A052898 -id:A052898 - cp's OEIS frontend

A020543 a(0) = 1, a(1) = 1, a(n+1) = (n+1)*a(n) + n.

1, 1, 3, 11, 47, 239, 1439, 10079, 80639, 725759, 7257599, 79833599, 958003199, 12454041599, 174356582399, 2615348735999, 41845579775999, 711374856191999, 12804747411455999, 243290200817663999, 4865804016353279999

Offset: 0

Simon Plouffe

nonn

First Bernoulli polynomial evaluated at x=n! and multiplied by 2.
From Jaroslav Krizek, Jan 23 2010: (Start)
a(0) = 1, for n >= 1: a(n) = numbers m for which there is one iteration {floor(r/k)} for k = n, n-1, n-2, ... 1 with property r mod k = k-1 starting at r = m.
For n = 5: a(5) = 239;
floor(239/5) = 47, 239 mod 5 = 4;
floor( 47/4) = 11,  47 mod 4 = 3;
floor( 11/3) =  3,  11 mod 3 = 2;
floor(  3/2) =  1,   3 mod 2 = 1;
floor(  1/1) =  1,   1 mod 1 = 0. (End)
With offset 1, is the eigensequence of a triangle with the natural numbers (1, 2, 3, ...) as the right border, (1, 1, 2, 3, 4, ...) as the left border; and the rest zeros. - Gary W. Adamson, Aug 01 2016

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Daniel Birmajer, Juan B. Gil, David S. Kenepp, and Michael D. Weiner, Restricted generating trees for weak orderings, arXiv:2108.04302 [math.CO], 2021.
Index entries for sequences related to Bernoulli numbers.

Cf. A052898(n) - 2.

Cf. sequences of the type k*n!-1: A033312 (k=1), this sequence, A173323 (k=3), A173321 (k=4), A173317 (k=5), A173316 (k=6).

[2*Factorial(n)-1: n in [0..25]]; // Vincenzo Librandi, Sep 30 2013

lst={1};s=1;Do[s+=(n+=s*n);AppendTo[lst, s], {n, 0, 5!}];lst (* Vladimir Joseph Stephan Orlovsky, Oct 10 2008 *)
FoldList[#1*#2 + #2 - 1 &, 1, Range[20]] (* Robert G. Wilson v, Jul 07 2012 *)
Table[2 n! - 1, {n, 0, 25}] (* Vincenzo Librandi, Sep 30 2013 *)

E.g.f.: (-2 + exp(x) - x*exp(x))/(1-x). - Ralf Stephan, Feb 18 2004
a(n) = 2*n! - 1. - Gary W. Adamson, Jan 07 2008
a(0) = a(1) = 1, a(n) = a(n-1) * n + (n-1) for n >= 2. - Jaroslav Krizek, Jan 23 2010
a(n) ~ 2*sqrt(2*Pi*n)*n^n/exp(n). - Ilya Gutkovskiy, Aug 02 2016

Better description from Benoit Cloitre, Dec 29 2001

A173314 a(n) = 6*n!+1.

Original entry on oeis.org

7, 7, 13, 37, 145, 721, 4321, 30241, 241921, 2177281, 21772801, 239500801, 2874009601, 37362124801, 523069747201, 7846046208001, 125536739328001, 2134124568576001, 38414242234368001, 729870602452992001

Offset: 0

Vincenzo Librandi, Feb 16 2010

nonn

			For n=0, a(0)=7; n=1,a(1)=7; n=2, a(2)=13; n=3, a(3)=37; n=4, a(4)=145.

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. sequences of the type k*n!+1: A038507 (k=1), A052898 (k=2), A173324 (k=3), A173322 (k=4), A173319 (k=5), this sequence (k=6).

[6*Factorial(n) + 1: n in [0..25]]; // Vincenzo Librandi, Sep 29 2013

[7] cat [n eq 1 select n+6 else n*Self(n-1)-n+1: n in [1..25] ]; // Vincenzo Librandi, Sep 29 2013

Table[6 n! + 1, {n, 0, 25}] (* Vincenzo Librandi, Sep 29 2013 *)

a(0)=7, a(n) = n*a(n-1)-n+1. - Vincenzo Librandi, Sep 29 2013

A173322 a(n) = 4*n! + 1.

Original entry on oeis.org

5, 5, 9, 25, 97, 481, 2881, 20161, 161281, 1451521, 14515201, 159667201, 1916006401, 24908083201, 348713164801, 5230697472001, 83691159552001, 1422749712384001, 25609494822912001, 486580401635328001, 9731608032706560001, 204363768686837760001

Offset: 0

Vincenzo Librandi, Feb 16 2010

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. sequences of the type k*n!+1: A038507 (k=1), A052898 (k=2), A173324 (k=3), this sequence, A173319 (k=5), A173314 (k=6).

[4*Factorial(n) + 1: n in [0..25]]; // Vincenzo Librandi, Sep 29 2013

[5] cat [n eq 1 select n+4 else n*Self(n-1)-n+1: n in [1..25] ]; // Vincenzo Librandi, Sep 29 2013

a:= proc(n) if n=0 then 5 else a(n) := n*a(n-1)-n+1 fi end: seq (a(n), n=0..25);  # Sergei N. Gladkovskii, Jul 04 2012

4*Range[0,20]!+1 (* Harvey P. Dale, Jun 26 2012 *)
Table[4 n! + 1, {n, 0, 21}] (* Vincenzo Librandi, Sep 29 2013 *)

a(0) = 5, a(n) = n*a(n-1)-n+1. - Sergei N. Gladkovskii, Jul 04 2012

A173324 a(n) = 3*n! + 1.

