A173321 -id:A173321 - 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

A173316 a(n) = 6*n! - 1.

Original entry on oeis.org

5, 5, 11, 35, 143, 719, 4319, 30239, 241919, 2177279, 21772799, 239500799, 2874009599, 37362124799, 523069747199, 7846046207999, 125536739327999, 2134124568575999, 38414242234367999, 729870602452991999, 14597412049059839999, 306545653030256639999

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: A033312 (k=1), A020543 (k=2), A173323 (k=3), A173321 (k=4), A173317 (k=5).

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

[5] cat [n eq 1 select n+4 else n*Self(n-1)+n-1: n in [1..25] ]; // Vincenzo Librandi, Sep 30 2013

Table[6 n! - 1, {n, 0, 25}] (* Vincenzo Librandi, Sep 30 2013 *)

a(0)=5, a(n) = n*a(n-1)+n-1. - Vincenzo Librandi, Sep 30 2013

A173317 a(n) = 5*n! - 1.

Original entry on oeis.org

4, 4, 9, 29, 119, 599, 3599, 25199, 201599, 1814399, 18143999, 199583999, 2395007999, 31135103999, 435891455999, 6538371839999, 104613949439999, 1778437140479999, 32011868528639999, 608225502044159999

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: A033312 (k=1), A020543 (k=2), A173323 (k=3), A173321 (k=4), this sequence, A173316 (k=6).

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

[4] cat [n eq 1 select n+3 else n*Self(n-1)+n-1: n in [1..25] ]; // Vincenzo Librandi, Sep 30 2013

Table[5 n! - 1, {n, 0, 25}] (* Vincenzo Librandi, Sep 30 2013 *)

a(n) = 5*A000142(n)-1.

a(0)=4, a(n) = n*a(n-1)+n-1. - Vincenzo Librandi, Sep 30 2013

a(16) corrected from Vincenzo Librandi, Sep 30 2013

A173323 a(n) = 3*n! - 1.

Original entry on oeis.org

2, 2, 5, 17, 71, 359, 2159, 15119, 120959, 1088639, 10886399, 119750399, 1437004799, 18681062399, 261534873599, 3923023103999, 62768369663999, 1067062284287999, 19207121117183999, 364935301226495999, 7298706024529919999, 153272826515128319999, 3372002183332823039999

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: A033312 (k=1), A020543 (k=2), this sequence, A173321 (k=4), A173317 (k=5), A173316 (k=6).

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

[2] cat [n eq 1 select n+1 else n*Self(n-1)+n-1: n in [1..25] ]; // Vincenzo Librandi, Sep 30 2013

Table[3 n! - 1, {n, 0, 25}] (* Vincenzo Librandi, Sep 30 2013 *)
3*(Range[0,20]!)-1 (* Harvey P. Dale, Oct 27 2015 *)

a(0)=2, a(n) = n*a(n-1)+n-1. - Vincenzo Librandi, Sep 30 2013
D-finite with recurrence a(n) +(-n-2)*a(n-1) +(2*n-1)*a(n-2) +(-n+2)*a(n-3)=0. - R. J. Mathar, Mar 07 2022
E.g.f.: 3/(1 - x) - exp(x). - Stefano Spezia, Oct 14 2024

A229828 a(n) = 7*n! - 1.

Original entry on oeis.org

6, 6, 13, 41, 167, 839, 5039, 35279, 282239, 2540159, 25401599, 279417599, 3353011199, 43589145599, 610248038399, 9153720575999, 146459529215999, 2489811996671999, 44816615940095999, 851515702861823999, 17030314057236479999, 357636595201966079999

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: A033312 (k=1), A020543 (k=2), A173323 (k=3), A173321 (k=4), A173317 (k=5), A173316 (k=6).

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

[6] cat [n eq 1 select n+5 else n*Self(n-1)+n-1: n in [1..25] ];

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

a(0)=6, a(n) = n*a(n-1)+n-1.

A333924 Smallest prime of the form 4k + 3 that is a divisor of 4n! - 1.

Original entry on oeis.org

3, 3, 7, 23, 19, 479, 2879, 19, 179, 2551, 14515199, 159667199, 26246663, 47, 3007159, 85303, 43, 455999, 13099, 311369011223, 7791519641878751, 59, 50207, 149709500816123, 71, 61651424911, 1146111319366855507, 3902575987, 27963070149883187169101323, 3262754470190705587633531

Offset: 0

Bernard Schott, Apr 10 2020

nonn

Every integer equal to 4*n!-1 (A173321) has a prime factor > n of the form 4*k+3; this is one of the proofs which show that there are infinitely many primes of the form 4*k+3 (A002145).

			4*11!-1 = 159667199 that is prime of the form 4*k+3, hence a(11) = 159667199.
4*13!-1 = 24908083199 = 47 * 2963 * 178859, these 3 prime factors are all of the form 4*k+3, the smallest one is 47 hence a(13) = 47.
4*14!-1 = 348713164799 = 61 * 1901 * 3007159, only 3007159 is a prime of the form 4*k+3, hence a(14) = 3007159.

Cf. A002144, A173321.

Subsequence of A002145.

a[n_] := Min[Select[First /@ FactorInteger[4*n! - 1], Mod[#, 4] == 3 &]]; Array[a, 30, 0] (* Amiram Eldar, Apr 10 2020 *)

a(n) = {my(f=factor(4*n!-1)[,1]); for(i=1, #f, if(f[i]%4==3, return(f[i]))); } \\ Jinyuan Wang, Apr 10 2020

a(23) corrected by and more terms from Jinyuan Wang, Apr 10 2020

cp's OEIS Frontend

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

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A173316 a(n) = 6*n! - 1.

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A173317 a(n) = 5*n! - 1.

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Magma

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A173323 a(n) = 3*n! - 1.

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Magma

Magma

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A229828 a(n) = 7*n! - 1.

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Magma

Magma

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A333924 Smallest prime of the form 4*k + 3 that is a divisor of 4*n! - 1.

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A333924 Smallest prime of the form 4k + 3 that is a divisor of 4n! - 1.