A055976 -id:A055976 - cp's OEIS frontend

A061007 a(n) = -(n-1)! mod n.

0, 1, 1, 2, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0

Offset: 1

Henry Bottomley, Apr 12 2001

The following sequences all appear to have the same parity (with an extra zero term at the start of A010051): A010051, A061007, A035026, A069754, A071574. - Jeremy Gardiner, Aug 09 2002

In particular, this is identical to the isprime function A010051 except for a(4) = 2 instead of 0. This is equivalent to Wilson's theorem, (n-1)! == -1 (mod n) iff n is prime. If n = p*q with p, q > 1, then p, q < n-1 and (n-1)! will contain the two factors p and q, unless p = q = 2 (if p = q > 2 then also 2p < n-1, so there are indeed two factors p in (n-1)!), whence (n-1)! == 0 (mod n). - M. F. Hasler, Jul 19 2024

			a(4) = 2 since -(4 - 1)! = -6 = 2 mod 4.
a(5) = 1 since -(5 - 1)! = -24 = 1 mod 5.
a(6) = 0 since -(6 - 1)! = -120 = 0 mod 6.

Antti Karttunen, Table of n, a(n) for n = 1..10000

Positive for all but the first term of A046022.

Cf. A000040 (the primes), A000142, A010051 (isprime function), A055976, A061006, A061008, A061009.

Table[Mod[-(n - 1)!, n], {n, 100}] (* Alonso del Arte, Mar 20 2014 *)

A061007(n) = ((-((n-1)!))%n); \\ Antti Karttunen, Aug 27 2017

apply( {A061007(n) = !(n-1)!%n}, [0..99]) \\ M. F. Hasler, Jul 19 2024

from sympy import isprime
def A061007(n): return 2 if n == 4 else int(isprime(n)) # Chai Wah Wu, Mar 22 2023

a(4) = 2, a(p) = 1 for p prime, a(n) = 0 otherwise. Apart from n = 4, a(n) = A010051(n) = A061006(n)/(n-1).

A093521 Runs of 1's of lengths 1, prime(1), prime(2), prime(3), ... separated by 0's.

Original entry on oeis.org

1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0

Offset: 1

Robert G. Wilson v, Mar 29 2004

nonn

Carl Sagan's "Contact" sequence.

Zeros occur at positions given by 1+A110895(k). - Antti Karttunen, Nov 08 2018

W. A. Dembski and J. M. Kushiner, Signs of Intelligence, Baker Book House Co., Grand Rapids, MI, p30-31, 2001,
Carl Sagan, Contact, Simon and Schuster, Chapter 4 "Prime Numbers," pp. 68-82, NY, 1985.

Antti Karttunen, Table of n, a(n) for n = 1..24235 (first 101 runs)
Robin Dougherty, Robert Zemeckis' Contact, Salon.
Alex Kasman, Mathematical Fiction, Contact (1985).
Alex Kasman, How 'Contact' by Carl Sagan Ends.
Index entries for characteristic functions

Cf. A000040, A000042, A005171, A031974, A055976, A056051, A066247, A091247, A110895, A175851, A175856.

a = Table[1, {100}]; Do[ a[[Sum[Prime[i], {i, n}] + n]] = 0, {n, 1, 8}]; a

up_to = 111;
A093521list(up_to) = { my(v=vector(up_to), i=2, j); v[1] = 1; v[2] = 0; forprime(p=2, oo, j=p; while(j, if(i==up_to, return(v), i++; v[i] = 1; j--)); if(i==up_to, return(v), i++; v[i] = 0)); };
v093521 = A093521list(up_to);
A093521(n) = v093521[n];

Data section extended up to n=111 by Antti Karttunen, Nov 08 2018

A301316 a(n) = ((n-1)! + 1) mod n^2.

Original entry on oeis.org

0, 2, 3, 7, 0, 13, 35, 49, 64, 81, 11, 1, 0, 57, 1, 1, 85, 1, 38, 1, 1, 133, 184, 1, 1, 521, 1, 1, 522, 1, 589, 1, 1, 885, 1, 1, 259, 381, 1, 1, 656, 1, 559, 1, 1, 553, 282, 1, 1, 1, 1, 1, 1802, 1, 1, 1, 1, 2553, 1593, 1, 3416, 993, 1, 1, 1, 1, 804

Offset: 1

Stanislav Sykora, Mar 18 2018

nonn

By definition, when n > 1, a(n) = 0 then n is a Wilson prime (A007540).
For a(n) to equal 1, (n-1)! must be divisible by n^2 which is the prevailing case for large n. For example, all n which are a product of more than two distinct primes belong to this category. So do all proper powers of primes except 2^2, 2^3, and 3^2. Obviously, when a(n) = 1, then also A055976(n) = 1.
The cases of a(n) > 1 include, for example, all primes other than Wilson's and all numbers of the form n=2*p, where p is a prime.

			From _Muniru A Asiru_, Mar 20 2018: (Start)
((1-1)! + 1) mod 1^2 = (0! +1) mod 1 = 2 mod 1 = 0.
((2-1)! + 1) mod 2^2 = (1! +1) mod 4 = 2 mod 4 = 2.
((3-1)! + 1) mod 3^2 = (2! +1) mod 9 = 3 mod 9 = 3.
((4-1)! + 1) mod 4^2 = (3! +1) mod 16 = 7 mod 16 = 7.
((5-1)! + 1) mod 5^2 = (4! +1) mod 25 = 25 mod 25 = 0.
... (End)

Stanislav Sykora, Table of n, a(n) for n = 1..10000
Wikipedia, Wilson prime

Cf. A000290, A038507, A007540, A055976, A301317.

