A002981 -id:A002981 - cp's OEIS frontend

A109914 Product of all composite numbers k such that n! < k < prime(r) where prime(r-1)< n!.

1, 1, 1, 491400, 3546112878000, 143424700959632400, 10691567972893973348743970911396896000, 210948344078434820704169472200928966427054605885088717074131707385374604732966434908020301638860800000

Offset: 1

Amarnath Murthy, Jul 16 2005

nonn

k divides n!.

If n is in A002981, then a(n) is - by definition - 1. If not, then none of the numbers n!+1, n!+2, ... n!+n will be prime, which gives us the lower bound a(n) > (n!+1)^n. - Stefan Steinerberger, Mar 14 2006

			a(4) = 25*26*27*28 =491400.

Cf. A109915.

Table[Product[i, {i, n! + 1, Prime[PrimePi[n! ] + 1] - 1}], {n, 1, 8}] (* Stefan Steinerberger, Mar 14 2006 *)

More terms from Stefan Steinerberger, Mar 14 2006

A231549 Least k>0 such that k!*n!+1 is a prime, or 0 if no such k exists.

Original entry on oeis.org

1, 1, 1, 4, 2, 8, 3, 3, 3, 4, 1, 2, 3, 5, 8, 4, 10, 2, 11, 9, 5, 5, 7, 3, 14, 18, 1, 40, 24, 5, 5, 18, 8, 20, 2, 49, 1, 3, 5, 28, 1, 17, 38, 27, 11, 16, 10, 3, 24, 270, 2, 45, 2, 15, 175, 64, 17, 6, 4, 3, 8, 18, 13, 17, 65, 32, 12, 7, 72, 13, 21, 33, 1, 24, 36, 76, 1

Offset: 1

Alex Ratushnyak, Nov 15 2013

nonn

Indices of 1's: A002981.

Cf. A002981, A035093, A057217, A075757, A075758, A231901.

Table[k = 1; While[! PrimeQ[k!*n! + 1], k++]; k, {n, 100}] (* T. D. Noe, Nov 18 2013 *)

A241423 Largest number k > 0 such that n + k! is prime, or 0 if no such k exists.

Original entry on oeis.org

1, 2, 1, 4, 1, 6, 0, 2, 1, 10, 1, 6, 0, 2, 1, 11, 1, 14, 0, 2, 1, 16, 0, 3, 0, 2, 1, 20, 1, 22, 0, 0, 0, 4, 1, 33, 0, 2, 1, 25, 1, 38, 0, 2, 1, 44, 0, 6, 0, 2, 1, 52, 0, 4, 0, 2, 1, 27, 1, 50, 0, 0, 0, 4, 1, 64, 0, 2, 1, 55, 1, 67, 0, 0, 0, 6, 1, 73, 0, 2, 1, 68, 0, 4, 0, 2, 1, 52, 0, 6

Offset: 2

Derek Orr, Aug 08 2014

nonn

If k >= n, then n + k! is divisible by n and is not prime.
a(n) < A020639(n), because if prime p divides n then p divides n + k! for k >= p. - Robert Israel, Aug 10 2014
There is no term for n = 1 since factorial primes 1 + k! can probably be arbitrarily large (A002981 shows k values). - Jens Kruse Andersen, Aug 13 2014

Jens Kruse Andersen, Table of n, a(n) for n = 2..1000

Cf. A245714, A125162.

a:= proc(n)
local k;
for k from min(numtheory:-factorset(n)) to 1 by -1 do
  if isprime(n+k!)  then return(k) fi
od:
0
end proc:
seq(a(n),n=2..100); # Robert Israel, Aug 10 2014

a[n_] := Module[{k}, For[k = FactorInteger[n][[1, 1]], k >= 1, k--, If[PrimeQ[n + k!], Return[k]]]; 0];
a /@ Range[2, 100] (* Jean-François Alcover, Jul 27 2020, after Maple *)

a(n)=forstep(k=n,1,-1,if(ispseudoprime(n+k!),return(k)))
n=2;while(n<150,print1(a(n),", ");n++)

A359180 Numbers k such that k!^2 / 2 + 1 is prime.

Original entry on oeis.org

2, 3, 6, 18, 19, 82, 1298, 3139, 3687, 4637

Offset: 1

Arsen Vardanyan, Dec 18 2022

			3!^2 / 2 + 1 = 6^2/2 + 1 = 19, a prime number, so 3 is a term.

