A003422 -id:A003422 - cp's OEIS frontend

A014288 a(n) = floor(Sum_{k=0..n} k!/2), or floor( A003422(n+1)/2 ).

0, 1, 2, 5, 17, 77, 437, 2957, 23117, 204557, 2018957, 21977357, 261478157, 3374988557, 46964134157, 700801318157, 11162196262157, 189005910310157, 3390192763174157, 64212742967590157, 1280663747055910157, 26826134832910630157, 588826498721714470157

Offset: 0

N. J. A. Sloane

nonn

The first term a(0) would be a fraction if the floor( ... ) function were omitted; for n >= 2, all terms from A003422 are even. - M. F. Hasler, Dec 16 2007

Alois P. Heinz, Table of n, a(n) for n = 0..200

Cf. A003422, A007489, A067078.

[Floor((&+[Factorial(j): j in [0..n]])/2): n in [0..30]]; // G. C. Greubel, Sep 05 2022

a:= proc(n) a(n):= `if`(n<3, n, (n+1)*a(n-1)-n*a(n-2)) end:
seq(a(n), n=0..25);  # Alois P. Heinz, Feb 01 2013

f[x_] := {Floor[1 + (n - x[[2]])*x[[1]]], x[[2]] + 1};
a[0] = 0; a[n_] := Nest[f, {1, 0}, n][[1]]/2 (* Joseph E. Cooper III (easonrevant(AT)gmail.com), Aug 19 2008 *) (* updated by Jean-François Alcover, Jun 01 2015 *)
a[n_]:=-(1/2) Subfactorial[-1]-1/2(-1)^n Gamma[2+n] Subfactorial[-2-n]; Table[a[n] //FullSimplify,{n,0,25}] (* Gerry Martens, May 29 2015 *)

A014288(n)=sum(k=0,n,k!)>>1 \\ M. F. Hasler, Dec 16 2007

from math import factorial
def A014288(n): return sum(factorial(k) for k in range(n+1))>>1 # Chai Wah Wu, Nov 01 2023

[sum(factorial(j) for j in (0..n))//2 for n in (0..30)] # G. C. Greubel, Sep 05 2022

a(0)=0, a(1)=1, a(2)=2, a(n) = (n+1)*a(n-1) - n*a(n-2). - Benoit Cloitre, Sep 07 2002
a(0) = 0, a(n) = (1/2)*floor(1 + 1*floor(1 + 2*floor(1 + ... + (n-1)*floor(1+n*floor(1))). - Joseph E. Cooper III (easonrevant(AT)gmail.com), Aug 19 2008
G.f.: G(0)/(1-x)/2 -1/2, where G(k)= 1 + (2*k + 1)*x/( 1 - 2*x*(k+1)/(2*x*(k+1) + 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 24 2013
G.f.: A(x) = (Sum_{n>=0} x^n*n!)/(2-2*x) - 1/2 = G(0)/(4*(1-x)) - 1/2, where G(k) = 1 + 1/(1 - x/(x + 1/(k+1)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 02 2013
a(n) ~ n!/2. - Vaclav Kotesovec, Aug 10 2013
E.g.f.: -1/2 + (exp(x)/2)*Sum_{k>=0} (k! - k*Gamma(k,x)). - Robert Israel, Jun 01 2015
a(n) = ((n+1)!*ExpIntegral(n+2,-1)+Ei(1)+Pi*i)/(2*e). - Ammar Khatab, Aug 14 2020

Edited by M. F. Hasler, Dec 16 2007

A025016 Final digits of !n = Sum_{i=0..n} i! (A003422) for very large n, read from right.

Original entry on oeis.org

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

Offset: 0

David W. Wilson

Reversed digits of 10-adic sum of all factorials.

More generally, the 10-adic sum: B(n) = Sum_{k>=0} k^n*k! is given by: B(n) = A014182(n)*B(0) + A014619(n) for n>=0, where B(0) is the 10-adic sum of factorials (this constant). - Paul D. Hanna, Aug 12 2006

			!20 = 256132749111820314, !30 = 16158688114800553828940314 ... .

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Index entries for sequences related to final digits of numbers

Cf. A000142, A003422, A014182, A014619, A065356.

a[n_] := Module[{x, f=1}, While[Mod[f!, 10^(n+1)]>0, f += 1]; x = Sum[ Mod[k!, 10^(n+1)], {k, 0, f}]; Quotient[10*Mod[x, 10^(n+1)], 10^(n+1)]]; Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Nov 18 2015, after Paul D. Hanna *)

{a(n)=local(x,f=1);while(f!%10^(n+1)>0,f+=1); x=sum(k=0,f,k!%10^(n+1));(10*(x%10^(n+1)))\10^(n+1)} \\ Paul D. Hanna, Aug 12 2006

A101752 Table (read by rows) giving the coefficients of sum formulas of n-th Left factorials (A003422).

