%I #21 Dec 15 2022 12:24:38

%S 1,3,4,5,7,11,13,17,19,23,25,29,30,31,37,41,43,47,49,53,59,61,67,71,

%T 73,79,83,89,97,101,103,107,109,113,121,127,131,137,139,149,151,157,

%U 163,167,169,173,179,181,191,193,197,199,211,223,227,229,233,239,241,251,257

%N Numbers k such that the k-th Motzkin number == 1 (mod k).

%C All odd primes and all squares of primes are in the sequence. First composite (and not square of prime) are : 30, 464, 902, 21475, ... (A081736). [_Scott R. Shannon_ points out that this comment is wrong, since 9 is missing. Are there other errors? The comment needs to checked and corrected. - _N. J. A. Sloane_, Dec 15 2022]

%H Charles R Greathouse IV, <a href="/A081735/b081735.txt">Table of n, a(n) for n = 1..1000</a>

%t motzkin[0] = 1; motzkin[n_] := motzkin[n] = motzkin[n - 1] + Sum[motzkin[k] * motzkin[n - k - 2], {k, 0, n - 2}]; Select[Range[250], # == 1 || Mod[motzkin[#], #] == 1 &] (* _Amiram Eldar_, May 23 2022 *)

%o (PARI) a001006(n) = polcoeff((1-x-sqrt((1-x)^2-4*x^2+x^3*O(x^n)))/(2*x^2), n);

%o for(n=1, 1e3, if((a001006(n)-1) % n == 0, print1(n, ", "))); \\ _Altug Alkan_, Jan 07 2016

%Y Cf. A001006, A081736.

%K nonn

%O 1,2

%A _Benoit Cloitre_, Apr 06 2003

%E First term 1 prepended by _Altug Alkan_, Jan 07 2016