A002982 -id:A002982 - cp's OEIS frontend

A121859 Numbers n such that n!*3^n - 1 is prime.

1, 2, 12, 20, 38, 45, 46, 105, 117, 143, 214, 1835, 2931, 7703, 11542, 19225

Offset: 1

Alexander Adamchuk, Sep 10 2006

nonn

Corresponding primes of the form n!*3^n - 1 are a(n)!*3^a(n) - 1 ={2,17,254561089305599,8483004771271882804592639999,...}.

a(17) > 25000. - Robert Price, Jul 22 2013

Cf. A091415, A002982.

Do[f=n!*3^n-1;If[PrimeQ[f],Print[{n,f}]],{n,1,25000}]

a(12)-a(16) from Robert Price, Jul 22 2013

A240160 Least number k such that (k!-n)/k is prime, or 0 if no such k exists.

Original entry on oeis.org

0, 0, 0, 4, 5, 0, 7, 8, 0, 0, 0, 4, 13, 0, 15, 4, 0, 0, 0, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0, 0, 31, 0, 33, 34, 5, 0, 0, 0, 39, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 5, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 7, 6, 0, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 5

Offset: 1

Derek Orr, Aug 01 2014

nonn

For a(n) > 0, a(n) is a divisor of n.

If a(n) = n, then n - 1 is in A002982.

Jens Kruse Andersen, Table of n, a(n) for n = 1..2500

Cf. A002982, A240161, A245756.

a(n)=for(k=1,n,s=(k!-n)/k;if(floor(s)==s,if(ispseudoprime(s),return(k))))
vector(150, n, a(n))

Typo in PARI code fixed by Colin Barker, Aug 02 2014

A270780 Let p_i = the i-th prime. a(i) is the smallest n>1 such that p_i divides n!-1.

Original entry on oeis.org

3, 5, 9, 11, 5, 17, 4, 10, 15, 35, 39, 41, 45, 15, 18, 42, 48, 35, 17, 77, 41, 21, 43, 99, 96, 53, 22, 111, 125, 129, 120, 69, 25, 75, 155, 161, 83, 171, 177, 179, 189, 90, 195, 81, 105, 111, 82, 227, 101, 28, 239, 125, 255, 261, 267, 135, 236, 279, 141, 291

Offset: 3

Peter Shor, Mar 22 2016

Since p divides (p-2)!-1, the i-th term a(i) cannot be much larger than i log i.

			For i=3, the third prime is 5, and 5 divides 3!-1.
The 7th prime is 17, and 17 divides 5!-1, so a(7)=5.

Alois P. Heinz, Table of n, a(n) for n = 3..5000

Cf. A000040, A002982, A270838.

a:= proc(n) local k, p; p:=ithprime(n);
      for k from 2 do if irem(k!, p)=1 then return k fi od
    end:
seq(a(n), n=3..100);  # Alois P. Heinz, Mar 23 2016

snpd[p_]:=Module[{n=2},While[!Divisible[n!-1,p],n++];n]; Table[snpd[p],{p,Prime[Range[3,70]]}] (* Harvey P. Dale, Jun 06 2017 *)

More terms from Alois P. Heinz, Mar 23 2016

A325703 If n = prime(i_1)^j_1 * ... * prime(i_k)^j_k, then a(n) is the denominator of the reciprocal factorial sum j_1/i_1! + ... + j_k/i_k!.

Original entry on oeis.org

1, 1, 2, 1, 6, 2, 24, 1, 1, 6, 120, 2, 720, 24, 3, 1, 5040, 1, 40320, 6, 24, 120, 362880, 2, 3, 720, 2, 24, 3628800, 3, 39916800, 1, 120, 5040, 24, 1, 479001600, 40320, 720, 6, 6227020800, 24, 87178291200, 120, 6, 362880, 1307674368000, 2, 12, 3, 5040, 720

Offset: 1

Gus Wiseman, May 18 2019

Alternatively, if n = prime(i_1) * ... * prime(i_k), then a(n) is the denominator of 1/i_1! + ... + 1/i_k!.

Robert Israel, Table of n, a(n) for n = 1..3168
Gus Wiseman, Sequences counting and ranking integer partitions by their reciprocal sums

Factorial numbers: A000142, A002982, A011371, A022559, A071626, A076934, A108731, A325272, A325508, A325709.

Reciprocal sum: A002966, A316855, A316856, A316857, A318573, A318574, A325618, A325619, A325620, A325621, A325622, A325623, A325624, A325704.

f:= proc(n) local F,t;
    F:= ifactors(n)[2];
    denom(add(t[2]/numtheory:-pi(t[1])!,t=F))
end proc:
map(f, [$1..100]); # Robert Israel, Oct 13 2024

Table[Total[Cases[If[n==1,{},FactorInteger[n]],{p_,k_}:>k/PrimePi[p]!]],{n,100}]//Denominator

a(n) = A318574(A325709(n)).

A336418 Numbers with a factorial number of divisors.

Original entry on oeis.org

1, 2, 3, 5, 7, 11, 12, 13, 17, 18, 19, 20, 23, 28, 29, 31, 32, 37, 41, 43, 44, 45, 47, 50, 52, 53, 59, 61, 63, 67, 68, 71, 73, 75, 76, 79, 83, 89, 92, 97, 98, 99, 101, 103, 107, 109, 113, 116, 117, 124, 127, 131, 137, 139, 147, 148, 149, 151, 153, 157, 163

Offset: 1

Gus Wiseman, Jul 27 2020

nonn

Positive integers n such that A000005(n) belongs to A000142.

A000005 counts divisors.
A048656 counts squarefree divisors of n!.
A325617 counts permutations of the multiset of prime factors of n!.
A336414 counts divisors of n! with distinct prime exponents.
Cf. A000142, A001221, A001222, A002982, A007489, A022559, A027423, A032741, A054991, A090630, A108731, A181796, A251753, A336416.

nn=5;Select[Range[nn!],MemberQ[Array[Factorial,nn],Length[Divisors[#]]]&]

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

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

A236169 Numbers n such that n!*3^n + 1 is prime.

Original entry on oeis.org

0, 2, 3, 8, 13, 19, 124, 199, 288, 359, 692, 3610, 5316, 5372, 7442, 8251, 14288, 16883, 17131

Offset: 1

Robert Price, Jan 19 2014

a(20) > 25000.

Corresponding primes are all verified prime (i.e., not probable prime): 2, 19, 163, 264539521, 9927882482918401, 141383412854531380076544001, ...

Cf. A091415, A002982, A121859.

Do[f=n!*3^n+1;If[PrimeQ[f],Print[{n,f}]],{n,0,25000}]

isok(n) = isprime(n!*3^n + 1); \\ Michel Marcus, Jul 23 2017

a(1)=0 prepended by Robert Price, Jul 22 2017

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A121859 Numbers n such that n!*3^n - 1 is prime.

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A240160 Least number k such that (k!-n)/k is prime, or 0 if no such k exists.

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A270780 Let p_i = the i-th prime. a(i) is the smallest n>1 such that p_i divides n!-1.

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A325703 If n = prime(i_1)^j_1 * ... * prime(i_k)^j_k, then a(n) is the denominator of the reciprocal factorial sum j_1/i_1! + ... + j_k/i_k!.

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A336418 Numbers with a factorial number of divisors.

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