Original entry on oeis.org

1, 0, 1, 5, -16, 8, 69, -767, 1314, 117, 1774, -30405, 78914, 69024

Offset: 1

André F. Labossière, Dec 17 2004

The k-th row (k>=1) contains T(i,k) for i=1 to k+1, where k=[2*n+1+(-1)^(n-1)]/4 and T(i,k) satisfies !n = Sum_{i=1..k+1} T(i,k) * n^(k-i+1) / k!.

			!7 = 874; substituting n=7 in the formula of the k-th row we obtain k=4 and the coefficients T(i,4) will be the following: 117,1774,-30405,78914,69024, => !7 = [ 117*7^4 +1774*7^3 -30405*7^2 +78914*7 +69024 ]/4! = 874.

Cf. A094216.

A102411 Even triangle !n. This table read by rows gives the coefficients of sum formulas of n-th Left factorials (A003422). The k-th row (6>=k>=1) contains T(i,k) for i=1 to k+2, where k=[2n+1+(-1)^(n-1)]/4 and T(i,k) satisfies !n = Sum_{i=1..k+2} T(i,k) n^(i-1) / (2*k-2)!.

Original entry on oeis.org

0, 1, 0, -16, 5, 1, 0, 5256, -3068, 276, 32, 0, 2070720, 2367420, -912150, 53220, 3510, 0, -36031524480, 15327895296, -40587120, -387492840, 21414120, 758184, 840, -212459319878400, -75473246681280, 38182549456800, -2562251680800, -195611371200, 13639812480, 285616800, 453600

Offset: 1

André F. Labossière, Jan 07 2005

The sum of signed coefficients for each k-th row is divisible by (2*k-2)!. Moreover, another variant (but an incomplete one, and sorted differently) of the above sequence is presented in A101752.

			Triangle starts:
0, 1, 0;
-16, 5, 1, 0;
5256, -3068, 276, 32, 0;
2070720, 2367420, -912150, 53220, 3510, 0;
-36031524480, 15327895296, -40587120, -387492840, 21414120, 758184, 840;
...
!11=4037914; substituting n=11 in the formula of the k-th row we obtain k=6 and the coefficients T(i,6) are those needed for computing !11.
=> !11 = [ -212459319878400 -75473246681280*11 +38182549456800*11^2 -2562251680800*11^3 -195611371200*11^4 +13639812480*11^5 +285616800*11^6 +453600*11^7 ]/10! = 4037914.

Cf. A102412, A094638, A094216, A003422, A008276, A101752, A102409, A102410, A101751, A000142, A101559, A101032, A099731.

A102412 Odd triangle !n. This table read by rows gives the coefficients of sum formulas of n-th Left factorials (A003422). The k-th row (6>=k>=1) contains T(i,k) for i=1 to k+1, where k=[2n+3+(-1)^n]/4 and T(i,k) satisfies !n = Sum_{i=1..k+1} T(i,k) n^(i-1) / (2*k-2)!.

Original entry on oeis.org

0, 1, -4, 4, 0, 96, -396, 108, 0, 1012320, -192900, -64890, 11460, 90, -2038014720, 1977810240, -304486560, -12131280, 2792160, 21840, -33190735737600, 4445760574080, 2334485260800, -394554283200, 2330344800, 1198048320, 8215200

Offset: 1

André F. Labossière, Jan 07 2005

Incidentally, the sum of signed coefficients for each k-th row is divisible by (2*k-2)!.

			Triangle starts:
0, 1;
-4, 4, 0;
96, -396, 108, 0;
1012320, -192900, -64890, 11460, 90;
-2038014720, 1977810240, -304486560, -12131280, 2792160, 21840;
...
!11=4037914; substituting n=11 in the formula of the k-th row we obtain k=6 and the coefficients T(i,6) are those needed for computing !11.
=> !11 = [ -33190735737600 +4445760574080*11 +2334485260800*11^2 -394554283200*11^3 +2330344800*11^4 +1198048320*11^5 +8215200*11^6 ]/10! = 4037914.

Cf. A102411, A094638, A094216, A003422, A008276, A101752, A102409, A102410, A101751, A000142, A101559, A101032, A099731.

A275608 Numbers that divide no nonzero terms of A003422.

