The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation. Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A086748 Numbers m such that when C(2k, k) == 1 (mod m) then k is necessarily even. 0
 3, 5, 9, 15, 21, 25, 27, 33, 35, 39, 45, 51, 55, 57, 63, 65, 69, 75, 81, 85, 87, 93, 95, 99, 105, 111, 115, 117, 123, 125, 129, 135, 141, 145, 147, 153, 155, 159, 165, 171, 175, 177, 183, 185, 189, 195, 201, 205, 207, 213, 215, 219, 225, 231, 235, 237, 243, 245 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS From Jinyuan Wang, Apr 05 2020: (Start) All terms are odd, because C(2k, k) is always divisible by 2. If m is a term, then m*t is also a term for odd numbers t. Theorem 1: if C(2k, k) == 1 (mod 3) then k is necessarily even. If C(2k, k) == 2 (mod 3) then k is necessarily odd. Proof: for k < 6 it is correct. We have C(6r, 3r) == C(2r, r) (mod 3) and C(6r+4, 3r+2) == C(2r, r)*C(4, 2) == 0 (mod 3). Suppose k is the least value such that theorem 1 is incorrect, then k must be of the form 3r+1. But C(6r+2, 3r+1) == C(2r, r)*C(2, 1) (mod 3), which means that r is a smaller counterexample, a contradiction! Theorem 2: if C(2k, k) == 1 or 4 (mod 5) then k is necessarily even. If C(2k, k) == 2 or 3 (mod 5) then k is necessarily odd. Note that C(10r, 5r) == C(2r, r) (mod 5), C(10r+2, 5r+1) == C(2r, r)*C(2, 1) (mod 5), C(10r+4, 5r+2) == C(2r, r)*C(4, 2) (mod 5), C(10r+6, 5r+3) == C(2r, r)*C(6, 3) (mod 5) and C(10r+8, 5r+4) == C(2r, r)*C(8, 4) (mod 5). The proof is similar to that of theorem 1. (End) Up to m < 1000, all odd values are either terms, because of the form 3*(2t-1) or 5*(2t-1) (as proved by Jinyuan Wang), or there exist an odd k <= 7412629 such that C(2k, k) == 1 (mod m). - Giovanni Resta, Apr 05 2020 LINKS CROSSREFS Cf. A000984. Sequence in context: A029533 A018685 A107994 * A014957 A014876 A045602 Adjacent sequences:  A086745 A086746 A086747 * A086749 A086750 A086751 KEYWORD nonn AUTHOR Benoit Cloitre, Jul 30 2003 EXTENSIONS 13 removed and offset changed by Jinyuan Wang, Apr 04 2020 23 removed and more terms added by Giovanni Resta, Apr 05 2020 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified May 30 15:21 EDT 2020. Contains 334726 sequences. (Running on oeis4.)