A002981 -id:A002981 - cp's OEIS frontend

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

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

A204658 Numbers n such that n!10-1 is prime.

Original entry on oeis.org

3, 4, 6, 8, 12, 20, 40, 48, 60, 62, 70, 84, 88, 168, 240, 258, 372, 760, 932, 1010, 2110, 2464, 2490, 2702, 3180, 4744, 6024, 8858, 9060, 10322, 13382, 15778, 19322, 22372, 22928, 25344, 28050, 40604, 42282, 45884, 52428, 58250, 81220, 93612, 108650

Offset: 1

M. F. Hasler, Jan 17 2012

n!10 = product( n-10k, 0 <= k < n/10 ).
See also links in A156165.
a(1)-a(40) are proved prime by deterministic tests of pfgw. - Robert Price, Jun 11 2012
a(41) > 50000. - Robert Price, Jun 11 2012

Ken Davis, Status of Search for Multifactorial Primes.
Ken Davis, Results for n!10-1.

Cf. A204657, A204659, A204660, A204661, A204662, A204663, A204664, A156165, A156167, A085150, A085148, A085146, A037083, A080778, A002981.

MultiFactorial[n_, k_] := If[n < 1, 1, n*MultiFactorial[n - k, k]];
Select[Range[1000], PrimeQ[MultiFactorial[#, 10] - 1] & ] (* Robert Price, Apr 19 2019 *)

for(n=0,9999,isprime(prod(i=0,(n-2)\10,n-10*i)-1)& print1(n","))

a(26)-a(40) from Robert Price, Jun 11 2012

a(41)-a(45) from Ken Davis link entered by Robert Price, Apr 19 2019

A204661 Numbers n such that n!8+1 is prime (for n!8 see A114800).

Original entry on oeis.org

0, 1, 2, 4, 6, 28, 30, 46, 60, 72, 86, 90, 112, 154, 162, 206, 280, 354, 400, 512, 606, 614, 678, 790, 938, 1054, 1092, 1148, 1582, 1788, 2088, 2206, 2598, 2912, 3672, 4642, 6272, 6428, 7084, 7604, 8580, 9464, 12762, 18386, 24910, 30448, 31696, 40288, 41682, 45730

Offset: 1

M. F. Hasler, Jan 17 2012

n!8 = A114800(n).

No other terms < 50000. - Robert Price, Jul 29 2012

Ken Davis, Status of Search for Multifactorial Primes.
Ken Davis, Results for n!8+1.

Cf. A204657, A204658, A204659, A204660, A204662, A204663, A204664, A156165, A156167, A085150, A085148, A085146, A037083, A080778, A002981.

MultiFactorial[n_, k_] := If[n < 1, 1, n*MultiFactorial[n - k, k]];
Select[Range[0, 1000], PrimeQ[MultiFactorial[#, 8] + 1] & ] (* Robert Price, Apr 19 2019 *)
Select[Range[0,46000],PrimeQ[Times@@Range[#,1,-8]+1]&] (* Harvey P. Dale, Apr 12 2022 *)

for(n=0,9999,isprime(prod(i=0,(n-2)\8,n-8*i)+1)& print1(n","))

a(35)-a(50) from Robert Price, Jul 29 2012

A204662 Numbers n such that n!8-1 is prime.

Original entry on oeis.org

3, 4, 6, 8, 10, 12, 14, 16, 18, 22, 28, 30, 42, 48, 58, 68, 80, 86, 92, 108, 110, 112, 130, 198, 220, 230, 322, 432, 460, 478, 686, 706, 714, 842, 950, 1010, 1090, 1314, 1904, 2264, 2804, 3164, 3324, 4740, 4824, 4918, 5086, 5442, 6994, 7898, 8236, 8684, 10088, 13990, 15320, 17570, 18218, 21564, 22198, 22684, 24314, 24780, 25790, 38726

Offset: 1

M. F. Hasler, Jan 17 2012

n!8 = A114800(n).

No other terms < 50000. - Robert Price, Aug 15 2012

Ken Davis, Status of Search for Multifactorial Primes.
Ken Davis, Results for n!8-1.

Cf. A204657, A204658, A204659, A204660, A204661, A204663, A204664, A156165, A156167, A085150, A085148, A085146, A037083, A080778, A002981.

MultiFactorial[n_, k_] := If[n < 1, 1, n*MultiFactorial[n - k, k]];
Select[Range[0, 1000], PrimeQ[MultiFactorial[#, 8] - 1] & ] (* Robert Price, Apr 19 2019 *)

for(n=0,9999,isprime(prod(i=0,(n-2)\8,n-8*i)-1)& print1(n","))

a(39)-a(64) from Robert Price, Aug 15 2012

A204663 Numbers n such that n!8 + 2 is prime.

