A039716 -id:A039716 - cp's OEIS frontend

A239942 a(n) = prime(n)! - prime(n - 1)!.

4, 114, 4920, 39911760, 6187104000, 355681201075200, 121289412980736000, 25851895093784567808000, 8841761967887685215658639360000, 8213996892184183115771019264000000, 13763753083003506392138056763855339520000000

Offset: 2

Norman Koch, Mar 29 2014

nonn

			a(3) = Prime(3)! - Prime(2)! = 5! - 3! = 120 - 6 = 114.

Michael De Vlieger, Table of n, a(n) for n = 2..87

Cf. A000040, A039716.

A239942:=n->ithprime(n)!-ithprime(n-1)!: seq(A239942(n), n=2..15); # Wesley Ivan Hurt, Aug 03 2014

a239942[n_Integer] := Prime[n]! - Prime[n - 1]!; Table[a239942[n], {n, 2, 87}] (* Michael De Vlieger, Aug 03 2014 *)

a(n)=prime(n)! - prime(n-1)!;
vector(22,n,a(n+1)) \\ Joerg Arndt, Mar 31 2014

#!/usr/bin/perl
use strict;
use warnings;
use feature 'say';
use Math::Prime::XS qw(is_prime);
use Memoize;
use Math::BigInt;
memoize('factorial');
use Data::Dumper;
my @primes = ();
for (2 .. 200) {
        if(is_prime($_)) {
                push @primes, $_;
        }
}
for (1 .. $#primes) {
        say factorial($primes[$]) - factorial($primes[$ - 1]);
}
sub factorial {
        my $x = Math::BigInt->new(shift);
        return $x if $x == 1;
        return factorial($x - 1) * $x;
}

from gmpy2 import mpz,fac
from sympy import prime
def A239942(n):
    return fac(mpz(prime(n))) - fac(mpz(prime(n-1))) # Chai Wah Wu, Aug 06 2014

a(n) = A039716(n) - A039716(n-1).

A261457 a(n) = prime(n)! - n!.

Original entry on oeis.org

1, 4, 114, 5016, 39916680, 6227020080, 355687428090960, 121645100408791680, 25852016738884976277120, 8841761993739701954543612371200, 8222838654177922817725562840083200, 13763753091226345046315979581580901920998400

Offset: 1

Altug Alkan, Aug 19 2015

			The 3rd prime is 5, a(3)= 5! - 3! = 114.

Cf. A000142, A014689, A039716.

[Factorial(NthPrime(n)) - Factorial(n): n in [1..15]]; // Vincenzo Librandi, Aug 20 2015

Array[Prime[#]! - #! &, {12}] (* Michael De Vlieger, Aug 19 2015 *)

first(m)=vector(m,i,prime(i)!-i!) \\ Anders Hellström, Aug 20 2015

a(n) = prime(n)! - n! = A039716(n) - A000142(n).

A261523 a(n) = prime(n+1)! / (prime(n+1) - prime(n))!.

Original entry on oeis.org

6, 60, 2520, 1663200, 3113510400, 14820309504000, 60822550204416000, 1077167364120207360000, 12280224991305141603532800000, 4111419327088961408862781440000000, 19116323737814368119883304974417920000000, 1393855275548491962840419252226697986048000000000, 30207631531686917818677566034256998753632256000000000

Offset: 1

Altug Alkan, Aug 23 2015

The author comments that this sequence is related to the product A058077(n) * A039716(n). - Charles R Greathouse IV, Aug 29 2015

Cf. A058077, A039716.

[Factorial(NthPrime(n+1)) / Factorial((NthPrime(n+1) - NthPrime(n))): n in [1..15]]; // Vincenzo Librandi, Aug 23 2015

A261523:=n->ithprime(n+1)!/(ithprime(n+1)-ithprime(n))!: seq(A261523(n), n=1..15); # Wesley Ivan Hurt, Aug 23 2015

Table[Prime[i + 1]!/(Prime[i + 1] - Prime[i])!, {i, 15}] (* Vincenzo Librandi, Aug 23 2015 *)

vector( 15,  n,  prime(n+1)! / (prime(n+1) - prime(n))! )

vector( 15,  n,  binomial(prime(n+1), prime(n)) * prime(n)! )

a(n) = prime(n+1)! / (prime(n+1) - prime(n))! = A058077(n) * A039716(n).

