A062701 -id:A062701 - cp's OEIS frontend

A088332 Primes of the form k! + 1.

2, 3, 7, 39916801, 10888869450418352160768000001, 13763753091226345046315979581580902400000001, 33452526613163807108170062053440751665152000000001

Offset: 1

Cino Hilliard, Nov 06 2003

nonn

The next term is too large to include.
Of course 2 = 0! + 1 = 1! + 1 has two such representations.
Prime numbers that are the sum of two factorial numbers. - Juri-Stepan Gerasimov, Nov 08 2010

			3! + 1 = 7 is prime.

James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 118.

Amiram Eldar, Table of n, a(n) for n = 1..15 (terms 1..11 from T. D. Noe)
Romeo Meštrović, Euclid's theorem on the infinitude of primes: a historical survey of its proofs (300 BC--2012) and another new proof, arXiv preprint arXiv:1202.3670 [math.HO], 2012-2023. - From _N. J. A. Sloane_, Jun 13 2012

Cf. A002981 (values of k), A038507, A062701.

lst={};Do[p=n!+1;If[PrimeQ[p],AppendTo[lst,p]],{n,0,3*5!}];lst (* Vladimir Joseph Stephan Orlovsky, Jan 27 2009 *)
Select[Range[50]!+1,PrimeQ] (* Harvey P. Dale, May 17 2025 *)

factp1prime(n)=for(x=1,n,xf=x!+1; if(isprime(xf),print1(xf",")))

a(n) = A038507(A002981(n+1)). - Elmo R. Oliveira, Apr 16 2025

A136437 a(n) = prime(n) - k! where k is the greatest number such that k! <= prime(n).

Original entry on oeis.org

0, 1, 3, 1, 5, 7, 11, 13, 17, 5, 7, 13, 17, 19, 23, 29, 35, 37, 43, 47, 49, 55, 59, 65, 73, 77, 79, 83, 85, 89, 7, 11, 17, 19, 29, 31, 37, 43, 47, 53, 59, 61, 71, 73, 77, 79, 91, 103, 107, 109, 113, 119, 121, 131, 137, 143, 149, 151, 157, 161, 163, 173, 187, 191, 193, 197, 211, 217, 227, 229, 233, 239, 247

Offset: 1

Ctibor O. Zizka, Apr 02 2008

How many times does each prime appear in this sequence?
The only value (prime(n) - k!) = 0 is at n=1, where k=2.
Are n=2, k=2 and n=4, k=3 the only occurrences of (prime(n) - k!) = 1?
There exist infinitely many solutions of the form (prime(n) - k!) = prime(n-t), t < n.
Are there infinitely many solutions of the form (prime(n) - k!) = prime(r_1)*...*prime(r_i); r_i < n for all i?
From Bernard Schott, Jul 16 2021: (Start)
Answer to the second question is no: 18 other occurrences (n,k) of (prime(n) - k!) = 1 are known today; indeed, every k > 1 in A002981 that satisfies k! + 1 is prime gives an occurrence, but only a third pair (n, k) is known exactly; and this comes for n = 2428957, k = 11 because (prime(2428957) - 11!) = 1.
The next occurrence corresponds to k = 27 and n = X where prime(X) = 1+27! = 10888869450418352160768000001 but index X is not yet available (see A062701).
For the occurrences of (prime(m) - k!) = 1, integers k are in A002981 \ {0, 1}, corresponding indices m are in A062701 \ {1} (only 3 indices are known today) and prime(m) are in A088332 \ {2}. (End)

			a(1)  = prime(1)  - 2! =  2 -  2 =  0;
a(2)  = prime(2)  - 2! =  3 -  2 =  1;
a(3)  = prime(3)  - 2! =  5 -  2 =  3;
a(4)  = prime(4)  - 3! =  7 -  6 =  1;
a(5)  = prime(5)  - 3! = 11 -  6 =  5;
a(6)  = prime(6)  - 3! = 13 -  6 =  7;
a(7)  = prime(7)  - 3! = 17 -  6 = 11;
a(8)  = prime(8)  - 3! = 19 -  6 = 13;
a(9)  = prime(9)  - 3! = 23 -  6 = 17;
a(10) = prime(10) - 4! = 29 - 24 =  5.

Jinyuan Wang, Table of n, a(n) for n = 1..10000

Cf. A135996, A000040, A000142, A212598, A212266.

Cf. also A002981, A062701, A088332, A346425 (gives k).

f:=proc(n) local p,i; p:=ithprime(n); for i from 0 to p do if i! > p then break; fi; od; p-(i-1)!; end;
[seq(f(n),n=1..70)]; # N. J. A. Sloane, May 22 2012

a[n_] := Module[{p, k},p = Prime[n];k = 1;While[Factorial[k] <= p, k++];p - Factorial[k - 1]] (* James C. McMahon, May 05 2025 *)

a(n) = my(k=1, p=prime(n)); while (k! <= p, k++); p - (k-1)!; \\ Michel Marcus, Feb 19 2019

a(n) = prime(n)- k! where k is the greatest number for which k! <= prime(n).
a(n) = A212598(prime(n)). - Michel Marcus, Feb 19 2019
a(n) = A000040(n) - A346425(n). - Bernard Schott, Jul 16 2021

More terms from Jinyuan Wang, Feb 18 2019

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

cp's OEIS Frontend

A088332 Primes of the form k! + 1.

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A136437 a(n) = prime(n) - k! where k is the greatest number such that k! <= prime(n).

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

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