Cf. A002981, A046029, A358805, A358878.

isok(k) = (k>1) && isprime(k!^2 / 2 + 1); \\ Michel Marcus, Jan 15 2023

a(7) from Michael S. Branicky, Dec 18 2022

a(8)-a(10) from Michael S. Branicky, Apr 10 2023

A062702 Index of factorial primes of form m!-1.

Original entry on oeis.org

3, 9, 128, 675, 25306287, 3610490805

Offset: 1

Labos Elemer, Jul 11 2001

			The exact subscript of 7th prime [=30!-1=265252859812191058636308479999999] is not yet available.

Cf. A002981, A002982, A055490.

a(n) = PrimePi(A002982(n)!-1) = A000720(A055490(n)).

Offset 1 from Michel Marcus, Aug 29 2019

A084914 Numbers k such that k^k*k! - 1 is prime.

Original entry on oeis.org

2, 4, 30, 94, 113, 162, 296, 3243

Offset: 1

Farideh Firoozbakht, Jul 14 2003

a(1)-a(8) have been proved to correspond to primes. No other terms less than 10000. - Robert Price, May 19 2012

			4 is in the sequence because 4!*4^4 - 1 = 6143 is prime.

Cf. A002981, A002982.

Do[If[PrimeQ[n^n*n!-1], Print[n]], {n, 700}]

a(8) = 3243 from Robert Price, May 19 2012

A088412 A088258 indexed by A000142.

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 11, 12, 14, 27, 30, 32, 33, 37, 38, 41, 73, 77, 94, 116, 154, 166, 320, 324, 340, 379, 399, 427, 469, 546, 872, 974, 1477, 1963, 3507, 3610, 6380, 6917, 21480, 26951, 34790, 94550, 103040, 110059, 147855, 150209, 208003

Offset: 1

Ray Chandler, Sep 29 2003

nonn

Union of A002981 and A002982, except 0. - Andrey Zabolotskiy, Aug 25 2016

Terms correspond to indices m where A056752(m)=1, excepting m=2. - Bill McEachen, May 20 2025

Hisanori Mishima, Primes near n!, Dec 2008.

Cf. A000142, A088258.

select(t -> isprime(t!-1) or isprime(t!+1), [$1..600]); # Robert Israel, Aug 25 2016

Select[Range[10^3], Or @@ PrimeQ@ {# - 1, # + 1} &[#!] &] (* Michael De Vlieger, Aug 25 2016 *)

a(n) is such positive k that A088258(n) = A000142(k).

Values 320 to 546 extracted from Mishima's table by R. J. Mathar, Mar 05 2010

More terms and correction of the initial term from Andrey Zabolotskiy, Aug 25 2016

A100013 Number of prime factors in n!+7 (counted with multiplicity).

Original entry on oeis.org

3, 3, 2, 1, 1, 1, 1, 3, 3, 3, 3, 2, 3, 3, 4, 2, 2, 3, 3, 5, 5, 5, 3, 4, 3, 2, 4, 5, 5, 4, 7, 6, 4, 4, 7, 2, 5, 4, 7, 4, 5, 3, 4, 6, 5, 4, 3, 3, 5, 6, 3, 5, 6, 3, 3, 7, 4, 5, 5, 2, 4, 4, 5, 4, 2, 4, 3, 5, 2, 5, 7, 4, 7, 5, 5, 3, 5, 4, 6, 6, 8, 5

Offset: 0

Jonathan Vos Post, Nov 18 2004

			Example 1!+7 = 2^3 so a(1) = 3.
a(3) = a(4) = a(5) = a(6) = 1 because 3!+1 = 13, 4!+7 = 31, 5!+1 = 127, 6!+7 = 727 and these are all primes. a(11) = a(15) = a(16) = a(25) = a(35) = a(59) = 2 because 11!+7 = 39916807 = 7 * 5702401, 15!+7 = 1307674368007 = 7 * 186810624001, 16!+7 = 20922789888007 = 7 * 2988969984001, 25!+7 = 15511210043330985984000007 = 7 * 2215887149047283712000001, 35!+7 = 10333147966386144929666651337523200000007 = 7 *
1476163995198020704238093048217600000001 and 59!+7 = 138683118545689835737939019720389406345902876772687432540821294940160000000000007 = 7 * 19811874077955690819705574245769915192271839538955347505831613562880000000000001 are all semiprimes.

