A071828 -id:A071828 - cp's OEIS frontend

A005165 Alternating factorials: n! - (n-1)! + (n-2)! - ... 1!.

0, 1, 1, 5, 19, 101, 619, 4421, 35899, 326981, 3301819, 36614981, 442386619, 5784634181, 81393657019, 1226280710981, 19696509177019, 335990918918981, 6066382786809019, 115578717622022981, 2317323290554617019, 48773618881154822981

Offset: 0

N. J. A. Sloane

Conjecture: for n > 2, smallest prime divisor of a(n) > n. - Gerald McGarvey, Jun 19 2004
Rebuttal: This is not true; see Zivkovic link (Math. Comp. 68 (1999), pp. 403-409) has demonstrated that 3612703 divides a(n) for all n >= 3612702. - Paul Jobling, Oct 18 2004
Conjecture: For n>1, a(n) is the number of lattice paths from (0,0) to (n+1,0) that do not cross above y=x or below the x-axis using up-steps +(1,a) and down-steps +(1,-b) where a and b are positive integers. For example, a(3) = 5: [(1,1)(1,1)(1,1)(1,-3)], [(1,1)(1,-1)(1,3)(1,-3)], [(1,1)(1,-1)(1,2)(1,-2)], [(1,1)(1,-1)(1,1)(1,-1)] and [(1,1)(1,1)(1,-1)(1,-1)]. - Nicholas Ham, Aug 23 2015
Ham's claim is true for n=2. We proceed with a proof for n>2 by induction. On the j-th step, from (j-1,y) to (j,y'), there are j options for y': 0, 1, ..., y-1, y+1, ..., j. Thus there are n! possible paths from (0,0) to x=n that stay between y=0 and y=x. (Then the final step is determined.) However, because +(1,0) is not an allowable step, we cannot land on (n,0) on the n-th step. Therefore, the number of acceptable lattice paths is n! - a(n-1). - Danny Rorabaugh, Nov 30 2015

Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section B10, pp. 152-153.
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
Richard K. Guy, Letter to N. J. A. Sloane, Sep 25 1986.
Richard K. Guy, Letter to N. J. A. Sloane, 1987
Richard K. Guy, The strong law of small numbers. Amer. Math. Monthly 95 (1988), no. 8, 697-712.
Richard K. Guy, The strong law of small numbers. Amer. Math. Monthly 95 (1988), no. 8, 697-712. [Annotated scanned copy]
Hisanori Mishima, Factorizations of many number sequences: 103 and 130.
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.
Eric Wegrzynowski, Séries de factorielles.
Eric Weisstein's World of Mathematics, Alternating Factorial and Factorial.
Miodrag Živković, 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.

Cf. A000142, A001272, A003422, A071828, A303697, A343187, A359808.

List([0..30],n->Sum([1..n],i->(-1)^(n-i)*Factorial(i))); # Muniru A Asiru, Jun 01 2018

a005165 n = a005165_list !! n
a005165_list = 0 : zipWith (-) (tail a000142_list) a005165_list
-- Reinhard Zumkeller, Jul 21 2013

A005165 := proc(n) local i; add((-1)^(n-i)*i!,i=1..n); end;

nn=25;With[{fctrls=Range[nn]!},Table[Abs[Total[Times@@@Partition[ Riffle[ Take[ fctrls,n],{1,-1}],2]]],{n,nn}]] (* Harvey P. Dale, Dec 10 2011 *)
a[0] = 0; a[n_] := n! - a[n - 1]; Array[a, 26, 0] (* Robert G. Wilson v, Aug 06 2012 *)
RecurrenceTable[{a[n] == n! - a[n - 1], a[0] == 0}, a, {n, 0, 20}] (* Eric W. Weisstein, Jul 27 2017 *)
AlternatingFactorial[Range[0, 20]] (* Eric W. Weisstein, Jul 27 2017 *)
a[n_] = (-1)^n (Exp[1]((-1)^n Gamma[-1-n,1] Gamma[2+n] - ExpIntegralEi[-1]) - 1)
Table[a[n] // FullSimplify, {n, 0, 20}] (* Gerry Martens, May 22 2018 *)

a(n)=if(n<0,0,sum(k=0,n-1,(-1)^k*(n-k)!))

