A090786 -id:A090786 - cp's OEIS frontend

A005095 a(n) = n! + n.

1, 2, 4, 9, 28, 125, 726, 5047, 40328, 362889, 3628810, 39916811, 479001612, 6227020813, 87178291214, 1307674368015, 20922789888016, 355687428096017, 6402373705728018, 121645100408832019, 2432902008176640020, 51090942171709440021

Offset: 0

N. J. A. Sloane

Every infinite, increasing, integer arithmetic progression meets this sequence infinitely often. - John Abbott (abbott(AT)dima.unige.it), Mar 06 2003
Sum(A010051(k): A038507(n) < k <= a(n)) = 0. - Reinhard Zumkeller, Jul 10 2009
Largest k such that (k!-n!)/(k-n) is an integer. - Derek Orr, Apr 02 2014

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Aria Chen, Tyler Cummins, Rishi De Francesco, Jate Greene, Tanya Khovanova, Alexander Meng, Tanish Parida, Anirudh Pulugurtha, Anand Swaroop, and Samuel Tsui, Card Tricks and Information, arXiv:2405.21007 [math.HO], 2024. See p. 5.
Index entries for sequences related to factorial numbers

Cf. A135723.

Cf. A090786. - Reinhard Zumkeller, Jul 10 2009

[Factorial(n) + n: n in [0..20]]; // Vincenzo Librandi, Jun 08 2013

lst={};Do[AppendTo[lst, n!+n], {n, 3*4!}];lst (* Vladimir Joseph Stephan Orlovsky, Nov 20 2008 *)
Table[n! + n, {n, 0, 20}] (* Vincenzo Librandi, Jun 08 2013 *)

A005095(n):= n!+n$
makelist(A005095(n),n,0,30); /* Martin Ettl, Nov 03 2012 */

E.g.f.: x*exp(x) + 1/(1-x). - Len Smiley, Dec 05 2001
Row sums of triangle A135723. - Gary W. Adamson, Nov 25 2007
(n-1)*(n-3)*a(n) -n*(n^2-3*n+1)*a(n-1) +n*(n-1)*(n-2)*a(n-2)=0. - R. J. Mathar, Oct 30 2015
a(n) +(-n-3)*a(n-1) +3*(n)*a(n-2) +(-3*n+5)*a(n-3) +(n-3)*a(n-4)=0. - R. J. Mathar, Oct 30 2015

A301427 Least nonnegative integer k such that n! - n - k is prime.

Original entry on oeis.org

0, 1, 2, 5, 10, 23, 4, 1, 2, 1, 10, 3, 32, 37, 42, 23, 82, 11, 10, 51, 66, 49, 124, 11, 16, 73, 2, 49, 30, 131, 14, 159, 78, 91, 60, 41, 34, 43, 90, 37, 66, 65, 8, 43, 32, 55, 10, 47, 128, 15, 6, 73, 6, 405, 220, 51, 78, 79, 10, 9, 38, 295, 62, 251, 124, 183, 34, 27, 680, 91, 300

Offset: 3

Seiichi Manyama, Mar 21 2018

nonn

The (n-1) consecutive numbers n!-n, ... , n!-2 (for n > 3) are not prime.

			a(3)=0 because 3! - 3 - 0 =   3 is prime.
a(4)=1 because 4! - 4 - 1 =  19 is prime and 20 is not.
a(5)=2 because 5! - 5 - 2 = 113 is prime and 114 and 115 are not prime.

Seiichi Manyama, Table of n, a(n) for n = 3..500

Cf. A037155, A073443, A090786.

f:= proc(n) local r; r:= n!-n;
  r - prevprime(r)
end proc:
f(3):= 0:
seq(f(i),i=3..100); # Robert Israel, Mar 23 2018

a[n_] := n! - NextPrime[n! - 1, -1] - n;
a /@ Range[3, 100] (* Jean-François Alcover, Oct 26 2020 *)

a(n) = apply(x->(x-precprime(x)), n!-n);
vector(99, n, a(n+2)) \\ Altug Alkan, Mar 21 2018

a(n) = A037155(n) - n.

A301431 Least nonnegative integer k such that (n!)^2 + n + k + 1 is prime.