Original entry on oeis.org

3, 6, 8, 9, 12, 13, 15, 16, 18, 20, 21, 24, 25, 26, 27, 28, 29, 30, 32, 33, 35, 36, 39, 40, 42, 43, 44, 45, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 63, 64, 65, 66, 67, 68, 69, 70, 72, 75, 76, 78, 79, 80, 81, 83, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100

Offset: 1

Robert Israel, Nov 14 2016

nonn

Numbers k such that A013584(k) = 0.

If k is in the sequence, then so is every multiple of k.

			3 is in the sequence because A003422(1)=1 and A003422(2)=2 are not divisible by 3, and A003422(k) == 1 (mod 3) for k >= 3.
4 is not in the sequence because A003422(3) = 4 is divisible by 4.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A003422, A013584.

Complement of A049045.

filter:= proc(n) local t,r,m;
  r:= 1; t:= 1;
  for m from 1 do
    r:= r*m mod n;
    if r = 0 then return true fi;
    t:= t + r mod n;
    if t = 0 then return false fi;
  od;
end proc:
select(filter, [$2..100]);

okQ[n_] := Module[{t, r, m}, r = 1; t = 1; For[m = 1, True, m++, r = Mod[r*m, n]; If[r == 0, Return[True]]; t = Mod[t + r, n]; If[t == 0, Return[False]]]];
Select[Range[2, 100], okQ] (* Jean-François Alcover, Apr 12 2019, after Robert Israel *)

A049041 Least k > 0 such that A049042(n) | A003422(k-1).

Original entry on oeis.org

2, 4, 6, 6, 5, 7, 7, 12, 22, 16, 55, 54, 42, 24, 25, 86, 97, 133, 64, 94, 72, 58, 49, 69, 19, 78, 14, 208, 167, 138, 80, 59, 63, 142, 41, 110, 22, 286, 39, 84, 215, 80, 14, 305, 188, 151, 53, 187, 180, 44, 32, 83, 92, 300, 16, 421, 146, 507, 28, 243, 119, 429, 239, 415

Offset: 1

David W. Wilson

nonn

F. Smarandache, Definitions solved and unsolved problems, conjectures and theorems in number theory and geometry, edited by M. Perez, Xiquan Publishing House, 2000.

F. Smarandache, Definitions, Solved and Unsolved Problems, Conjectures, ...
Eric Weisstein's World of Mathematics, Smarandache-Kurepa Function.

Cf. A049042, A049043, A003422.

A049044 Least k > 0 such that A049045(n) | A003422(k-1).

Original entry on oeis.org

1, 2, 3, 4, 6, 4, 6, 6, 5, 7, 6, 7, 12, 5, 22, 7, 16, 7, 55, 12, 54, 42, 22, 6, 16, 24, 25, 86, 55, 97, 133, 54, 42, 6, 64, 94, 72, 58, 24, 49, 69, 19, 25, 78, 86, 14, 208, 167, 138, 80, 97, 59, 133, 63, 142, 41, 110, 64, 22, 94, 286, 72, 39, 58, 84, 215, 80, 14, 49, 305, 69

Offset: 1

David W. Wilson

nonn

Cf. A049041, A049045, A049046, A003422.

A102639 Combinatorial triangle !n. This table read by rows gives the coefficients of general sum formulas of n-th left factorials (A003422). The k-th row (k>=1) contains T(i,k) for i=1 to 2k and k=1 to n-2, where T(i,k) satisfies !n = n + Sum_{k=1..n-2} Sum_{i=1..2k} T(i,k) * C(n-k-1,i).

Original entry on oeis.org

1, 1, 3, 8, 8, 3, 9, 46, 101, 114, 65, 15, 33, 272, 975, 1935, 2289, 1615, 630, 105, 153, 1796, 9175, 26795, 49474, 60080, 48104, 24535, 7245, 945, 873, 13424, 90255, 353507, 902164, 1582455, 1953272, 1700860, 1025927, 408870, 97020, 10395, 5913

Offset: 1

André F. Labossière, Feb 01 2005

The coefficients T(i,k) along the i-th columns of the triangle are the consecutive partial sums of those found in table A094216.

			!7 = 7 + 1*C(7-2,1) + 1*C(7-2,2) + 3*C(7-3,1) + ... + 33*C(7-5,1) + 272*C(7-5,2) + 153*C(7-6,1) = 7 + 5 + 10 + 12 + 8*C(4,2) + 8*C(4,3) + 3*C(4,4) + 9*C(3,1) + 46*C(3,2) + 101*C(3,3) + 66 + 272 + 153 = 7 + 5 + 10 + 12 + 48 + 32 + 3 + 27 + 138 + 101 + 66 + 272 + 153 = 874.

