A320167 Regular triangle where T(n,k) = Sum (-1)^i, where the sum is over all factorizations of n into i factors that are all > 1 and <= k.

1, 0, -1, 0, 0, -1, 0, 1, 1, 0, 0, 0, 0, 0, -1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, -1, 0, -1, -1, 0, 0, 0, 0, -1, 0, 0, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, -1, 0, 0, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0

Offset

1

Links

Table of n, a(n) for n=1..84.

Formula

T(1,k) = 1, T(n,k) = -Sum_{d|n, 1 < d <= k} T(n/d,d).

Example

Triangle begins:
  1
  0 -1
  0  0 -1
  0  1  1  0
  0  0  0  0 -1
  0  0  1  1  1  0
  0  0  0  0  0  0 -1
  0 -1 -1  0  0  0  0 -1
  0  0  1  1  1  1  1  1  0
  0  0  0  0  1  1  1  1  1  0
  0  0  0  0  0  0  0  0  0  0 -1
  0  0 -1  0  0  1  1  1  1  1  1  0
  0  0  0  0  0  0  0  0  0  0  0  0 -1
  0  0  0  0  0  0  1  1  1  1  1  1  1  0
  0  0  0  0  1  1  1  1  1  1  1  1  1  1  0
  0  1  1  1  1  1  1  2  2  2  2  2  2  2  2  1
  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 -1
  0  0 -1 -1 -1  0  0  0  1  1  1  1  1  1  1  1  1  0
  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 -1
  0  0  0  0  0  0  0  0  0  1  1  1  1  1  1  1  1  1  1  0

Mathematica

u[n_, k_]:=If[n==1, 1, -Sum[u[n/d, d], {d, Select[Rest[Divisors[n]], #<=k&]}]]
Table[u[n, k], {n, 20}, {k, n}]

Crossrefs

Cf. A001055, A045778, A066032, A114592, A162247, A316441, A319237, A319238, A320835, A320836.

Cf. A195017, A281118, A317145, A317176, A330935.

Sequence in context: A129569 A266611 A286752 * A030658 A187950 A112539

Adjacent sequences: A320164 A320165 A320166 * A320168 A320169 A320170

Keyword

sign,tabl

Author

Gus Wiseman, Oct 22 2018

Status

approved