9 because in (1)(2)(3), (1)(23), (12)(3), (13)(2), (123), (132) we have 3+2+2+1+1+0=9 such cycles. - _Emeric Deutsch_, Jul 14 2009
%C A007489 Conjectured to be the length of the shortest word over {1,...,n} that contains each of the n! permutations as a factor (cf. A180632) [see Johnston]. - _N. J. A. Sloane_, May 25 2013
%C A007489 The above conjecture has been disproven for n>=6. See A180632 and the Houston 2014 reference. - _Dmitry Kamenetsky_, Mar 07 2016
%C A007489 a(n) is also the number of compositions of n if cardinal values do not matter but ordinal rankings do. Since cardinal values do not matter, a sequence of k summands summing to n can be represented as (s(1),...,s(k)), where the s's are positive integers and the numbers in parentheses are the initial ordinal rankings. The number of compositions of these summands are equal to k!, with k ranging from 1 to n. - _Gregory L. Simay_, Jul 31 2016
%C A007489 When the numbers denote finite permutations (as row numbers of A055089) these are the circular shifts to the left. Compare array A211370 for circular shifts to the left in a broader sense. Compare sequence A001563 for circular shifts to the right. - _Tilman Piesk_, Apr 29 2017
%C A007489 Since a(n) = (1!+2!+3!+...+n!) = 3(1+3!/3+4!/3+...+n!/3) is a multiple of 3 for n>2, the only prime in this sequence is a(2) = 3. - _Eric W. Weisstein_, Jul 15 2017
%C A007489 Generalization of 2nd comment: a(n) for n>=4 is never a perfect power (A007916) (Chentzov link). - _Bernard Schott_, Jan 26 2023
%D A007489 R. K. Guy, Unsolved Problems in Number Theory, 3rd ed., Section B44, Springer 2010.
%D A007489 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A007489 Carauleanu Marc, <a href="/A007489/b007489.txt">Table of n, a(n) for n = 0..212</a> (first 100 terms from T. D. Noe)
%H A007489 N. N. Chentzov, D. O. Shklarsky, and I. M. Yaglom, <a href="https://archive.org/details/ussr_olympiad_problem_book/page/n106/mode/1up">The USSR Olympiad Problem Book, Selected Problems and Theorems of Elementary Mathematics</a>, problem 115, pp. 28 and 201-202, Dover publications, Inc., New York, 1993.
%H A007489 R. K. Guy, <a href="/A005728/a005728.pdf">Letter to N. J. A. Sloane, 1987</a>
%H A007489 Robin Houston, <a href="http://arxiv.org/abs/1408.5108">Tackling the Minimal Superpermutation Problem</a>, arXiv:1408.5108 [math.CO], 2014.
%H A007489 Nathaniel Johnston, <a href="http://www.njohnston.ca/2013/04/the-minimal-superpermutation-problem/">The minimal superpermutation problem</a> (2013)
%H A007489 Nathaniel Johnston, <a href="http://dx.doi.org/10.1016/j.disc.2013.03.024">Non-uniqueness of minimal superpermutations</a>, Discrete Math. 313 (2013), no. 14, 1553--1557. MR3047396
%H A007489 S. Legendre and P. Paclet, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL14/Legendre/legendre5.html">On the Permutations Generated by Cyclic Shift </a>, J. Int. Seq. 14 (2011) # 11.3.2.
%H A007489 Hisanori Mishima, <a href="http://www.asahi-net.or.jp/~KC2H-MSM/mathland/matha1/matha132.htm">Factorizations of many number sequences</a>
%H A007489 Hisanori Mishima, <a href="http://www.asahi-net.or.jp/~KC2H-MSM/mathland/matha1/matha103.htm">Factorizations of many number sequences</a>
%H A007489 Alexsandar Petojevic, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL5/Petojevic/petojevic5.html">The Function vM_m(s; a; z) and Some Well-Known Sequences</a>, Journal of Integer Sequences, Vol. 5 (2002), Article 02.1.7.
%H A007489 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Factorial.html">Factorial</a>
%H A007489 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LeftFactorial.html">Left Factorial</a>
%H A007489 G. Xiao, Sigma Server, <a href="http://wims.unice.fr/~wims/en_tool~analysis~sigma.en.html">Operate on "n!"</a>
%H A007489 Jun Yan, <a href="https://arxiv.org/abs/2404.07958">Results on pattern avoidance in parking functions</a>, arXiv:2404.07958 [math.CO], 2024. See p. 5.
