A000522 -id:A000522 - cp's OEIS frontend

A072456 Annihilating primes for A000522.

3, 7, 11, 17, 47, 53, 61, 67, 73, 79, 89, 101, 139, 151, 157, 191, 199, 229, 233, 241, 263, 269, 277, 283, 311, 317, 337, 347, 359, 367, 379, 397, 433, 449, 467, 487, 503, 521, 541, 563, 569, 571, 577, 593, 607, 613, 619, 647, 659, 673, 683, 691, 727, 743, 769, 773, 809, 823, 827, 911, 919, 929, 953, 971, 991

Offset: 1

N. J. A. Sloane, Aug 02 2002

nonn

Primes p such that A072453(p) = 0.

Amiram Eldar, Table of n, a(n) for n = 1..3000
C. Cobeli and A. Zaharescu, Promenade around Pascal Triangle-Number Motives, Bull. Math. Soc. Sci. Math. Roumanie, Tome 56(104) No. 1, 2013, pp. 73-98. - From _N. J. A. Sloane_, Feb 16 2013
Lorenz Halbeisen and Norbert Hungerbuehler, Number theoretic aspects of a combinatorial function, Notes on Number Theory and Discrete Mathematics 5 (1999) 138-150. (ps, pdf)
Wikipedia, Annihilating primes

Cf. A000522, A072453.

use warnings;
  use strict;
  use ntheory ":all";
  use Math::GMPz;
  use Memoize;  memoize 'a000522';
  sub a000522 {
    my($n, $sum, $fn) = (shift, 0, Math::GMPz->new(1));
    do {  $sum += $fn;  $fn *= ($n-$_);  } for 0 .. $n;
    $sum;
  }
  sub a072453 {
    my $n = shift;
    vecsum( map { a000522($_) % $n == 0 } 0 .. $n-1 );
  }
  forprimes { print "$\n" unless a072453($) } 1000;
# Dana Jacobsen, Feb 16 2016

More terms from Vladeta Jovovic, Aug 02 2002

Offset corrected by Amiram Eldar, May 15 2020

A124781 a(n) = gcd(A093101(n), A093101(n+2)) where A093101(n) = gcd(n!, A(n)) and A(n) = A000522(n) = Sum_{k=0..n} n!/k!.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 10, 1, 2, 1, 2, 5, 2, 1, 2, 1, 10, 1, 2, 1, 2, 5, 26, 1, 2, 1, 10, 1, 2, 1, 2, 5, 2, 1, 2, 13, 10, 1, 2, 1, 2, 5, 2, 1, 2, 1, 10, 1, 26, 1, 2, 5, 2, 1, 2, 1, 10, 1, 2, 1, 2, 65, 2, 1, 2, 1, 10, 1, 2, 1, 74, 5, 2, 1, 26, 1, 10, 1, 2, 1, 2, 5, 2, 1, 2, 1, 10, 13, 2, 1, 2, 5, 2, 1, 2, 1

Offset: 0

Jonathan Sondow, Nov 07 2006

nonn

a(n) divides n+3 because A(n+2) = (n+2)(n+1)*A(n) + n+3.

			a(3) = gcd(d(3),d(5)) = gcd(gcd(3!,16), gcd(5!,326)) = gcd(2,2) = 2.

Antti Karttunen, Table of n, a(n) for n = 0..4096
J. Sondow, A geometric proof that e is irrational and a new measure of its irrationality, Amer. Math. Monthly 113 (2006) 637-641.
J. Sondow, A geometric proof that e is irrational and a new measure of its irrationality, arXiv:0704.1282 [math.HO], 2007-2010.
J. Sondow and K. Schalm, Which partial sums of the Taylor series for e are convergents to e? (and a link to the primes 2, 5, 13, 37, 463), II, Gems in Experimental Mathematics (T. Amdeberhan, L. A. Medina, and V. H. Moll, eds.), Contemporary Mathematics, vol. 517, Amer. Math. Soc., Providence, RI, 2010.
Index entries for sequences related to factorial numbers

Cf. A000522, A093101, A123899, A123900, A123901, A124779, A124780, A124782.