Original entry on oeis.org

0, 0, 0, 1, 6, 1, 4, 3, 4, 13, 6, 1, 46, 9, 16, 7, 24, 41, 48, 9, 10, 81, 366, 35, 82, 21, 100, 39, 152, 71, 66, 377, 4, 27, 8, 25, 10, 225, 70, 13, 158, 125, 294, 3, 86, 81, 26, 133, 208, 141, 50, 31, 26, 127, 112, 173, 802, 363, 374, 47, 910, 437, 74, 213, 1044, 13, 1962, 41, 160, 169, 296, 29

Offset: 0

Seiichi Manyama, Mar 21 2018

nonn

The (n-1) consecutive numbers (n)^2! + 2, ..., (n!)^2 + n (for n >= 2) are not prime powers (cf. A246655).

			a(0)=0 because (0!)^2 + 0 + 0 + 1 =   2 is prime.
a(1)=0 because (1!)^2 + 1 + 0 + 1 =   3 is prime.
a(2)=0 because (2!)^2 + 2 + 0 + 1 =   7 is prime.
a(3)=1 because (3!)^2 + 3 + 1 + 1 =  41 is prime and 40 is not prime.
a(4)=6 because (4!)^2 + 4 + 6 + 1 = 587 is prime and 581, 582, ... , 586 are not prime.

Seiichi Manyama, Table of n, a(n) for n = 0..500

Cf. A046029, A090786, A246655.

a(n) = apply(x->(nextprime(x)-x), (n!)^2+n+1); \\ Altug Alkan, Mar 21 2018

A090975 Least integer k such that n!+1-k is prime.

Original entry on oeis.org

0, 0, 0, 0, 2, 8, 2, 2, 32, 14, 12, 0, 2, 24, 2, 48, 54, 60, 42, 102, 32, 32, 74, 90, 74, 150, 38, 0, 102, 32, 2, 62, 2, 2, 194, 114, 128, 0, 2, 74, 84, 0, 80, 110, 110, 54, 90, 80, 104, 60, 98, 180, 68, 60, 128, 62, 462, 278, 110, 138, 140, 72, 72, 102, 360, 128, 318, 192

Offset: 0

Frederick Magata (frederick.magata(AT)uni-muenster.de), Feb 28 2004

nonn

The (n-1) consecutive numbers n!+2,...,n!+n (for n>=2) are not prime. This fact implies that there are arbitrarily large gaps in the distribution of the prime numbers. n!+1 itself may be a prime number as in the case of n=3, 11, 27 (see A002981 for all such n). Now a(n) measures, when the first prime number previous to n!+2 appears. Thus a(n)=8 means that n!+1-3 is prime and so on. Obviously, the values of a(n) are always even numbers. Conjectures: |a(n)-1| is either 1 or a prime number. Is the growth of b(n) := sum(a(k),k=0..n) quadratic, that is b(n)=O(n^2)?

			a(3)=0 because 3!+1-0=7 is prime.
a(4)=2 because 4!+1-2=23 is prime and 24 and 25 are not.

Cf. A090786, A002981, A005095.

a := proc(n) option remember;n!+1-prevprime(n!+2); end;

A343593 a(n) is the smallest number k > 0 such that n! + n + k is prime.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 4, 15, 8, 1, 6, 17, 54, 5, 28, 7, 14, 19, 70, 9, 10, 9, 74, 107, 16, 33, 20, 39, 194, 77, 96, 47, 4, 63, 26, 95, 274, 5, 58, 13, 20, 55, 28, 3, 194, 55, 186, 5, 34, 11, 220, 1, 18, 169, 16, 93, 50, 225, 234, 211, 708, 69, 208, 3, 128, 217

Offset: 0

Ventsislav D. Tsenov, Apr 21 2021

nonn

			For n = 2: the smallest value of k such that 2! + 2 + k = 4 + k is prime is 1.

Cf. A000040, A037153, A090786.

a:= n-> (t-> nextprime(t)-t)(n!+n):
seq(a(n), n=0..80);  # Alois P. Heinz, Apr 21 2021

a[n_] := NextPrime[(m = n! + n)] - m; Array[a, 100, 0] (* Amiram Eldar, Apr 21 2021 *)

a(n) = my(s=n!+n); nextprime(s+1) - s; \\ Michel Marcus, Apr 21 2021

a(n) = A037153(n) - n for n >= 1.

a(n) = A090786(n) + 1.

More terms from Alois P. Heinz, Apr 21 2021

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A005095 a(n) = n! + n.

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A301427 Least nonnegative integer k such that n! - n - k is prime.

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A301431 Least nonnegative integer k such that (n!)^2 + n + k + 1 is prime.

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A090975 Least integer k such that n!+1-k is prime.

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A343593 a(n) is the smallest number k > 0 such that n! + n + k is prime.

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