A262204 a(n) = (2*prime(n))! / prime(n)!.

Original entry on oeis.org

12, 120, 30240, 17297280, 28158588057600, 64764752532480000, 830034394580628357120000, 4299578163927654889881600000, 212850788988365112429784203264000000, 265847614191284935213187014536606662000640000000

Offset: 1

Altug Alkan, Sep 15 2015

Inspired by simplicity of sequence formula that is (p + p)!/p! where p is n-th prime number.

			For n=1, a(n) = (2*prime(n))! / prime(n)! = 4!/2! = 3*4 = 12.
For n=2, a(n) = (2*prime(n))! / prime(n)! = 6!/3! = 4*5*6 = 120.
For n=3, a(n) = (2*prime(n))! / prime(n)! = 10!/5! = 6*7*8*9*10 = 30240.

Cf. A001813, A039716, A100484, A075069, A262206.

[Factorial(NthPrime(n)+NthPrime(n)) / Factorial(NthPrime(n)): n in [1..10]]; // Vincenzo Librandi, Sep 16 2015

a(n) = (2*prime(n))!/prime(n)!;
vector(10, n, a(n))

a(n) = 2 * A075069(n).
a(n) = A001813(prime(n)). - Michel Marcus, Sep 20 2015
a(n) mod A039716(n) = 0.

A381665 Integers k such that prime(k)!/k^k is an integer.

Original entry on oeis.org

1, 12, 24, 36, 40, 45, 48, 60, 72, 80, 90, 96, 120, 144, 160, 180, 192, 210, 216, 224, 240, 252, 270, 280, 288, 315, 320, 336, 360, 378, 420, 432, 448, 480, 504, 540, 560, 576, 630, 640, 672, 720, 756, 840, 864, 896, 945, 960, 1008, 1080, 1120, 1134, 1152, 1200, 1260, 1280, 1296

Offset: 1

Michel Marcus, Mar 03 2025

nonn

Michel Marcus, Table of n, a(n) for n = 1..211

Cf. A000312, A039716.

Select[Range[1296],IntegerQ[Prime[#]!/#^#]&] (* James C. McMahon, Mar 03 2025 *)

PARI
```
isok(k) = Mod(prime(k)!, k^k) == 0;
```

from collections import Counter
from itertools import count, islice
from sympy import prime, factorint
def A381665_gen(): # generator of terms
    c, p = Counter(), 1
    for k in count(1):
        q, m = prime(k), Counter({a:b*k for a, b in factorint(k).items()})
        c += sum((Counter(factorint(i)) for i in range(p+1,q+1)),start=Counter())
        if m<=c:
            yield k
        p = q
A381665_list = list(islice(A381665_gen(),57)) # Chai Wah Wu, Mar 03 2025

A131491 a(n) = 2*prime(n)!.

Original entry on oeis.org

4, 12, 240, 10080, 79833600, 12454041600, 711374856192000, 243290200817664000, 51704033477769953280000, 17683523987479403909087232000000, 16445677308355845635451125760000000, 27527506182452690092631959163161804800000000

Offset: 1

Giovanni Teofilatto, Jul 29 2007

Cf. A000040, A000142, A039716.

2*Prime[Range[15]]! (* Harvey P. Dale, Dec 19 2011 *)

a(n) = 2*A039716(n) = 2*A000142(A000040(n)). - R. J. Mathar, Jul 31 2007

More terms from Harvey P. Dale, Dec 19 2011

A262398 a(n) = prime(n)! mod prime(n!).

Original entry on oeis.org

0, 0, 3, 56, 511, 194, 46976, 104633, 546681, 41130177, 643108140, 7034542959, 65748733699, 1518781632657, 35097481516962, 396029533782911, 4146710666095789, 159899356955923308, 3662069108121609141, 109629928744379590001, 828180977946159463007

Offset: 1

Altug Alkan, Sep 21 2015

nonn

Inspired by A261997.
a(n) = n and a(n) = prime(n-1) for n=3.
a(n) = 0 only for n=1 and n=2. What is the minimum value of a(n) for n > 2? Is there a possibility of observing that a(n) = 1 or a(n) = 2?

			a(1) = prime(1)! mod prime(1!) = 2 mod 2 = 0.
a(2) = prime(2)! mod prime(2!) = 6 mod 3 = 0.
a(3) = prime(3)! mod prime(3!) = 120 mod 13 = 3.