%H A007489 <a href="/index/Fa#factorial">Index entries for sequences related to factorial numbers</a>
%F A007489 a(n) = Sum_{k=1..n} P(n, k)/C(n, k). - _Ross La Haye_, Sep 21 2004
%F A007489 a(n) = 3*A056199(n) for n>=2. - _Philippe Deléham_, Feb 10 2007
%F A007489 a(n) = !(n+1)-1=A003422(n+1)-1. - _Artur Jasinski_, Nov 08 2007 [corrected by _Werner Schulte_, Oct 20 2021]
%F A007489 Starting (1, 3, 9, 33, 153, ...), = row sums of triangle A137593 - _Gary W. Adamson_, Jan 28 2008
%F A007489 a(n) = a(n-1) + n! for n >= 1. - _Jaroslav Krizek_, Jun 16 2009
%F A007489 E.g.f. A(x) satisfies to the differential equation A'(x)=A(x)+x/(1-x)^2+1. - _Vladimir Kruchinin_, Jan 22 2011
%F A007489 a(0)=0, a(1)=1, a(n) = (n+1)*a(n-1)-n*a(n-2). - _Sergei N. Gladkovskii_, Jul 05 2012
%F A007489 G.f.: W(0)*x/(2-2*x) , where W(k) = 1 + 1/( 1 - x*(k+2)/( x*(k+2) + 1/W(k+1) )); (continued fraction). - _Sergei N. Gladkovskii_, Aug 19 2013
%F A007489 G.f.: x /(1-x)/Q(0),m=+2, where Q(k) = 1 - 2*x*(2*k+1) - m*x^2*(k+1)*(2*k+1)/( 1 - 2*x*(2*k+2) - m*x^2*(k+1)*(2*k+3)/Q(k+1) ) ; (continued fraction). - _Sergei N. Gladkovskii_, Sep 24 2013
%F A007489 E.g.f.: exp(x-1)*(Ei(1) - Ei(1-x)) - exp(x) + 1/(1 - x), where Ei(x) is the exponential integral. - _Ilya Gutkovskiy_, Nov 27 2016
%F A007489 a(n) = sqrt(a(n-1)*a(n+1)-a(n-2)*n*n!), n >= 2. - _Gary Detlefs_, Oct 26 2020
%F A007489 a(n) ~ n!. - _Ridouane Oudra_, Jun 11 2025
%e A007489 a(4) = 1! + 2! + 3! + 4! = 1 + 2 + 6 + 24 = 33. - _Michael B. Porter_, Aug 03 2016
%p A007489 A007489 := proc(n) local i; add(i!,i=1..n); end proc;
%t A007489 FoldList[Plus, 0, (Range@ 21)! ] (* _Robert G. Wilson v_, Sep 21 2007 *)
%t A007489 Table[Sum[i!, {i, 1, n}], {n, 0, 21}] (* _Zerinvary Lajos_, Jul 12 2009 *)
%t A007489 Accumulate[Range[50]!]  (* _Harvey P. Dale_, Apr 30 2011 *)
%t A007489 Table[Plus@@(Range[n]!), {n, 20}] (* _Alonso del Arte_, Jul 18 2011 *)
%o A007489 (PARI) a(n)=sum(k=1,n,k!) \\ _Charles R Greathouse IV_, Jul 25 2011
%o A007489 (Haskell)
%o A007489 a007489 n = a007489_list !! n
%o A007489 a007489_list = scanl (+) 0 $ tail a000142_list
%o A007489 -- _Reinhard Zumkeller_, Aug 29 2014
%o A007489 (Magma) [0] cat [&+[Factorial(i): i in [1..n]]: n in [1..25]]; // _Vincenzo Librandi_, Sep 02 2016
%o A007489 (GAP) List([1..20],n->Sum([1..n],Factorial)); # _Muniru A Asiru_, Jan 31 2018
%Y A007489 Equals A003422(n+1) - 1.
%Y A007489 Cf. A000142, A000670, A001597, A007916, A137593, A143122, A161128, A180632.
%Y A007489 Column k=0 of A120695.
%K A007489 nonn,easy,nice
%O A007489 0,3
%A A007489 _N. J. A. Sloane_, _Robert G. Wilson v_