A067348 Even numbers n such that binomial(n, [n/2]) is divisible by n.

2, 12, 30, 56, 84, 90, 132, 154, 182, 220, 252, 280, 306, 312, 340, 374, 380, 408, 418, 420, 440, 456, 462, 476, 532, 552, 598, 616, 624, 630, 644, 650, 660, 690, 756, 828, 840, 858, 870, 880, 884, 900, 918, 920, 936, 952, 966, 986, 992, 1020, 1054, 1102

Offset: 1

Author

Dean Hickerson, Jan 16 2002

Keywords

• nonn

Comments

This sequence has a surprisingly large overlap with A080385(n); a few values, 2, 420, 920 are exceptional. This means that usually A080383(A067348(n))=7. - Labos Elemer, Mar 17 2003

Conjecture: sequence contains most of 2*A000384(k). Exceptions are k = 8, 18, 20, 23, 35, ... - Ralf Stephan, Mar 15 2004

Links

• David A. Corneth, Table of n, a(n) for n = 1..12000

Crossrefs

Subsequence of A042996.

Cf. A000984, A067315, A080385.

Programs

Mathematica

Select[Range[2, 1200, 2], Mod[Binomial[ #, #/2], # ]==0&]

PARI

val(n, p) = my(r=0); while(n, r+=n\=p);r
is(n) = {if(valuation(n, 2) == 0, return(0)); my(f = factor(n)); for(i=1, #f~, if(val(n, f[i, 1]) - 2 * val(n/2, f[i, 1]) - f[i, 2] < 0, return(0))); return(1)} \\ David A. Corneth, Jul 29 2017

Formula

a(n) = 2*A002503(n-2) + 2.

Appears to be 2*A058008(n). - Benoit Cloitre, Mar 21 2003

Extensions

Name clarified by Peter Luschny, Aug 04 2017