A002981 -id:A002981 - cp's OEIS frontend

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

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;

A109132 Numbers n such that n! + 1 is a Chen prime. n! + 1 is then called a factorial Chen prime.

Original entry on oeis.org

0, 1, 2, 3, 11, 77

Offset: 1

Jason Earls, Aug 17 2005

Conjecture: 77! + 1 is the largest factorial Chen prime.

For n>2, n is in the sequence iff both numbers n!/3+1 and n!+1 are primes. - Farideh Firoozbakht, Mar 24 2006

Cf. A002981, A109611.

A110094 Startorial primes.

Original entry on oeis.org

2, 3, 5, 7, 23, 719, 5039, 1451521, 2903041, 5806081, 46448639, 92897281, 371589121, 10032906239, 30098718719, 270888468479, 812665405439, 7313988648961, 21941965946881, 89874292518420479

Offset: 1

Jonathan Vos Post, Sep 04 2005

These are primes of the form A109834 startorials (base 10) +1 or -1. This is by analogy to factorial primes (A002981), superfactorial primes (A073828), hyperfactorial primes, ultrafactorial primes (comment in A046882), subfactorial primes (A100015), double factorial primes (A080778), multifactorial primes (A037083).

Cf. A000040, A002981, A073828, A080778, A037083, A100015, A103317, A046882, A109834.

{a(n)} = {A109834(k)+1 an element of A000040, or A109834(k)-1 an element of A000040, for some k}.

A121931 Numbers k such that (k!)^8 + 1 is prime.

Original entry on oeis.org

0, 1, 2, 58, 75, 347

Offset: 1

Alexander Adamchuk, Sep 10 2006

Corresponding primes of the form (k!)^8 + 1 are {2,2,257,...}.

a(7) > 7000. - Robert Price, Aug 26 2014

Cf. A060890, A051855, A046029, A002981.

Do[f=(n!)^8+1;If[PrimeQ[f],Print[{n,f}]],{n,1,75}]

a(6) from Ryan Propper, Jan 03 2008

A172114 Partial sums of factorial primes A088054.

Original entry on oeis.org

2, 5, 10, 17, 40, 759, 5798, 39922599, 518924198, 87697215397, 10888869450418352248465215398, 265263748681641476988556945215397, 263396100682375171644206569105215396, 8946713719494261667162400970385215395

Offset: 1

Jonathan Vos Post, Jan 25 2010

nonn

The primes in this sequence begin 2, 5, 17; where 5 is itself a factorial prime 3!-1. What is the next prime in the sequence?

Cf. A000040, A000142, A002981, A002982.

a(n) = SUM[i=1..n] A088054(i) = SUM[i=1..n] {primes which are within 1 of a factorial number}.

A176049 Primes of the form n!(n+1)!(n+2)! - 1 or n!(n+1)!(n+2)! + 1.

Original entry on oeis.org

3, 11, 13, 2073601, 146313215999, 52563198423859200001, 709885457731229765106401279999999, 15120395453651827088974983182763034097693491200000000001

Offset: 1

Jonathan Vos Post, Apr 07 2010

Primes of the form A010790(k)-1 or A010790(k)+1. This is the 3rd sequence in the supersequence whose first member is factorial primes, A002981 UNION A002982, and whose 2nd member is A176038 Primes of the form n!*(n+1)! - 1 or n!*(n+1)! + 1.
a(9) has already 486 digits and is not listed for that reason. The sequence is generated by the n-values 0, 1, 1, 4, 6, 9, 13, 19, 101, 196,... [From R. J. Mathar, Oct 03 2010]
a(9) also ends with 72 nines. - Harvey P. Dale, Jan 05 2013

			a(2) = 11 because 1!*(1+1)!*(1+2)! - 1 = 11 is prime. a(4) = 2073601 because 4!*(4+1)!*(4+2)! + 1 = 2073601 is prime. a(7) because 13!*(13+1)!*(13+2)! - 1 = 709885457731229765106401279999999 is prime.

Cf. A000040, A000142, A002981, A002982, A010790, A176038.

