A002982 -id:A002982 - cp's OEIS frontend

A258357 Numbers n such that cyclotomic polynomial value Phi(7,n!) is prime.

0, 1, 2, 3, 13, 470, 2957

Offset: 1

Robert Price, May 27 2015

Except for the values 0,1,2 and 3, terms correspond to probable primes.
a(8) > 6502.
Also, numbers n such that n! belongs to A100330. - Michel Marcus, May 30 2015

			3 is in the sequence because Phi(7,3!) = 1 + 6 + 6^2 + 6^3 + 6^4 + 6^5 + 6^6 = 55987 is prime.

Cf. A002981, A002982, A046029, A200906.

Select[Range[0, 6502], PrimeQ[Cyclotomic[7, #!]] &]

A331547 Numbers k such that k and k! - 1 have the same number of divisors.

Original entry on oeis.org

3, 7, 8, 10, 26, 27, 34, 85, 93, 104, 143, 152

Offset: 1

Matthew Niemiro, Jan 20 2020

The sequence also includes: 143, 152, 186, 230, 379, 381, 543, 573, 602. - Daniel Suteu, Jan 21 2020
The sequence also includes 2881. Even though the complete factorization of 136!-1 is not known, 136 is not a term, since 136!-1 is known to be the product of 2 distinct primes and a composite number, so it has at least 4 prime factors and 3 distinct prime factors, thus the number of divisors >= 12, whereas 136 has 8 divisors. - Chai Wah Wu, Feb 26 2020
Similar reasoning (considering only small prime factors of k! - 1) shows that the next terms (> a(12) = 152) can only be within the set {154, 160, 162, 164, 176, 180, 182, 186, 187, 188, 192, 195, 196, 198, 204, ...}. - M. F. Hasler, Feb 26 2020

factordb.com, Status of 154!-1.

Supersequence of A103317.

Cf. A000005, A002982, A064145.

Select[Range[50], DivisorSigma[0, #] - DivisorSigma[0, Factorial[#] - 1] == 0 &]

isok(k) = k>1 && numdiv(k)==numdiv(k!-1); \\ Jinyuan Wang, Jan 20 2020

{is(n)=my(f); n>2&& numdiv(n)>=numdiv(f=factor(n!-1,0))&& if( ispseudoprime(vecmax(f[,1])), numdiv(n)==numdiv(f), numdiv(n)<2*numdiv(f), 0, numdiv(n)==numdiv(n!-1))} \\ Avoids complete factorization if possible. - M. F. Hasler, Feb 26 2020

A331547 = { n > 1 | A000005(n) = A064145(n) }. - M. F. Hasler, Feb 26 2020

a(8)-a(9) from Jinyuan Wang, Jan 20 2020
a(10) from Amiram Eldar, Jan 20 2020
a(11)-a(12) from Chai Wah Wu, Feb 26 2020

A344991 Numbers k such that (2k+1)k!-1 is prime.

Original entry on oeis.org

1, 3, 5, 15, 20, 44, 45, 73, 80, 93, 295, 5395

Offset: 1

Reza K Ghazi, Jun 07 2021

a(13) > 10^4.

Cf. A002982, A344963.

select(k -> isprime((2*k+1)*k!-1), [$1 .. 300])[];

Do[If[PrimeQ[(2*k + 1)*Factorial[k] - 1], Print[k]], {k, 1, 3000}]

for(k=1, 3000, if(isprime((2*k+1)*k!-1), print1(k", ")))

for k in range(1, 3000):
    if is_prime((2*k+1)*factorial(k) - 1):
        print(k)

A051857 Numbers n such that (n!)^2-n!+1 is prime.

Original entry on oeis.org

2, 3, 5, 7, 38, 2319, 2996, 3321, 3892

Offset: 1

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

a(1)-a(9) are verified primes using BLS option in pfgw. - Robert Price, Aug 24 2014

a(10) > 15000. - Robert Price, Aug 24 2014

C. K. Caldwell, The Prime Pages
C. Nash, Prime Form [?Broken link]
M. Oakes, Re: Gaussian primorial and factorial primes, Primeform, Dec 21 2010
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, A051856.

isok(n) = isprime((n!)^2-n!+1); \\ Michel Marcus, Aug 26 2013

a(7)-a(9) from Robert Price, Aug 24 2014

A084830 Numbers k such that (k!)^2 + k! - 1 is prime.