Original entry on oeis.org

0, 1, 3, 5, 9, 13, 15, 21, 23, 27, 33, 35, 45, 53, 55, 57, 75, 79, 109, 197, 221, 227, 267, 333, 413, 545, 695, 703, 801, 967, 1029, 1329, 1351, 1475, 1549, 1757, 2173, 2861, 3161, 3167, 3885, 4681, 4965, 6277, 6655, 8477, 9821, 9959, 10269, 17999, 23349, 29347, 29477, 30181, 34133, 36687, 40985, 43395, 47499

Offset: 1

M. F. Hasler, Jan 17 2012

n!8 = A114800(n).
See also links in A156165.
For odd k, n!k +-2 is even for all n > k and thus cannot be prime.
a(60) > 50000. - Robert Price, Aug 19 2012

C. Caldwell and H. Dubner (Eds): The top ten prime numbers: from the unpublished collections of R. Ondrejka (May 2001), Table 21 F, p. 75.
Ken Davis, Status of Search for Multifactorial Primes.

Cf. A204657, A204658, A204659, A204660, A204661, A204662, A204664, A156165, A156167, A085150, A085148, A085146, A037083, A080778, A002981.

Select[Range[0,9999], PrimeQ[Product[# - 8i,{i, 0, Floor[(# - 2)/8]}] + 2] &] (* Indranil Ghosh, Mar 13 2017 *)

for(n=0,9999,isprime(prod(i=0,(n-2)\8,n-8*i)+2)& print1(n","))

a(39)-a(59) from Robert Price, Aug 19 2012

A204664 Numbers n such that n!8-2 is prime.

Original entry on oeis.org

4, 5, 7, 9, 11, 15, 17, 25, 27, 33, 47, 59, 63, 77, 87, 89, 93, 95, 107, 119, 127, 133, 139, 193, 201, 217, 269, 291, 369, 373, 435, 445, 669, 803, 831, 859, 907, 1271, 1705, 1743, 1849, 3087, 3189, 3497, 4221, 4475, 5119, 6013, 8023, 9237, 12755, 16501, 16747, 17021, 17309, 20671, 21539, 28377, 33625, 35645, 36831, 54663, 56223, 65299, 66159, 68121, 69339, 70579, 73511, 77745, 94601

Offset: 1

M. F. Hasler, Jan 17 2012

n!8 = A114800(n).
See also links in A156165.
For odd k, n!k +- 2 is even for all n > k and thus cannot be prime.
a(62) > 50000. - Robert Price, Aug 27 2012
The first 10 associated primes: 2, 3, 5, 7, 31, 103, 151, 3823, 16927, 126223. - Robert Price, Mar 10 2017
a(72) > 10^5. - Robert Price, Apr 24 2017

C. Caldwell and H. Dubner (Eds): The top ten prime numbers: from the unpublished collections of R. Ondrejka (May 2001), Table 21 F, p. 75
Ken Davis, Status of Search for Multifactorial Primes.

Cf. A204657, A204658, A204659, A204660, A204661, A204662, A204663, A156165, A156167, A085150, A085148, A085146, A037083, A080778, A002981.

MultiFactorial[n_, k_] := If[n < 1, 1, n*MultiFactorial[n - k, k]];
Select[Range[4, 50000], PrimeQ[MultiFactorial[#, 8] - 2] &] (* Robert Price, Mar 10 2017 *)

for(n=0,9999,isprime(prod(i=0,(n-2)\8,n-8*i)-2)& print1(n","))

a(46)-a(61) from Robert Price, Aug 27 2012

a(62)-a(71) from Robert Price, Apr 24 2017

A073308 Numbers k such that k! + k + 1 is prime.

Original entry on oeis.org

0, 1, 2, 4, 6, 10, 52, 6822, 30838

Offset: 1

Rick L. Shepherd, Jul 24 2002

Clearly, for k>2, k != 2 (mod 3).
Often m! + 2, m! + 3, ..., m! + m is cited as a constructed sequence of m-1 consecutive composite numbers.
Except for 0, k+1 is prime. - Robert Israel, Jan 13 2015

J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 52, p. 20, Ellipses, Paris 2008.