Select[Union[Flatten[Times@@#+{1,-1}&/@Partition[Range[0,30]!,3,1]]], PrimeQ] (* Harvey P. Dale, Jan 05 2013 *)

a(8) from R. J. Mathar, Oct 03 2010

A177502 Prime numbers p such that (p-2)! + 1 is prime.

Original entry on oeis.org

2, 3, 5, 13, 29, 43, 79, 401, 26953, 150211, 288467, 422431

Offset: 1

Artur Jasinski, May 10 2010

			a(1)=2 because (2-2)! + 1 = 0! + 1 = 1 + 1 = 2.

Titus Piezas III, Solving Solvable Sextics Using Polynomial Decomposition, 2004.

Cf. A088054, A002981.

aa = {}; Do[p = Prime[n]; If[PrimeQ[(p - 2)! + 1], AppendTo[aa, p]], {n, 1, 10000}]; aa

New term added by Artur Jasinski, May 19 2010

a(10)-a(12) from Amiram Eldar, Oct 24 2024

A193814 Least k such that n! + k + 1 is a prime.

Original entry on oeis.org

0, 0, 0, 0, 4, 6, 6, 10, 22, 16, 10, 0, 28, 66, 18, 42, 22, 30, 36, 88, 28, 30, 30, 96, 130, 40, 58, 0, 66, 222, 106, 126, 78, 36, 96, 60, 130, 0, 42, 96, 52, 0, 96, 70, 46, 238, 100, 232, 52, 82, 60, 270, 52, 70, 222, 70, 148, 106, 282, 292, 270, 768, 130, 270, 66

Offset: 0

Michel Lagneau, Aug 06 2011

nonn

			a(5) = 6 because 5! + 6 + 1 = 127 is prime.

Cf. A002981

a={};Do[k = 0; While[ !PrimeQ[n! + k + 1], k++ ]; AppendTo[a, k], {n, 0, 70} ];a
ld[n_]:=Module[{f=n!},NextPrime[f]-f-1]; Array[ld,70,0] (* Harvey P. Dale, Jul 13 2014 *)

a(n) = my(k=0); while (!isprime(n!+k+1), k++); k; \\ Michel Marcus, Oct 10 2019

a(n) = 0 for n in A002981. - Michel Marcus, Oct 10 2019

A258355 Numbers n such that cyclotomic polynomial value Phi(9,n!) is prime.

Original entry on oeis.org

0, 1, 2, 98, 775

Offset: 1

Robert Price, May 27 2015

All values correspond to certified primes.

a(6) > 5944.

			2 is in the sequence because Phi(9,2!) = 1 + 2^3 + 2^6 = 73 is prime.

Cf. A002981, A002982, A046029, A200906.

Select[Range[0, 5944], PrimeQ[Cyclotomic[9, #!]] &]

A258356 Numbers n such that cyclotomic polynomial value Phi(10,n!) is prime.

Original entry on oeis.org

2, 36, 101, 107, 267, 316

Offset: 1

Robert Price, May 27 2015

All values except 2 are probable primes.
a(7) > 7560.
That is, numbers n such that n! belongs to A246392. - Michel Marcus, May 30 2015

			2 is in the sequence because Phi(10,2!) = 1 - 2 + 2^2 - 2^3 + 2^4 = 11 is prime.

Cf. A002981, A002982, A046029, A200906.

Select[Range[0, 7560], PrimeQ[Cyclotomic[10, #!]] &]

cp's OEIS Frontend

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

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A109132 Numbers n such that n! + 1 is a Chen prime. n! + 1 is then called a factorial Chen prime.

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A110094 Startorial primes.

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A121931 Numbers k such that (k!)^8 + 1 is prime.

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A172114 Partial sums of factorial primes A088054.

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A176049 Primes of the form n!*(n+1)!*(n+2)! - 1 or n!*(n+1)!*(n+2)! + 1.

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A177502 Prime numbers p such that (p-2)! + 1 is prime.

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A193814 Least k such that n! + k + 1 is a prime.

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A258355 Numbers n such that cyclotomic polynomial value Phi(9,n!) is prime.

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A258356 Numbers n such that cyclotomic polynomial value Phi(10,n!) is prime.

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A176049 Primes of the form n!(n+1)!(n+2)! - 1 or n!(n+1)!(n+2)! + 1.