(A[n_] := Sum[n!/k!, {k,0,n}]; d[n_] := GCD[n!,A[n]]; Table[GCD[d[n],d[n+2]], {n,0,100}])
(* Second program, faster: *)
Table[GCD @@ Map[GCD[#!, Floor[E*#!] - Boole[# == 0]] &, n + {0, 2}], {n, 0, 96}] (* Michael De Vlieger, Jul 12 2017 *)

A000522(n) = sum(k=0, n, binomial(n, k)*k!); \\ This function from Joerg Arndt, Dec 14 2014
A093101(n) = gcd(n!,A000522(n));
m1=m2=1; for(n=0,4096,m=m1; m1=m2; m2 = A093101(n+2); m124781 = gcd(m,m2); write("b093101.txt", n, " ", m); write("b124781.txt", n, " ", m124781); write("b123901.txt", n, " ", (n+3)/m124781)); \\ Antti Karttunen, Jul 12 2017

a(n) = gcd(A093101(n), A093101(n+2)) = (n+3)/A123901(n).

a(n) = gcd(A(n), A(n+2), n!) where A(n)=1+n+n(n-1)+...+n!. - Jonathan Sondow, Nov 13 2006

Replaced d(n) in the name with A093101(n). - Antti Karttunen, Jul 12 2017

A026243 a(n) = A000522(n) - 2.

Original entry on oeis.org

0, 3, 14, 63, 324, 1955, 13698, 109599, 986408, 9864099, 108505110, 1302061343, 16926797484, 236975164803, 3554627472074, 56874039553215, 966858672404688, 17403456103284419, 330665665962403998, 6613313319248079999, 138879579704209680020, 3055350753492612960483

Offset: 1

N. J. A. Sloane, based on a message from a correspondent who wishes to remain anonymous, Dec 21 2003

nonn

Number of operations of addition and multiplication needed to evaluate a determinant of order n by cofactor expansion.

			To calculate a determinant of order 3:
    |a b c|       |e f|       |d f|       |d e|
D = |d e f| = a * |h i| - b * |g i| + c * |g h| =
    |g h i|
   = a * (e*i - f*h) - b * (d*i - f*g) + c * (d*h - e*g).
There are 9 multiplications * and 5 additions (+ or -), so 14 operations and a(3) = 14. - _Bernard Schott_, Apr 21 2019

Alois P. Heinz, Table of n, a(n) for n = 1..170
C. Dubbs and D. Siegel, Computing determinants, College Math. J., 18 (1987), 48-49.
A. R. Pargeter, The vanishing coffee morning, Math. Gaz., 76 (1992), 386-387.
P. G. Sawtelle, The ubiquitous e, Math. Mag., 49 (1976), 244-245. [_N. J. A. Sloane_, Jan 29 2009]

Cf. A000522, A007526. Equals A033312 + A038156.

Cf. A001339.

a:= proc(n) a(n):= n*(a(n-1)+2)-1: end: a(1):= 0:
seq (a(n), n=1..30);  # Alois P. Heinz, May 25 2012

Table[E*Gamma[n+1, 1] - 2, {n, 1, 30}] (* Jean-François Alcover, May 18 2018 *)

a(n) = n*(a(n-1)+2)-1 for n>1, a(1) = 0. - Alois P. Heinz, May 25 2012
Conjecture: a(n) +(-n-2)*a(n-1) +(2*n-1)*a(n-2) +(-n+2)*a(n-3)=0. - R. J. Mathar, Jun 23 2013 [Confirmed by Altug Alkan, May 18 2018]
a(n) = floor(e*n!) - 2. - Bernard Schott, Apr 21 2019

A072453 Shadow transform of A000522.

