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 A301316 a(n) = ((n-1)! + 1) mod n^2. 2
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS 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. LINKS Stanislav Sykora, Table of n, a(n) for n = 1..10000 Wikipedia, Wilson prime FORMULA 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 EXAMPLE 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) MAPLE seq((factorial(n-1)+1) mod n^2, n=1..60); # Muniru A Asiru, Mar 20 2018 MATHEMATICA Array[Mod[(# - 1)! + 1, #^2] &, 67] (* Michael De Vlieger, Apr 21 2018 *) PROG (PARI) a(n) = ((n-1)! + 1) % n^2; \\ Michel Marcus, Mar 18 2018 (GAP) List([1..60], n->(Factorial(n-1)+1) mod n^2); # Muniru A Asiru, Mar 20 2018 CROSSREFS Cf. A000290, A038507, A007540, A055976, A301317. Sequence in context: A263501 A203143 A249523 * A023048 A083521 A104691 Adjacent sequences:  A301313 A301314 A301315 * A301317 A301318 A301319 KEYWORD nonn AUTHOR Stanislav Sykora, Mar 18 2018 STATUS approved

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Last modified November 25 08:58 EST 2020. Contains 338623 sequences. (Running on oeis4.)