Chris Zheng, Jeffrey Zheng, Triangular Numbers and Their Inherent Properties, Variant Construction from Theoretical Foundation to Applications, Springer, Singapore, 51-65.

Cf. A094216, A102411, A102412, A101752, A003422, A094638, A008276.

A124375 Numbers k such that A003422(k+1)/2 is prime.

Original entry on oeis.org

2, 3, 4, 7, 8, 9, 10, 29, 75, 162, 270, 272, 353, 720, 1795, 3732, 4768, 9315, 12220, 41531

Offset: 1

Alexander Adamchuk, Oct 28 2006

Sum_{i=0..k} i! = k! + !k = A003422(k+1), where !k is left factorial !k = Sum_{i=0..k-1} i! = A003422(k). Left factorials are even for k > 1. Corresponding primes of the form (k! + !k)/2 = (a(n)! + !a(n))/2 are listed in A124374(n) = {2, 5, 17, 2957, 23117, 204557, 2018957, 4578979328975537786697650470157, ...}.

A near-duplicate of A100614: a(n) = A100614(n) - 1. - Ryan Propper, Feb 07 2008

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

Cf. A003422, A124374.

f=0;Do[f=f+n!;If[PrimeQ[f/2],Print[{n,f/2}]],{n,0,353}]
Flatten[Position[Accumulate[(Range[0,12220]!)]/2,?PrimeQ]]-1 (* _Harvey P. Dale, Jul 02 2019 *)

More terms from Ryan Propper, Feb 07 2008

a(20) from Jinyuan Wang, Mar 20 2021

cp's OEIS Frontend

A014288 a(n) = floor(Sum_{k=0..n} k!/2), or floor( A003422(n+1)/2 ).

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A025016 Final digits of !n = Sum_{i=0..n} i! (A003422) for very large n, read from right.

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A101752 Table (read by rows) giving the coefficients of sum formulas of n-th Left factorials (A003422).

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A102411 Even triangle !n. This table read by rows gives the coefficients of sum formulas of n-th Left factorials (A003422). The k-th row (6>=k>=1) contains T(i,k) for i=1 to k+2, where k=[2*n+1+(-1)^(n-1)]/4 and T(i,k) satisfies !n = Sum_{i=1..k+2} T(i,k) * n^(i-1) / (2*k-2)!.

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A102412 Odd triangle !n. This table read by rows gives the coefficients of sum formulas of n-th Left factorials (A003422). The k-th row (6>=k>=1) contains T(i,k) for i=1 to k+1, where k=[2*n+3+(-1)^n]/4 and T(i,k) satisfies !n = Sum_{i=1..k+1} T(i,k) * n^(i-1) / (2*k-2)!.

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A275608 Numbers that divide no nonzero terms of A003422.

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A049041 Least k > 0 such that A049042(n) | A003422(k-1).

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A049044 Least k > 0 such that A049045(n) | A003422(k-1).

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A102639 Combinatorial triangle !n. This table read by rows gives the coefficients of general sum formulas of n-th left factorials (A003422). The k-th row (k>=1) contains T(i,k) for i=1 to 2*k and k=1 to n-2, where T(i,k) satisfies !n = n + Sum_{k=1..n-2} Sum_{i=1..2*k} T(i,k) * C(n-k-1,i).

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A102411 Even triangle !n. This table read by rows gives the coefficients of sum formulas of n-th Left factorials (A003422). The k-th row (6>=k>=1) contains T(i,k) for i=1 to k+2, where k=[2n+1+(-1)^(n-1)]/4 and T(i,k) satisfies !n = Sum_{i=1..k+2} T(i,k) n^(i-1) / (2*k-2)!.

A102412 Odd triangle !n. This table read by rows gives the coefficients of sum formulas of n-th Left factorials (A003422). The k-th row (6>=k>=1) contains T(i,k) for i=1 to k+1, where k=[2n+3+(-1)^n]/4 and T(i,k) satisfies !n = Sum_{i=1..k+1} T(i,k) n^(i-1) / (2*k-2)!.

A102639 Combinatorial triangle !n. This table read by rows gives the coefficients of general sum formulas of n-th left factorials (A003422). The k-th row (k>=1) contains T(i,k) for i=1 to 2k and k=1 to n-2, where T(i,k) satisfies !n = n + Sum_{k=1..n-2} Sum_{i=1..2k} T(i,k) * C(n-k-1,i).