Original entry on oeis.org

2, 3, 4, 5, 6, 14, 17, 50, 111, 254, 506, 6613, 7475

Offset: 1

Farideh Firoozbakht, Jul 12 2003

			5 is in the sequence because (5!)^2 + 5! - 1 = 14519 is prime.

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

Dorin Andrica, George C. Ţurkaş, An elliptic Diophantine equation from the study of partitions, Stud. Univ. Babeş-Bolyai Math. (2019) Vol. 64, No. 3, 349-356.

Cf. A002981, A002982.

Do[If[PrimeQ[n!^2+n!-1], Print[n]], {n, 600}]

a(12)-a(13) from Michael S. Branicky, May 28 2025

A084898 Numbers k such that k^k*k! + 1 is prime.

Original entry on oeis.org

1, 3, 7, 13, 23, 55, 90, 337, 2313, 8767

Offset: 1

Farideh Firoozbakht, Jul 14 2003

All terms under 500 correspond to certified primes (Primo 2.2.0 beta). a(10) > 2500. - Ryan Propper, Apr 05 2006

a(11) > 10000. - Eric Snyder, Jun 03 2022

			7 is in the sequence because 7^7*7! + 1 = 4150656721 is prime.

Cf. A002981, A002982.

Do[If[PrimeQ[n^n*n!+1], Print[n]], {n, 600}]

a(9) from Ryan Propper, Apr 05 2006

a(10) from Eric Snyder, Jun 03 2022

A085700 Numbers k such that (2k)! - (2k-2)! + (2k-4)! - ... + (-1)^k 0! is prime.

Original entry on oeis.org

2, 4, 26, 112, 365, 449, 453

Offset: 1

Farideh Firoozbakht, Jul 18 2003

There is no further term up to 1000. Consider that 3 divides k! + (k-1)! + (k-2)! + ... + 1! (k > 1), so this number is composite for k > 2. Also 5 divides k! - (k-1)! + ... + (-1)^k*1! for k > 2, so this number is composite for k > 3.

Data is complete as there are no further primes for k < 1398 and for all k >= 1398, the given alternating factorial sum is divisible by 2797. - Michael S. Branicky, Dec 22 2024

			4 is in the sequence because 8! - 6! + 4! - 2! + 1 = 39623 is prime.

Cf. A001272, A002981, A002982.

Do[If[PrimeQ[Sum[(-1)^(n-k)(2k)!, {k, 0, n}]], Print[n]], {n, 1000}]

A122724 Primes p such that (2p)! - 1 is also prime.

Original entry on oeis.org

2, 3, 7, 19, 47, 83, 487

Offset: 1

Alexander Adamchuk, Sep 23 2006

Corresponding primes of the form (2p)! - 1 are {23,719,87178291199,523022617466601111760007224100074291199999999,...}.
No other terms < 20663. - Robert Price, Mar 02 2012
No other terms < 104002 using A002982. - Michael S. Branicky, May 14 2025

Cf. A002982, A103317.

Select[Prime[Range[100]],PrimeQ[(2#)!-1]&] (* James C. McMahon, Nov 09 2024 *)

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

cp's OEIS Frontend

A258357 Numbers n such that cyclotomic polynomial value Phi(7,n!) is prime.

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A331547 Numbers k such that k and k! - 1 have the same number of divisors.

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A344991 Numbers k such that (2*k+1)*k!-1 is prime.

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A051857 Numbers n such that (n!)^2-n!+1 is prime.

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A084830 Numbers k such that (k!)^2 + k! - 1 is prime.

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A084898 Numbers k such that k^k*k! + 1 is prime.

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A085700 Numbers k such that (2k)! - (2k-2)! + (2k-4)! - ... + (-1)^k 0! is prime.

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A122724 Primes p such that (2p)! - 1 is also prime.

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A344991 Numbers k such that (2k+1)k!-1 is prime.

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