A307223 Irregular table T(n, k) read by rows: n-th row gives number of subsets of the divisors of n which sum to k for 1 <= k <= sigma(n).

1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1

Offset

1,24

Comments

T(n, k) > 0 for all values of k iff n is practical (A005153).

Links

Amiram Eldar, Table of n, a(n) for n = 1..8299 (rows 1..100 flattened)

Formula

T(n, n) = A033630(n).

T(n, A030057(n)) = 0 if there is a 0 in the n-th row, i.e. A030057(n) <= sigma(n) or n is not practical.

Example

Table begins as:
  1
  1,1,1
  1,0,1,1
  1,1,1,1,1,1,1
  1,0,0,0,1,1
  1,1,2,1,1,2,1,1,2,1,1,1
  1,0,0,0,0,0,1,1
  1,1,1,1,1,1,1,1,1,1,1,1,1,1,1
  1,0,1,1,0,0,0,0,1,1,0,1,1
  1,1,1,0,1,1,1,1,0,1,1,1,1,0,1,1,1,1

Mathematica

T[n_, k_] := Module[{d = Divisors[n]}, SeriesCoefficient[Series[Product[1 + x^d[[i]], {i, Length[d]}], {x, 0, k}], k]]; Table[T[n, k], {n, 1, 10}, {k, 1, DivisorSigma[1, n]}] // Flatten

Crossrefs

Cf. A000203 (row lengths), A307224 (row products).

Cf. A005153, A027750, A030057, A033630, A119348, A225561, A237287, A322860.

Sequence in context: A101428 A360171 A348528 * A321787 A379144 A331596

Adjacent sequences: A307220 A307221 A307222 * A307224 A307225 A307226

Keyword

nonn,tabf

Author

Amiram Eldar, Mar 29 2019

Status

approved