first(m)=vector(m,j,sum(i=0,j-1,((-1)^i)*(j-i)!)) \\ Anders Hellström, Aug 23 2015

a(n)=round((-1)^n*(exp(1)*(gamma(n+2)*incgam(-1-n,1)*(-1)^n +eint1(1))-1)) \\ Gerry Martens, May 22 2018

a = 0
f = 1
for n in range(1, 33):
    print(a, end=",")
    f *= n
    a = f - a
# Alex Ratushnyak, Aug 05 2012

a(0) = 0, a(n) = n! - a(n-1) for n > 0; also a(n) = n*a(n-2) + (n-1)*a(n-1) for n > 1. Sum_{n>=1} Pi^n/a(n) ~ 30.00005. - Gerald McGarvey, Jun 19 2004
E.g.f.: 1/(1-x) + exp(-x)*(e*(Ei(1,1)-Ei(1,1-x)) - 1). - Robert Israel, Dec 01 2015
a(n) = (-1)^n*(exp(1)*(gamma(n+2)*gamma(-1-n,1)*(-1)^n +Ei(1))-1). - Gerry Martens, May 22 2018
Sum_{n>=1} 1/a(n) = A343187. - Amiram Eldar, Jun 01 2023

A130308 Primes of the form [k!! - (k-1)!! + (k-2)!! - ... 1!!] - 1.

Original entry on oeis.org

5, 317, 1217216458429656137

Offset: 1

Paolo P. Lava and Giorgio Balzarotti, May 21 2007

The next term is too large to include.

The corresponding values of k are 4, 8, 32, 180, 264, 328, 1788, 2308, 5152, 7572, 13496, ... ; all these values are even since for k odd above 11 this form is divisible by 7. - Amiram Eldar, Jul 18 2019

			5 = 4!! - 3!! + 2!! - 1!! -1 = 8 - 3 + 2 - 1 - 1.

Amiram Eldar, Table of n, a(n) for n = 1..6

Cf. A071828, A130309.

P:=proc(n) local a,i,j,k,w; for i from 1 by 1 to n do a:=0; for j from i by -1 to 0 do k:=j; w:=j-2; while w>0 do k:=k*w; w:=w-2; od; a:=a+k*(-1)^j od; if isprime(abs(a)-1) then print(abs(a)-1); fi; od; end: P(1000);

f[n_] := Sum[(-1)^(n-k)*k!!, {k, 1, n}] - 1; Select[f/@Range[32], PrimeQ] (* Amiram Eldar, Jul 18 2019 *)

A130309 Primes of the form [k!! - (k-1)!! + (k-2)!! -....1!!] + 1.

Original entry on oeis.org

2, 3, 7, 67, 153979499670311863, 96139392052480758114443739387402080695373863

Offset: 1

Paolo P. Lava and Giorgio Balzarotti, May 21 2007

nonn

The next term is too large to include.

The corresponding values of k are 1, 2, 3, 4, 7, 31, 63, 263, 311, 371, 383, 10243, ... (1 and 2 give the same prime, 2). All these values except 2 and 4 are odd since for k even above 10 this form is divisible by 7. a(11) ~ 2.060 * 10^18317. - Amiram Eldar, Jul 18 2019

			2 = 1!! + 1 or 2!! - 1!! + 1.
7 = 4!! - 3!! + 2!! - 1!! +1 = 8 - 3 + 2 - 1 + 1.

Amiram Eldar, Table of n, a(n) for n = 1..10

Cf. A071828, A130308.

P:=proc(n) local a,i,j,k,w; for i from 1 by 1 to n do a:=0; for j from i by -1 to 0 do k:=j; w:=j-2; while w>0 do k:=k*w; w:=w-2; od; a:=a+k*(-1)^j od; if isprime(abs(a)+1) then print(abs(a)+1); fi; od; end: P(1000);

f[n_] := Sum[(-1)^(n-k)*k!!, {k, 1, n}] + 1; Select[f/@Range[2, 31], PrimeQ] (* Amiram Eldar, Jul 18 2019 *)

A361436 Primes of the form k! - Sum_{i=1..k-1} (-1)^(k-i)*i!.

Original entry on oeis.org

3, 7, 29, 139, 821, 5659, 44741, 515616581, 1389068025019, 2390389721955353653838200398484730341485707553165512827613149996957838364422981

Offset: 1

Jack Braxton, Mar 11 2023

Primes of the form k! + A005165(k - 1).

