A152983 -id:A152983 - cp's OEIS frontend

A152982 Sum of proper divisors of Motzkin number A001006(n).

0, 0, 1, 3, 4, 11, 21, 1, 37, 173, 1648, 3610, 1, 25125, 139086, 474576, 284493, 984021, 6536394, 24265740, 18678381, 96214041, 277799337, 1282283505, 2077807083, 1899874619, 19252363864, 44221482398, 1967547359, 29743945411, 1265868629

Offset: 0

Omar E. Pol, Dec 20 2008

nonn

			a(6)=21 because A001006(6)=51, having as proper divisors 1, 3 and 17. - _Emeric Deutsch_, Dec 31 2008

Amiram Eldar, Table of n, a(n) for n = 0..200

Cf. A001065, A001006, A152762, A152981, A152983.

with(numtheory): M := proc (n) options operator, arrow: sum(binomial(n, 2*k)*binomial(2*k, k)/(k+1), k = 0 .. n) end proc: seq(sigma(M(n))-M(n), n = 0 .. 30); # Emeric Deutsch, Dec 31 2008

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

a(n) = A001065(A001006(n)).

Extended by Emeric Deutsch, Dec 31 2008

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

Original entry on oeis.org

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

Omar E. Pol, Jan 07 2009

nonn

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.

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

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

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

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 *)

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

Extended by Emeric Deutsch, Jan 12 2009

cp's OEIS Frontend

A152982 Sum of proper divisors of Motzkin number A001006(n).

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