Cf. A073309 (corresponding primes), A002981 (n!+1 is prime), A073443 (n!-n-1 is prime), A092791.

f[n_]:=n!+n+1; lst={};Do[p=f[n];If[PrimeQ[p],AppendTo[lst,n]],{n,0,2*5!}];lst (* Vladimir Joseph Stephan Orlovsky, Jul 02 2009 *)

for(n=0,1960,if(isprime(n!+n+1),print1(n,",")))

a(n) = A092791(n) - 1. - Seiichi Manyama, Mar 19 2018

a(8) from T. D. Noe, Jan 18 2008

a(9) from Seiichi Manyama (by using the data calculated by Giovanni Resta, May 04 2013), Mar 19 2018

A076681 Numbers k such that 5*k! + 1 is prime.

Original entry on oeis.org

2, 3, 5, 10, 11, 12, 17, 34, 74, 136, 155, 259, 271, 290, 352, 479, 494, 677, 776, 862, 921, 932, 2211, 3927, 4688, 12567

Offset: 1

Phillip L. Poplin (plpoplin(AT)bellsouth.net), Oct 25 2002

a(26) > 4700. - Jinyuan Wang, Feb 04 2020

			k = 3 is here because 5*3! + 1 = 31 is prime.

Cf. A002981, A051915, A076679, A076680, A076682, A076683, A178488, A180626, A126896.

is(k) = ispseudoprime(5*k!+1); \\ Jinyuan Wang, Feb 04 2020

a(25) from Jinyuan Wang, Feb 04 2020

a(26) from Michael S. Branicky, Jul 03 2024

A051856 Numbers k such that (k!)^2 + k! + 1 is prime.

Original entry on oeis.org

0, 1, 2, 3, 4, 6, 76, 2837, 6001, 7076

Offset: 1

Andrew Walker (ajw01(AT)uow.edu.au), Dec 13 1999

			6 is in the sequence because (6!)^2+6!+1=519121 is prime.

H. Dubner, Factorial and primorial primes, J. Rec. Math., 19 (No. 3) (1987)

C. K. Caldwell, The Prime Pages
C. Nash, Prime Form [broken link]
M. Oakes, Re: Gaussian primorial and factorial primes, Primeform, Dec 21 2010 [broken link]
Mike Oakes, Andrew Walker, David Broadhurst, Gaussian primorial and factorial primes, digest of 7 messages in primeform Yahoo group, Dec 20 - Dec 21, 2010.

Cf. A002981, A002982, A046029.

Do[If[PrimeQ[n!^2+n!+1], Print[n]], {n, 600}] (* Farideh Firoozbakht, Jul 12 2003 *)

Edited by R. J. Mathar, Aug 08 2008

a(8)-a(10) from Serge Batalov, Nov 24 2011

A066856 a(n) = omega(n!+1), where omega is the number of distinct primes dividing n, A001221.

Original entry on oeis.org

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

Offset: 1

Robert G. Wilson v, Jan 21 2002

103!+1 = 27437*31084943*C153, so a(103) is unknown until this 153-digit composite is factored. a(104) = 4 and a(105) = 6. - Rick L. Shepherd, Jun 09 2003

Amiram Eldar, Table of n, a(n) for n = 1..139
William Gerst, A conjecture on the prime factorization of n!+1, arXiv:1809.07360 [math.GM], 2018.
Paul Leyland, Factors of n!+1 [Typo in URL corrected by R. J. Mathar, Nov 21 2008]
Hisanori Mishima, Appendix 1. Factorization results for n!+1
Hisanori Mishima, Bernoulli numbers (n = 2 to 114)

Cf. A054990 (bigomega(n!+1)), A002981 (n!+1 is prime), A064237 (n!+1 divisible by a square), A084846 (mu(n!+1)).

[#PrimeDivisors(Factorial(n) + 1): n in [1..55]]; // Vincenzo Librandi, Oct 11 2018

Table[ Length[ FactorInteger[ n! + 1]], {n, 1, 15}]
PrimeNu[Range[50]! + 1] (* Paolo Xausa, Feb 07 2025 *)

PARI
```
for(n=1,64,print1(omega(n!+1),","))
```

More terms from Rick L. Shepherd, Jun 09 2003

cp's OEIS Frontend

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

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A204658 Numbers n such that n!10-1 is prime.

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A204661 Numbers n such that n!8+1 is prime (for n!8 see A114800).

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A204662 Numbers n such that n!8-1 is prime.

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A204663 Numbers n such that n!8 + 2 is prime.

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A204664 Numbers n such that n!8-2 is prime.

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A073308 Numbers k such that k! + k + 1 is prime.

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