Original entry on oeis.org

4, 4, 7, 19, 73, 361, 2161, 15121, 120961, 1088641, 10886401, 119750401, 1437004801, 18681062401, 261534873601, 3923023104001, 62768369664001, 1067062284288001, 19207121117184001, 364935301226496001, 7298706024529920001, 153272826515128320001

Offset: 0

Vincenzo Librandi, Feb 16 2010

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. sequences of the type k*n!+1: A038507 (k=1), A052898 (k=2), this sequence, A173322 (k=4), A173319 (k=5), A173314 (k=6).

[3*Factorial(n) + 1: n in [0..25]]; // Vincenzo Librandi, Sep 29 2013

[4] cat [n eq 1 select n+3 else n*Self(n-1)-n+1: n in [1..25] ]; // Vincenzo Librandi, Sep 29 2013

a:= proc(n) if n=0 then 4 else a(n):= n*a(n-1)-n+1 fi end: seq (a(n), n=0..25);  # Sergei N. Gladkovskii, Jul 04 2012

Table[3 n! + 1, {n, 0, 30}] (* Vincenzo Librandi, Sep 29 2013 *)

a(0) = 4, a(n) = n*a(n-1)-n+1 for n>0. - Sergei N. Gladkovskii, Jul 04 2012

A173319 a(n) = 5*n! + 1.

Original entry on oeis.org

6, 6, 11, 31, 121, 601, 3601, 25201, 201601, 1814401, 18144001, 199584001, 2395008001, 31135104001, 435891456001, 6538371840001, 104613949440001, 1778437140480001, 32011868528640001, 608225502044160001, 12164510040883200001, 255454710858547200001

Offset: 0

Vincenzo Librandi, Feb 16 2010

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. sequences of the type k*n!+1: A038507 (k=1), A052898 (k=2), A173324 (k=3), A173322 (k=4), this sequence, A173314 (k=6).

[5*Factorial(n) + 1: n in [0..20]]; // Vincenzo Librandi, Sep 28 2013

[6] cat [n eq 1 select n+5 else n*Self(n-1)-n+1: n in [1..25] ]; // Vincenzo Librandi, Sep 28 2013

A173319:=n->5*n!+1: seq(A173319(n), n=0..20); # Wesley Ivan Hurt, Oct 10 2014

Table[5 n! + 1, {n, 0, 20}] (* Vincenzo Librandi, Sep 28 2013 *)

a(0)=6; for n>0, a(n) = n*a(n-1)-n+1. - Vincenzo Librandi, Sep 28 2013

(n-2)*a(n) - (n^2-n-1)*a(n-1) + (n-1)^2*a(n-2) = 0. [Bruno Berselli, Sep 28 2013]

A062698 Primes of form 2*k! + 1.

Original entry on oeis.org

3, 5, 13, 241, 958003201, 12804747411456001, 20666295932772289859333302675046400000001, 3102237506574764560448486032938606422126519440033972224000000000001, 8549766568120051128596027506778799299380687576733627449344000000000001

Offset: 1

Jason Earls, Jul 10 2001

nonn

Harry J. Smith, Table of n, a(n) for n=1,...,12

Cf. A051915, A052898.

lst={};Do[p=2*n!+1;If[PrimeQ[p],AppendTo[lst,p]],{n,0,5!}];lst (* Vladimir Joseph Stephan Orlovsky, Jan 28 2009 *)
Select[2*Range[200]!+1,PrimeQ] (* Harvey P. Dale, Oct 26 2013 *)

for(n=1,55, if(isprime(2*n!+1),print(2*n!+1)))

{ n=0; f=1; for (m=1, 10^5, f*=m; if(isprime(a=2*f + 1), write("b062698.txt", n++, " ", a); if (n==12, break)) ) } \\ Harry J. Smith, Aug 09 2009

a(n) = A052898(A051915(n+1)). - Elmo R. Oliveira, Apr 16 2025

One additional term from Harvey P. Dale, Oct 26 2013

A229554 a(n) = 7*n! + 1.

Original entry on oeis.org

8, 8, 15, 43, 169, 841, 5041, 35281, 282241, 2540161, 25401601, 279417601, 3353011201, 43589145601, 610248038401, 9153720576001, 146459529216001, 2489811996672001, 44816615940096001, 851515702861824001, 17030314057236480001, 357636595201966080001

Offset: 0

Vincenzo Librandi, Sep 30 2013

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. sequences of the type k*n!+1: A038507 (k=1), A052898 (k=2), A173324 (k=3), A173322 (k=4), A173319 (k=5), A173314 (k=6), this sequence (k=7).

Magma
```
[7*Factorial(n)+1: n in [0..25]];
```

[8] cat [n eq 1 select n+7 else n*Self(n-1)-n+1: n in [1..25]];

Mathematica
```
Table[7 n! + 1, {n, 0, 25}]
```

a(0)=8, a(n) = n*a(n-1)-n+1.

E.g.f.: 7/(1-x)+exp(x). - Geoffrey Critzer, Sep 30 2013

cp's OEIS Frontend

A020543 a(0) = 1, a(1) = 1, a(n+1) = (n+1)*a(n) + n.

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A173314 a(n) = 6*n!+1.

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A173322 a(n) = 4*n! + 1.

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A173324 a(n) = 3*n! + 1.

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A173319 a(n) = 5*n! + 1.

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Magma

Magma

Maple

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A062698 Primes of form 2*k! + 1.

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PARI

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A229554 a(n) = 7*n! + 1.

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