Original entry on oeis.org

0, 1, 1, 0, 1, 2, 0, 0, 1, 0, 2, 0, 0, 3, 0, 0, 1, 0, 0, 2, 2, 0, 0, 2, 0, 2, 3, 0, 0, 1, 0, 2, 1, 0, 0, 0, 0, 3, 2, 0, 2, 1, 0, 1, 0, 0, 2, 0, 0, 0, 2, 0, 3, 0, 0, 0, 0, 0, 1, 1, 0, 0, 2, 0, 1, 6, 0, 0, 0, 0, 0, 2, 0, 0, 3, 0, 2, 0, 0, 0, 2, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 0, 2, 0, 0, 4, 0, 1, 0, 0, 2, 0, 0, 1, 3

Offset: 0

N. J. A. Sloane, Aug 02 2002

Alois P. Heinz, Table of n, a(n) for n = 0..10000
Lorenz Halbeisen and Norbert Hungerbuehler, Number theoretic aspects of a combinatorial function, Notes on Number Theory and Discrete Mathematics 5 (1999) 138-150. (ps, pdf); see Definition 7 for the shadow transform.
OEIS Wiki, Shadow transform.
N. J. A. Sloane, Transforms.

Cf. A000522, A072456.

A000522 := proc(n)
    add(n!/k!,k=0..n) ;
end proc:
shadD := proc(a)
    local s,n ;
    s := {} ;
    for n from 0 to a-1 do
        if A000522(n) mod a = 0 then
            s := s union {n} ;
        end if;
    end do:
    s ;
end proc:
A072453 := proc(a)
    nops(shadD(a)) ;
end proc: # R. J. Mathar, Jun 24 2013
# second Maple program:
b:= proc(n) option remember; n*b(n-1)+1 end: b(0):=1:
a:= n-> add(`if`(irem(b(j), n)=0, 1, 0), j=0..n-1):
seq(a(n), n=0..150);  # Alois P. Heinz, Jun 28 2018

b[n_] := b[n] = n*b[n - 1] + 1 ; b[0] = 1;
a[n_] := Sum[If[Mod[b[j], n] == 0, 1, 0], {j, 0, n - 1}];
a /@ Range[0, 104] (* Jean-François Alcover, Jan 15 2020, after Alois P. Heinz *)

More terms from Christian G. Bower, Jun 08 2005

A073591 a(n) = A000522(n) + 1.

Original entry on oeis.org

2, 3, 6, 17, 66, 327, 1958, 13701, 109602, 986411, 9864102, 108505113, 1302061346, 16926797487, 236975164806, 3554627472077, 56874039553218, 966858672404691, 17403456103284422, 330665665962404001, 6613313319248080002

Offset: 0

Vladeta Jovovic, Aug 28 2002

nonn

a(n) is an upper bound on the Ramsey numbers in A003323. - D. G. Rogers, Aug 27 2006

There is a nice derivation of the recurrence relation given in the Walker reference.

Alois P. Heinz, Table of n, a(n) for n = 0..200 (28 terms from Vincenzo Librandi)
R. C. Walker, A graph coloring theorem, Math. Gaz., 60 (1976), 54-57.

Cf. A000522, A001339, A003323, A007526.

a:= proc(n) a(n):= `if`(n=0, 2, n*a(n-1)-n+2) end:
seq(a(n), n=0..30);  # Alois P. Heinz, Feb 17 2014

f[n_] := n*(f[n - 1] - 1) + 2;f[0]=2; ff[n_]:=(1/(1+n))(1+E*Gamma[1+n, 1]-E*(n^2)*Gamma[1+n, 1]+E*n*Gamma[2+n, 1]) (Spindler)
Table[FunctionExpand[Gamma[n, 1] E] + 1, {n, 2, 29}] (* Vincenzo Librandi, Feb 17 2014 *)

Conjecture: a(n) +(-n-2)*a(n-1) +(2*n-1)*a(n-2) +(-n+2)*a(n-3)=0. - R. J. Mathar, Feb 16 2014

a(n) = n*(a(n-1) - 1) + 2. - Georg Fischer, Dec 24 2023 [from the Walker reference, p. 55]

A124780 a(n) = gcd(A(n), A(n+2)) where A(n) = A000522(n) = Sum_{k=0..n} n!/k!.

