A222786 -id:A222786 - cp's OEIS frontend

A211202 Positive numbers n such that Lambda_n = A002336(n) is divisible by n.

1, 2, 3, 4, 5, 6, 7, 8, 12, 15, 16, 18, 20, 21, 22, 23, 24

Offset: 1

Raphie Frank, Feb 18 2013

nonn

Observations:
For all n in this sequence to n = 24, then y = Lambda_n/n follows form: y = (x^2 + x^k) - (floor[z^2/4]) or y = (x^2 + x^k) + (floor[z^2/4]); k = 1 or 2 and z = 0, 1, 3, 6 or 7. y (= A222786) gives the average number of spheres/dimension of the laminated lattice Kissing numbers in A222785.
e.g. Where T_x is the x-th triangular number = (1/2*(x^2 + x)), 2*T_x is the x-th pronic number = (x^2 + x) = floor[(2*x + 1)^2/4], and S_x is the x-th square = (x^2) = floor[(2*x)^2/4]:
For k = 1, z = 0 or 1, then n = {1, 4, 6, 8, 15, 20, 24}, x = {1, 2, 3, 5, 12, 29, 90}, and y = 2*T_x = {2, 6, 12, 30, 156, 870, 8190}.
For k = 2, z = 0 or 1, then n = {1, 5, 7, 23}, x = {1, 2, 3, 45}, and y = 2*T_x + 2*T_(-x) = 2*S_x = {2, 8, 18, 4050}.
For k = 1, z = 3, then n = {3, 7, 12, 16}, x = {2, 4, 7, 16}, and y = 2*T_x - 2*T_1 = {4, 18, 54, 270}.
For k = 1, z = 6, then n = {2, 18}, x = {3, 20}, and y = 2*T_x - S_3 = {3, 411}.
For k = 1, z = 7, then n = {5, 7, 8, 21}, x = {4, 5, 6, 36}, and y = 2*T_x - 2*T_3 = {8, 18, 30, 1320}.
For k = 1, z = 7, then n = {6, 7, 12, 22}, x = {0, 2, 6, 47}, and y = 2*T_x + 2*T_3 = {12, 18, 54, 2268}.
For the special case where k = 1 and z = 0 or 1, then all associated x values follow form (A216162(n) - A216162(n - 2)) [type 1] or (A216162(n) - A216162(n - 1)) [type II] for some n in N. Type II x values = {1, 2, 5, 90} (= A215797(n+1)) are associated with the positive Ramanujan-Nagell triangular numbers = {1, 3, 15, 4095} (= A076046(n+1)) by the formula 1/2*(x^2 + x) = T_x.

			Lambda_6/6 = 72/6 = 12, so 6 is in this sequence.
Lambda_12/12 = 648/12 = 54, so 12 is in this sequence.
Lambda_18/18 = 7398/18 = 411, so 18 is in this sequence.
Lambda_24/24 = 196560/24 = 8190, so 24 is in this sequence.
But...
Lambda_19/19 = 10668/19 = 561.47368..., so 19 is not in this sequence.

Cf. A222785, A222786, A002336, A216162

A222785 Terms A002336(k) that are divisible by k.

Original entry on oeis.org

2, 6, 12, 24, 40, 72, 126, 240, 648, 2340, 4320, 7398, 17400, 27720, 49896, 93150, 196560

Offset: 1

Raphie Frank, Mar 05 2013

Average number of spheres/dimension are given in A222786.

Associated dimensions are given in A211202.

Cf. A222786, A211202, A002336.

cp's OEIS Frontend

A211202 Positive numbers n such that Lambda_n = A002336(n) is divisible by n.

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