A261997 -id:A261997 - cp's OEIS frontend

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

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

A362056 Prime numbers of the form prime(k)! - prime(k!).

Original entry on oeis.org

3, 107, 4951, 39916141, 355687428046967

Offset: 1

Michael S. Branicky, Apr 06 2023

No other terms < prime(22)! - prime(22!) (using A062439).

Problem suggested by Carlos Rivera as a follow-on to Puzzle 1127 (see links).

			prime(2)! - prime(2!) = 3 is prime.
prime(3)! - prime(3!) = 107 is prime.
prime(4)! - prime(4!) = 4951 is prime.
prime(5)! - prime(5!) = 39916141 is prime.
prime(7)! - prime(7!) = 355687428046967 is prime.

Carlos Rivera, Puzzle 1127. Prime(11!)+Prime(11)!, The Prime Puzzles and Problems Connection.
Carlos Rivera, Puzzle 1128. Prime(n)!-Prime(n!), The Prime Puzzles and Problems Connection.

Cf. A000040, A000142, A062439.

Prime terms in A261997.

A261997(k) = A000142(A000040(k)) - A000040(A000142(k)) is in A000040.

cp's OEIS Frontend

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

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