Original entry on oeis.org

1, 2, 5, 2, 1, 2, 1, 10, 1, 2, 13, 2, 5, 2, 1, 2, 1, 10, 1, 2, 1, 2, 5, 26, 1, 2, 1, 10, 1, 2, 1, 2, 5, 2, 37, 2, 13, 10, 1, 2, 1, 2, 5, 2, 1, 2, 1, 10, 1, 26, 1, 2, 5, 2, 1, 2, 1, 10, 1, 2, 1, 2, 65, 2, 1, 2, 1, 10, 1, 2, 1, 74, 5, 2, 1, 26, 1, 10, 1, 2, 1, 2, 5, 2, 1, 2, 1, 10, 13, 2, 1, 2, 5, 2, 1, 2, 1

Offset: 0

Jonathan Sondow, Nov 07 2006

nonn

a(n) divides n+3 because A(n+2) = (n+2)(n+1)*A(n) + n+3.

			a(2) = gcd(A(2), A(4)) = gcd(5, 65) = 5.

Antti Karttunen, Table of n, a(n) for n = 0..4096
J. Sondow, A geometric proof that e is irrational and a new measure of its irrationality, Amer. Math. Monthly 113 (2006) 637-641.
J. Sondow, A geometric proof that e is irrational and a new measure of its irrationality, arXiv:0704.1282 [math.HO], 2007-2010.
J. Sondow and K. Schalm, Which partial sums of the Taylor series for e are convergents to e? (and a link to the primes 2, 5, 13, 37, 463), II, Gems in Experimental Mathematics (T. Amdeberhan, L. A. Medina, and V. H. Moll, eds.), Contemporary Mathematics, vol. 517, Amer. Math. Soc., Providence, RI, 2010.
Index entries for sequences related to factorial numbers

Cf. A000522, A093101, A123899, A123900, A123901, A124779, A124781, A124782.

(A[n_] := Sum[n!/k!, {k, 0, n}]; Table[GCD[A[n],A[n+2]], {n,0,100}])
GCD[#[[1]],#[[3]]]&/@Partition[Table[Sum[n!/k!,{k,0,n}],{n,0,100}],3,1] (* Harvey P. Dale, Jun 14 2022 *)

A000522(n) = sum(k=0, n, binomial(n, k)*k!); \\ This function from Joerg Arndt, Dec 14 2014
A124780(n) = gcd(A000522(n),A000522(n+2)); \\ Antti Karttunen, Jul 07 2017

a(n) = gcd(A000522(n), A000522(n+2)) = (n+3)/A124782(n)

A124782 a(n) = (n+3)/gcd(A(n), A(n+2)) where A(n) = A000522(n) = Sum_{k=0..n} n!/k!.

Original entry on oeis.org

3, 2, 1, 3, 7, 4, 9, 1, 11, 6, 1, 7, 3, 8, 17, 9, 19, 2, 21, 11, 23, 12, 5, 1, 27, 14, 29, 3, 31, 16, 33, 17, 7, 18, 1, 19, 3, 4, 41, 21, 43, 22, 9, 23, 47, 24, 49, 5, 51, 2, 53, 27, 11, 28, 57, 29, 59, 6, 61, 31, 63, 32, 1, 33, 67, 34, 69, 7, 71, 36, 73, 1, 15, 38, 77, 3, 79, 8, 81, 41

Offset: 0

Jonathan Sondow, Nov 07 2006

nonn

a(n) is an integer since A(n+2) = (n+2)(n+1)*A(n) + n+3.

			a(3) = (3+3)/gcd(A(3), A(5)) = 6/gcd(16, 326) = 6/2 = 3.

Antti Karttunen, Table of n, a(n) for n = 0..4096
J. Sondow, A geometric proof that e is irrational and a new measure of its irrationality, Amer. Math. Monthly 113 (2006) 637-641.
J. Sondow, A geometric proof that e is irrational and a new measure of its irrationality, [math.HO], 2007-2010.
J. Sondow and K. Schalm, Which partial sums of the Taylor series for e are convergents to e? (and a link to the primes 2, 5, 13, 37, 463), II, Gems in Experimental Mathematics (T. Amdeberhan, L. A. Medina, and V. H. Moll, eds.), Contemporary Mathematics, vol. 517, Amer. Math. Soc., Providence, RI, 2010.
Index entries for sequences related to factorial numbers

Cf. A000522, A093101, A123899, A123900, A123901, A124779, A124780, A124781.