			139 is in the sequence because it is 5! + (4! - 3! + 2! - 1!).

Cf. A005165 (alternating factorials), A071828, A361437 (the k's).

\\ here b(n) is n! + A005165(n-1).
b(n) = {n! - sum(i=1, n-1, (-1)^(n-i)*i!)}
{ for(k=1, 150, if(ispseudoprime(b(k)), print1(b(k), ", "))) } \\ Andrew Howroyd, Mar 12 2023

A361437 Numbers k such that k! - Sum_{i=1..k-1} (-1)^(k-i)*i! is prime.

Original entry on oeis.org

2, 3, 4, 5, 6, 7, 8, 12, 15, 58, 59, 102, 111, 118, 164, 291, 589, 685, 1671, 1900, 1945, 4905, 9564

Offset: 1

Jack Braxton, Mar 11 2023

Numbers k such that k! + A005165(k - 1) is prime.

a(23) > 7000. - Hugo Pfoertner, Mar 15 2023

			2 is in the sequence because 2! + 1! = 3.
3 is in the sequence because 3! + (2! - 1!) = 7.
4 is in the sequence because 4! + (3! - 2! + 1!) = 29.
5 is in the sequence because 5! + (4! - 3! + 2! - 1!) = 139.

Cf. A361436 (the corresponding primes).

Cf. A001272, A005165 (alternating factorials), A071828.

isok(k) = isprime(k! + sum(i=1, k-1, (-1)^(i+1)*(k-i)!)); \\ Michel Marcus, Mar 12 2023

Missing a(10) inserted and a(12)-a(18) from Andrew Howroyd, Mar 12 2023
a(19)-a(22) from Hugo Pfoertner, Mar 13 2023
a(23) from Michael S. Branicky, Oct 02 2024

A119555 Primes in the sequence f(n) = f(n-1)+((-1)^n)*n!, with f(0)=0.

Original entry on oeis.org

19, 619, 35899, 3301819, 468544077492065936712052044718939948687543330546977719976017418129955876663406131164377030450551575840099843957105136480237871017419158043635450756712088769133544426722033165168878328322819566779381528981882285541609256481166622331374702000809600061055686236758821446539362161635577019

Offset: 1

Paolo P. Lava and Giorgio Balzarotti, May 30 2006

f(n) = (-1)^n*A005165(n). The primes are those terms in A071828 which correspond to even n values in A001272: n = 4, 6, 8, 10, 160, 4998, 9158, 11164 (the last three are only probable primes). 3612703 divides f(n) for n >= 3612702, so the sequence is finite. - Jens Kruse Andersen, Jul 04 2014

			f(0)=0, f(1) = 0+((-1)^1)*1! = -1, f(2) = -1+((-1)^2)*2! = 1, f(3) = 1+((-1)^3)*3! = -5, f(4) = -5+((-1)^4)*4! = 19, which is prime, so 19 is the first term of the sequence.

G. Balzarotti and P. P. Lava, Le sequenze di numeri interi, Hoepli, 2008, p. 160.

Cf. A005165, A071828, A001272.

P:=proc(n) local i,j; j:=0; for i from 1 by 1 to n do j:=j+((-1)^i)*i!; if isprime(j) then print(j); fi; od; end: P(100);

nxt[{n_,a_}]:={n+1,a+(-1)^(n+1) (n+1)!}; Select[NestList[nxt,{0,0},200][[All,2]],#>0&&PrimeQ[#]&] (* Harvey P. Dale, Jan 22 2017 *)

Offset changed to 1 (this is a list) from Bruno Berselli, Feb 16 2012

Formula in name corrected by Jens Kruse Andersen, Jul 04 2014

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A005165 Alternating factorials: n! - (n-1)! + (n-2)! - ... 1!.

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A130308 Primes of the form [k!! - (k-1)!! + (k-2)!! - ... 1!!] - 1.

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A130309 Primes of the form [k!! - (k-1)!! + (k-2)!! -....1!!] + 1.

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A361436 Primes of the form k! - Sum_{i=1..k-1} (-1)^(k-i)*i!.

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A361437 Numbers k such that k! - Sum_{i=1..k-1} (-1)^(k-i)*i! is prime.

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A119555 Primes in the sequence f(n) = f(n-1)+((-1)^n)*n!, with f(0)=0.

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A290045 Primes in A290044.

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