Cf. A000040, A039716, A062439, A261997.

[Factorial(NthPrime(n)) mod NthPrime(Factorial(n)): n in [1..11]]; // Vincenzo Librandi, Sep 23 2015

Table[Mod[Prime[n]!, Prime[n!]], {n, 15}] (* Michael De Vlieger, Sep 21 2015 *)

a(n) = prime(n)! % prime(n!);
vector(11, n, a(n))

a(n) = A039716(n) mod A062439(n).

a(11)-a(15) from Michael De Vlieger, Sep 21 2015

A267897 a(n) = prime(n)! - prime(n).

Original entry on oeis.org

0, 3, 115, 5033, 39916789, 6227020787, 355687428095983, 121645100408831981, 25852016738884976639977, 8841761993739701954543615999971, 8222838654177922817725562879999969, 13763753091226345046315979581580902399999963

Offset: 1

Vincenzo Librandi, Jan 22 2016

Cf. A000040, A005096, A039716, A114574.

Magma
```
[Factorial(p)-p: p in PrimesUpTo(40)];
```

Array[Prime[#]! - Prime[#] &, 20]
#!-#&/@Prime[Range[20]] (* Harvey P. Dale, Oct 23 2024 *)

lista(nn) = forprime(p=2, nn, print1(p! - p, ", ")); \\ Altug Alkan, Jan 23 2016

a(n) = A039716(n) - A000040(n).

a(n) = A005096(A000040(n)).

A267898 a(n) = prime(n)! + prime(n).

Original entry on oeis.org

4, 9, 125, 5047, 39916811, 6227020813, 355687428096017, 121645100408832019, 25852016738884976640023, 8841761993739701954543616000029, 8222838654177922817725562880000031, 13763753091226345046315979581580902400000037

Offset: 1

Vincenzo Librandi, Jan 22 2016

Cf. A000040, A005095, A039716, A114574.

Magma
```
[Factorial(p)+p: p in PrimesUpTo(40)];
```

Array[Prime[#]! + Prime[#]&, 20]
#!+#&/@Prime[Range[20]] (* Harvey P. Dale, Aug 04 2016 *)

lista(nn) = forprime(p=2, nn, print1(p! + p, ", ")); \\ Altug Alkan, Jan 23 2016

a(n) = A039716(n) + A000040(n).

a(n) = A005095(A000040(n)).

A309789 Numbers n such that -1 + Sum_{k=1..n} prime(k)! is prime, where prime(k) is the k-th prime.

Original entry on oeis.org

2, 3, 4, 20, 25

Offset: 1

Metin Sariyar, Aug 17 2019

The sequence is finite. Proof: for n=112, the sum is 2! + 3! + 5! + ... + 601! + 607! - 1, which is divisible by the next prime, 613. All the factorials of the subsequent primes 613!, 617!, ... are obviously divisible by 613. So after n=112 the sum will always be divisible by 613. And from n=26 to n=112 there are no other primes. So this sequence will not produce any other primes. - Metin Sariyar, Aug 26 2019

			4 is a term because 2, 3, 5, 7 are the first 4 primes and 2! + 3! + 5! + 7! - 1 = 5167 is prime.

Cf. A000040, A000142, A002982, A111179, A039716.

p=-1; lst={}; Do[p+=Prime[n]!; If[PrimeQ[p], AppendTo[lst, n]], {n, 1000}]; lst
Position[Accumulate[Prime[Range[25]]!],?(PrimeQ[#-1]&)]//Flatten (* _Harvey P. Dale, May 03 2021 *)

isok(n) = isprime(sum(k=1, n, prime(k)!) - 1); \\ Michel Marcus, Aug 18 2019

cp's OEIS Frontend

A239942 a(n) = prime(n)! - prime(n - 1)!.

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A261457 a(n) = prime(n)! - n!.

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A261523 a(n) = prime(n+1)! / (prime(n+1) - prime(n))!.

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A262204 a(n) = (2*prime(n))! / prime(n)!.

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A381665 Integers k such that prime(k)!/k^k is an integer.

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A131491 a(n) = 2*prime(n)!.

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A262398 a(n) = prime(n)! mod prime(n!).

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