(A[n_] := Sum[n!/k!, {k,0,n}]; Table[(n+3)/GCD[A[n],A[n+2]], {n,0,80}])

A000522(n) = sum(k=0, n, binomial(n, k)*k!); \\ This function from Joerg Arndt, Dec 14 2014
A124780(n) = gcd(A000522(n),A000522(n+2));
A124782(n) = ((n+3)/A124780(n)); \\ Antti Karttunen, Jul 07 2017

a(n) = (n+3)/A124780(n) = (n+3)/gcd(A000522(n), A000522(n+2)).

A101799 a(n)= det[A000522(i+j+1)], i,j=0...n, is the Hankel determinant of order n+1 of the arrangements numbers, s. A000522; a(n) = product( (p!)^2,p=0..n )(n+1)!LaguerreL(n+1,0,-1), n=0,1..., where LaguerreL(n,lambda,x) are generalized Laguerre polynomials; a(n)=A055209(n)*A002720(n+1);.

Original entry on oeis.org

2, 7, 136, 30096, 128231424, 15917683507200, 81063451589345280000, 22675515428700722036736000000, 449302248871829829537656890982400000000, 790103237429135552913731284331032467210240000000000

Offset: 0

Karol A. Penson, Dec 16 2004

nonn

Cf. A000522, A055209, A002720.

a[n_] := Det[Table[E*Gamma[i+j+2, 1] // FunctionExpand, {i, 0, n}, {j, 0, n}]];
Table[a[n], {n, 0, 9}] (* Jean-François Alcover, May 23 2016 *)
Table[BarnesG[n+2]^2 * (n+1)! * LaguerreL[n+1, 0, -1], {n, 0, 12}] (* Vaclav Kotesovec, May 11 2021 *)

a(n) ~ 2^(n+1/2) * Pi^(n+1) * n^(n^2 + 3*n + 25/12) / (A^2 * exp(3*n^2/2 + 3*n - 2*sqrt(n) + 1/3)), where A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, May 11 2021

A121726 Sum sequence A000522 then subtract 0,1,2,3,4,5,...

Original entry on oeis.org

1, 2, 6, 21, 85, 410, 2366, 16065, 125665, 1112074, 10976174, 119481285, 1421542629, 18348340114, 255323504918, 3809950976993, 60683990530209, 1027542662934898, 18430998766219318, 349096664728623317, 6962409983976703317, 145841989688186383338, 3201192743180799343822

Offset: 1

Alford Arnold, Aug 17 2006

Let aut(p) denote the size of the centralizer of the partition p (see A339016 for the definition). Then a(n) = Sum_{p in P} n!/aut(p), where P are the partitions of n with largest part k and length n + 1 - k. - Peter Luschny, Nov 19 2020

			A000522 begins     1 2 5 16 65 326 ...
with sums          1 3 8 24 89 415 ...
so sequence begins 1 2 6 21 85 410 ...
.
From _Peter Luschny_, Nov 19 2020: (Start):
The combinatorial interpretation is illustrated by this computation of a(5):
5! / aut([5])             = 120 / A339033(5, 1) = 120/5   = 24
5! / aut([4, 1])          = 120 / A339033(5, 2) = 120/4   = 30
5! / aut([3, 1, 1])       = 120 / A339033(5, 3) = 120/6   = 20
5! / aut([2, 1, 1, 1])    = 120 / A339033(5, 4) = 120/12  = 10
5! / aut([1, 1, 1, 1, 1]) = 120 / A339033(5, 5) = 120/120 =  1
--------------------------------------------------------------
                                                Sum: a(5) = 85
(End)

Also the row sums of A092271.

