A188914 -id:A188914 - cp's OEIS frontend

A049984 Primes of the form n! - (n-1)! + 1.

2, 5, 19, 97, 601, 35281, 5748019201, 2311256907767808001, 594596384994354462720001, 5382999938946608755288342267304597177897268019200000000001, 136332557214406957166109544809874331662074014454506289616400595025920000000000001

Offset: 1

N. J. A. Sloane

nonn

Primes in A188914.

Cf. A188914. For more terms, see A049432.

f[n_]:=n!+n; lst={};Do[If[PrimeQ[f[n+1]-f[n]],AppendTo[lst,f[n+1]-f[n]]],{n,0,5!}];lst (* Vladimir Joseph Stephan Orlovsky, Jun 27 2009 *)

a(n) = A188914(A049432(n)). - Elmo R. Oliveira, Feb 18 2025

A362777 Triangular array read by rows: T(n,k) = n!*k + 1, n >= 1, 1 <= k <= n.

Original entry on oeis.org

2, 3, 5, 7, 13, 19, 25, 49, 73, 97, 121, 241, 361, 481, 601, 721, 1441, 2161, 2881, 3601, 4321, 5041, 10081, 15121, 20161, 25201, 30241, 35281, 40321, 80641, 120961, 161281, 201601, 241921, 282241, 322561, 362881, 725761, 1088641, 1451521, 1814401, 2177281, 2540161, 2903041, 3265921

Offset: 1

Joe B. Stephen, May 03 2023

These numbers are used in a simple proof of the infinitude of the primes: n!*i + 1 and n!*j + 1 are coprime for 1 <= i < j <= n, so for any n we get n coprime integers (greater than 1) and hence we get at least n distinct primes.

			Triangle T(n,k) begins:
  n\k  1    2    3    4    5    6 ...
  1    2
  2    3    5
  3    7   13   19
  4   25   49   73   97
  5  121  241  361  481  601
  6  721 1441 2161 2881 3601 4321
  ...

Cf. A362778, A362779.

Cf. A038507 (1st column), A188914 (right diagonal).

A346168 Primes p such that p*p! - 1 is prime.

Original entry on oeis.org

2, 3, 5, 7, 11, 397, 599, 2239

Offset: 1

Reza K Ghazi, Jul 08 2021

a(9) > 10^4.

Prime terms of A090704.

Cf. A001563, A090703, A188914, A346167.

select(p -> isprime(p) and isprime(p*factorial(p) - 1), [$2 .. 600])

Select[Range[2, 600], PrimeQ[#] && PrimeQ[#*#! - 1] &]
Select[Prime[Range[110]],PrimeQ[# #!-1]&] (* The program generates the first 7 terms of the sequence. *) (* Harvey P. Dale, Feb 12 2025 *)

a = List(); for(p=2, 600, if(isprime(p) && isprime(p*p!-1), listput(a, p))); a

[p for p in range(2, 600) if is_prime(p) and is_prime(p*factorial(p) - 1)]

cp's OEIS Frontend

A049984 Primes of the form n! - (n-1)! + 1.

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A362777 Triangular array read by rows: T(n,k) = n!*k + 1, n >= 1, 1 <= k <= n.

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A346168 Primes p such that p*p! - 1 is prime.

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