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proposed

Least k such that n divides s(k)-s(j) for some j satisfying 1<=j<k, where s(j)=2*j^2-j, the jth j-th hexagonal number.

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_Clark Kimberling (ck6(AT)evansville.edu), _, Jan 25 2012

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allocated Least k such that n divides s(k)-s(j) for Clark Kimberlingsome j satisfying 1<=j<k, where s(j)=2*j^2-j, the jth hexagonal number.

2, 3, 3, 5, 2, 5, 3, 9, 3, 5, 4, 6, 4, 3, 5, 17, 5, 7, 6, 6, 6, 4, 7, 11, 7, 11, 4, 11, 8, 5, 9, 33, 9, 14, 8, 9, 10, 6, 5, 13, 11, 12, 12, 5, 7, 7, 13, 18, 9, 14, 6, 12, 14, 8, 11, 13, 8, 23, 16, 6

1,1

See A204892 for a discussion and guide to related sequences.

s[n_] := s[n] = 2 n^2 - n; z1 = 500; z2 = 60;

Table[s[n], {n, 1, 30}] (* A000384, hexagonal numbers *)

u[m_] := u[m] = Flatten[Table[s[k] - s[j], {k, 2, z1}, {j, 1, k - 1}]][[m]]

Table[u[m], {m, 1, z1}] (* A205128 *)

v[n_, h_] := v[n, h] = If[IntegerQ[u[h]/n], h, 0]

w[n_] := w[n] = Table[v[n, h], {h, 1, z1}]

d[n_] := d[n] = First[Delete[w[n], Position[w[n], 0]]]

Table[d[n], {n, 1, z2}] (* A205129 *)

k[n_] := k[n] = Floor[(3 + Sqrt[8 d[n] - 1])/2]

m[n_] := m[n] = Floor[(-1 + Sqrt[8 n - 7])/2]

j[n_] := j[n] = d[n] - m[d[n]] (m[d[n]] + 1)/2

Table[k[n], {n, 1, z2}] (* A205130 *)

Table[j[n], {n, 1, z2}] (* A205131 *)

Table[s[k[n]], {n, 1, z2}] (* A205132 *)

Table[s[j[n]], {n, 1, z2}] (* A205133 *)

Table[s[k[n]] - s[j[n]], {n, 1, z2}] (* A205134 *)

Table[(s[k[n]] - s[j[n]])/n, {n, 1, z2}] (* A205135 *)

Cf. A000384, A204892, A205135.

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Clark Kimberling (ck6(AT)evansville.edu), Jan 25 2012

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editing

allocated for Clark Kimberling

allocated

approved