Cf. A000522, A006231, A002104, A339016, A339033.

f[list_] :=Total[list]!/Apply[Times, list]/Apply[Times, Map[Length, Split[list]]!]; Table[Total[Map[f, Select[Partitions[n], Count[#, Except[1]] == 1 &]]] + 1, {n, 1, 20}] (* Geoffrey Critzer, Nov 07 2015 *)

A000522(n)={ return( sum(k=0,n,n!/k!)) ; } A121726(n)={ return(sum(k=0,n-1,A000522(k))-n+1) ; } { for(n=1,25, print1(A121726(n),",") ; ) ; } \\ R. J. Mathar, Sep 02 2006

def A121726(n):
    def h(n, k):
        if n == k: return 1
        return factorial(n)//((n + 1 - k)*factorial(k - 1))
    return sum(h(n, k) for k in (1..n))
print([A121726(n) for n in (1..23)])
# Demonstrates the combinatorial view:
def A121726(n):
    if n == 0: return 1
    f = factorial(n); S = 0
    for k in (0..n):
        for p in Partitions(n, max_part=k, inner=[k], length=n+1-k):
            S += (f // p.aut())
    return S
print([A121726(n) for n in (1..23)]) # Peter Luschny, Nov 20 2020

a(n) = A006231(n) + 1 = A002104(n) - (n-1). - Franklin T. Adams-Watters, Aug 29 2006

E.g.f.: exp(x)*(log(1/(1-x)) - x + 1). - Geoffrey Critzer, Nov 07 2015

More terms from Franklin T. Adams-Watters, Aug 29 2006

More terms from R. J. Mathar, Sep 02 2006

A138761 a(n) is the smallest member of A000522 divisible by 2^n, where A000522(m) = total number of arrangements of a set with m elements.

Original entry on oeis.org

1, 2, 16, 16, 16, 330665665962404000, 4216377920843140187197325631474390438452208808916276571342090223552

Offset: 0

Jonathan Sondow, Apr 01 2008

nonn

a(n) < A000522(2^n) for n > 0; see Sondow and Schalm, Proposition A.13 part (ii).

			a(5) = A000522(19) = 330665665962404000 because that is the smallest member of A000522 divisible by 2^5.

J. Sondow and K. Schalm, Which partial sums of the Taylor series for e are convergents to e? (and a link to the primes 2, 5, 13, 37, 463), Gems in Experimental Mathematics (T. Amdeberhan, L. A. Medina, and V. H. Moll, eds.), Contemporary Mathematics, vol. 517, Amer. Math. Soc., Providence, RI, 2010.

Jean-François Alcover, Table of n, a(n) for n = 0..9
J. Sondow and K. Schalm, Which partial sums of the Taylor series for e are convergents to e? (and a link to the primes 2, 5, 13, 37, 463), II, arXiv:0709.0671 [math.NT], 2007-2009.
Index entries for sequences related to factorial numbers

Cf. A000522, A127014, A127015.

a522[n_] := E Gamma[n + 1, 1];
(* b = A127014 *)
b[1] = 1; b[n_] := b[n] = For[k = b[n - 1], True, k++, If[Mod[a522[k], 2^n] == 0, Return[k]]];
a[0] = 1; a[n_] := a522[b[n]];
Table[a[n], {n, 0, 6}] (* Jean-François Alcover, Feb 20 2019 *)

a(n) = A000522(A127014(n)) = Sum_{k=0..A127014(n)} A127014(n)!/k! for n > 0.

cp's OEIS Frontend

A072456 Annihilating primes for A000522.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Perl

Extensions

A124781 a(n) = gcd(A093101(n), A093101(n+2)) where A093101(n) = gcd(n!, A(n)) and A(n) = A000522(n) = Sum_{k=0..n} n!/k!.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A026243 a(n) = A000522(n) - 2.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A072453 Shadow transform of A000522.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

Extensions

A073591 a(n) = A000522(n) + 1.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A124780 a(n) = gcd(A(n), A(n+2)) where A(n) = A000522(n) = Sum_{k=0..n} n!/k!.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A124782 a(n) = (n+3)/gcd(A(n), A(n+2)) where A(n) = A000522(n) = Sum_{k=0..n} n!/k!.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A101799 a(n)= det[A000522(i+j+1)], i,j=0...n, is the Hankel determinant of order n+1 of the arrangements numbers, s. A000522; a(n) = product( (p!)^2,p=0..n )(n+1)!LaguerreL(n+1,0,-1), n=0,1..., where LaguerreL(n,lambda,x) are generalized Laguerre polynomials; a(n)=A055209(n)*A002720(n+1);.