C. Caldwell and H. Dubner, "Primorial, factorial and multifactorial primes," Math. Spectrum, 26:1 (1993/4) 1-7.

Chris Caldwell, multifactorial prime (another Prime Pages' Glossary entries).
Ken Davis, Status of Search for Multifactorial Primes.
Sean A. Irvine, Factorizations of n!+7

Cf. A002981, A037083, A037083, A085150, A085148, A085146, A062701, A055487, A062702.

More terms from Sean A. Irvine, Sep 20 2012

A176038 Primes of the form k!(k+1)! - 1 or k!(k+1)! + 1.

Original entry on oeis.org

2, 3, 11, 13, 2879, 86399, 114000816848279961600001, 2284848632399058501374484565150666260597460935294482959564800000000000001

Offset: 1

Jonathan Vos Post, Apr 07 2010

Primes of the form A010790(k)-1 or A010790(k)+1. This is the 2nd sequence in the supersequence whose first member is factorial primes, A002981 UNION A002982. No more through 20!*(20+1)! + 1.

a(9) has already 225 digits. The terms are generated by n= 0,1,2,2,4,5,14,32,76,166... [From R. J. Mathar, Aug 31 2010]

			a(6) = 86399 because 5!*(5+1)! - 1 = 86399 is prime. a(7) = 114000816848279961600001 because 14!*(14+1)! + 1 = 114000816848279961600001 is prime.

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

[{A010790(n)-1} INTERSECTION A000040] UNION [{A010790(n)+1} INTERSECTION A000040].

One more term from R. J. Mathar, Aug 31 2010

A230459 Ordered by increasing m with k < m, a(n) is the n-th record value of gcd(k!+1, m!+1).

Original entry on oeis.org

2, 7, 71, 661, 733, 2371, 3529, 13499, 46549, 98101, 163517, 197933, 1924217, 3322441, 5370731

Offset: 1

James G. Merickel, Oct 19 2013

The pairs (m,k) generating records are (1,0), (6,3), (9,7), (17,8), (89,51), (174,144), (349,228), (422,81), (650,406), (1415,1718), (1697,161), (1622,773), (1884,1219), (7003,2031) and (17057,660).
Heuristics in concert with a database of 'small' (less than, say, 10^12) prime factors of numbers of this kind would generate faster accurate results with near certainty, while any truly proving program is doomed to be relatively slow by comparison (and see following on a(15)).
Note: An auxiliary program employed a limit of 10^8--in lieu of a database--to generate the likely-but-not-certain value of a(15) shown last.

			a(1)=2, corresponding to m=1 and k=0.  7 is the first value other than 1 to be the greatest common divisor of two different numbers k!+1 and m!+1, where m is increasing and k is allowed to increase to m-1 for a given m (for m=6 and k=3, m!+1=7*103 and k!+1=7); so that a(2)=7.

Cf. A002583, A002981, A038507, A051301.

{
\\ The constant L here is arbitrary.\\
\\ This does not generate a(1).\\
rec=2;L=10000;F=vector(L);n=2;
for(k=1,L,n--;n*=k;n++;F[k]=n);
for(m=2,L,
  for(k=1,m-1,
    a=gcd(F[m],F[k]);if(a>rec,
      rec=a;print1(a": "m","k"\n"))))
}

cp's OEIS Frontend

A109914 Product of all composite numbers k such that n! < k < prime(r) where prime(r-1)< n!.

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A231549 Least k>0 such that k!*n!+1 is a prime, or 0 if no such k exists.

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A241423 Largest number k > 0 such that n + k! is prime, or 0 if no such k exists.

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A359180 Numbers k such that k!^2 / 2 + 1 is prime.

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PARI

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A062702 Index of factorial primes of form m!-1.

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A084914 Numbers k such that k^k*k! - 1 is prime.

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A088412 A088258 indexed by A000142.

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Maple

Mathematica

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A100013 Number of prime factors in n!+7 (counted with multiplicity).

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A176038 Primes of the form k!*(k+1)! - 1 or k!*(k+1)! + 1.

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A176038 Primes of the form k!(k+1)! - 1 or k!(k+1)! + 1.