A073830 a(n) = 4*((n-1)! + 1) + n (mod n*(n + 2)).
0, 2, 0, 8, 0, 10, 56, 12, 22, 14, 0, 16, 182, 18, 34, 20, 0, 22, 380, 24, 46, 26, 552, 28, 29, 30, 58, 32, 0, 34, 992, 36, 37, 38, 74, 40, 1406, 42, 82, 44, 0, 46, 1892, 48, 94, 50, 2256, 52, 53, 54, 106, 56, 2862, 58, 59, 60, 118, 62, 0, 64, 3782, 66, 67, 68, 134, 70
Offset: 1
Keywords
Links
- Amiram Eldar, Table of n, a(n) for n = 1..10000
- Chris Caldwell, Twin Prime
- P. A. Clement, Congruences for sets of primes, The American Mathematical Monthly, Vol. 56, No. 1 (1949), pp. 23-25.
- PlanetMath, Clement’s theorem on twin primes.
Programs
-
Magma
[(4*(Factorial(n-1)+1)+n) mod (n^2+2*n): n in [1..70]]; // Vincenzo Librandi, May 04 2014
-
Maple
A073830 := proc(n) 4*((n-1)!+1)+n ; modp(%,n*(n+2)) ; end proc: seq(A073830(n),n=1..60) ; # R. J. Mathar, Feb 21 2017
-
Mathematica
a[n_] := Mod[4 ((n - 1)! + 1) + n, n (n + 2)]; Array[a, 66] (* Jean-François Alcover, Feb 22 2018 *)
Formula
For n > 1: a(n) = 0 iff (n, n+2) are twin primes (Clement, 1949).
From Bernard Schott, Nov 16 2021: (Start)
If p is an odd prime, and p+2 is composite, then a(p) = p*(p+1).
If m is composite, and m+2 is prime, then a(m) = 2*(m+2).
If n even >= 4, a(n) = n + 4.
If p prime >= 5, a(p^2) = p^2 + 4. (End)
Comments