A154349 Sum of proper divisors minus the number of proper divisors of Motzkin number A001006(n).

0, 0, 0, 1, 2, 8, 18, 0, 34, 170, 1643, 3603, 0, 25118, 139063, 474559, 284490, 984006, 6536387, 24265729, 18678366, 96214018, 277799290, 1282283434, 2077807072, 1899874612, 19252363859, 44221482383, 1967547352, 29743945396, 1265868622

Offset: 0

Author

Omar E. Pol, Jan 07 2009

Keywords

• nonn

Comments

Note that, if a(n) != 0 then Motzkin number A001006(n) is a composite number (A002808), otherwise A001006(n) is a noncomposite number (A008578). See A152770.

Links

• Amiram Eldar, Table of n, a(n) for n = 0..201

Crossrefs

Cf. A001006, A001065, A002808, A032741, A008578, A152770, A152981, A152982, A152983, A152988, A152990.

Programs

Maple

with(numtheory): M := proc (n) options operator, arrow: (sum((-1)^j*binomial(n+1, j)*binomial(2*n-3*j, n), j = 0 .. floor((1/3)*n)))/(n+1) end proc: seq(sigma(M(n))-M(n)-tau(M(n))+1, n = 0 .. 30); # Emeric Deutsch, Jan 12 2009

Mathematica

mot[0] = 1; mot[n_] := mot[n] = mot[n - 1] + Sum[mot[k] * mot[n - 2 - k], {k, 0, n - 2}]; diff[n_] := DivisorSigma[1, n] - DivisorSigma[0, n] - n + 1; Table[diff[mot[n]], {n, 0, 30}] (* Amiram Eldar, Nov 26 2019 *)

Formula

a(n) = A001065(A001006(n)) - A032741(A001006(n)) = A152770(A001006(n)).

Extensions

Extended by Emeric Deutsch, Jan 12 2009