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A211202 Positive numbers n such that Lambda_n = A002336(n) is divisible by n. 2
1, 2, 3, 4, 5, 6, 7, 8, 12, 15, 16, 18, 20, 21, 22, 23, 24 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

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.

LINKS

Table of n, a(n) for n=1..17.

EXAMPLE

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.

CROSSREFS

Cf. A222785, A222786, A002336, A216162

Sequence in context: A240082 A321334 A238084 * A066418 A015845 A030702

Adjacent sequences:  A211199 A211200 A211201 * A211203 A211204 A211205

KEYWORD

nonn

AUTHOR

Raphie Frank, Feb 18 2013

STATUS

approved

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Last modified May 22 00:25 EDT 2019. Contains 323472 sequences. (Running on oeis4.)