List([1..60],n->(Factorial(n-1)+1) mod n^2); # Muniru A Asiru, Mar 20 2018

seq((factorial(n-1)+1) mod n^2,n=1..60); # Muniru A Asiru, Mar 20 2018

Array[Mod[(# - 1)! + 1, #^2] &, 67] (* Michael De Vlieger, Apr 21 2018 *)

a(n) = ((n-1)! + 1) % n^2; \\ Michel Marcus, Mar 18 2018

a(n) = ((n-1)! + 1) mod n^2. - Jon E. Schoenfield, Mar 18 2018

a(n) = A038507(n-1) mod A000290(n). - Michel Marcus, Mar 20 2018

A301317 a(n) = (n-1)! + 1 mod n^3.

Original entry on oeis.org

0, 2, 3, 7, 25, 121, 35, 433, 226, 881, 495, 1, 676, 1233, 2701, 2049, 4420, 1, 4009, 1, 2647, 6425, 4945, 1, 626, 15393, 1, 1, 13137, 1, 21731, 1, 13069, 2041, 1, 1, 23532, 19153, 50194, 1, 14104, 1, 41237, 1, 1, 76729, 86433, 1, 1, 1, 78031, 1, 77645

Offset: 1

Stanislav Sykora, Mar 18 2018

nonn

There is no known number n > 1 for which a(n)=0.
For a(n) to equal 1, (n-1)! must be divisible by n^3 which tends to be the most frequent case for large n. For example, all n which are a product of three or more distinct primes belong to this category. So do all proper powers of primes except 2^2, 2^3, 2^4, 3^2, and 5^2.
Obviously, when a(n) = 1, then also A055976(n) = 1 and A301316(n) = 1.
If n is prime, a(n) is divisible by n. - Robert Israel, Mar 20 2018

			From _Muniru A Asiru_, Mar 20 2018: (Start)
((1-1)! + 1) mod 1^3 = (0! +1) mod 1 = 2 mod 1 = 0.
((2-1)! + 1) mod 2^3 = (1! +1) mod 8 = 2 mod 8 = 2.
((3-1)! + 1) mod 3^3 = (2! +1) mod 27 = 3 mod 27 = 3.
((4-1)! + 1) mod 4^3 = (3! +1) mod 64 = 7 mod 64 = 7.
((5-1)! + 1) mod 5^3 = (4! +1) mod 125 = 25 mod 125 = 25.
... (End)

Stanislav Sykora, Table of n, a(n) for n = 1..10000
Wikipedia, Wilson prime

Cf. A000578, A038507, A055976, A301316.

List([1..60],n->(Factorial(n-1)+1) mod n^3); # Muniru A Asiru, Mar 20 2018

seq((factorial(n-1)+1) mod n^3,n=1..60); # Muniru A Asiru, Mar 20 2018

Array[Mod[(# - 1)! + 1, #^3] &, 53] (* Michael De Vlieger, Mar 19 2018 *)

a(n) = ((n-1)! + 1) % n^3; \\ Michel Marcus, Mar 18 2018

a(n) = ((n-1)! + 1) mod n^3. - Jon E. Schoenfield, Mar 18 2018

a(n) = A038507(n-1) mod A000578(n). - Michel Marcus, Mar 20 2018

A249355 Remainder of n!+2 divided by n+2.

Original entry on oeis.org

1, 0, 0, 3, 2, 3, 2, 2, 2, 3, 2, 3, 2, 2, 2, 3, 2, 3, 2, 2, 2, 3, 2, 2, 2, 2, 2, 3, 2, 3, 2, 2, 2, 2, 2, 3, 2, 2, 2, 3, 2, 3, 2, 2, 2, 3, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 3, 2, 3, 2, 2, 2, 2, 2, 3, 2, 2, 2, 3, 2, 3, 2, 2, 2, 2, 2, 3, 2, 2, 2, 3, 2, 2, 2, 2, 2, 3, 2, 2

Offset: 0

M. F. Hasler, Oct 26 2014

nonn

Antti Karttunen, Table of n, a(n) for n = 0..2048

Cf. A055976, A242707.

[(Factorial(n)+2) mod(n+2): n in [0..100]]; // Vincenzo Librandi, Oct 27 2014

Table[Mod[n!+2,n+2],{n,0,100}] (* Harvey P. Dale, Sep 01 2022 *)

PARI
```
a(n)=lift(prod(k=2,n,k,Mod(1,n+2))+2)
```

A249355(n)=if(n>2,isprime(n+2)+2,!n) \\ M. F. Hasler, Oct 31 2014

If n+2 = p > 4 is prime, then a(n) = 3. Indeed, it is known that (p-2)! = 1 (mod p) for all primes p. Thus n!+2 = 1+2 = 3 (mod n+2).

If n+2 is composite and n > 2 then a(n) = 2. There are two cases: n+2 = a*b with a < b <= n (so n! is divisible by a*b), or n+2 = a^2 with 2*a <= n (so n! is divisible by a*(2*a)). - Robert Israel, Oct 27 2014

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A061007 a(n) = -(n-1)! mod n.

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A093521 Runs of 1's of lengths 1, prime(1), prime(2), prime(3), ... separated by 0's.

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A301316 a(n) = ((n-1)! + 1) mod n^2.

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A301317 a(n) = (n-1)! + 1 mod n^3.

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A249355 Remainder of n!+2 divided by n+2.

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