A121895 -id:A121895 - cp's OEIS frontend

A122934 Triangle T(n,k) = number of partitions of n into k parts, with each part size divisible by the next.

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 3, 2, 2, 1, 1, 1, 1, 3, 2, 2, 1, 1, 1, 3, 2, 4, 2, 2, 1, 1, 1, 2, 4, 2, 4, 2, 2, 1, 1, 1, 3, 4, 5, 3, 4, 2, 2, 1, 1, 1, 1, 3, 4, 5, 3, 4, 2, 2, 1, 1, 1, 5, 4, 6, 5, 6, 3, 4, 2, 2, 1, 1, 1, 1, 5, 4, 6, 5, 6, 3, 4, 2, 2, 1, 1, 1, 3, 4, 7, 6, 7, 6, 6, 3, 4, 2, 2, 1, 1

Offset: 1

Franklin T. Adams-Watters, Sep 20 2006

			Triangle starts:
  1;
  1, 1;
  1, 1, 1;
  1, 2, 1, 1;
  1, 1, 2, 1, 1;
  1, 3, 2, 2, 1, 1;
  ...
T(6,3) = 2 because of the 3 partitions of 6 into 3 parts, [4,1,1] and [2,2,2] meet the definition; [3,2,1] fails because 2 does not divide 3.

Seiichi Manyama, Rows n = 1..140, flattened (Rows n = 1..50 from G. C. Greubel)
M. Benoumhani and M. Kolli, Finite topologies and partitions, JIS 13 (2010) # 10.3.5.

Column k=1..4 give A057427, A032741, A049822, A121895.

Row sums give A003238.

T[, 1] = 1; T[n, k_] := T[n, k] = DivisorSum[n, If[#==1, 0, T[#-1, k-1]]& ]; Table[T[n, k], {n, 1, 14}, {k, 1, n}] // Flatten (* Jean-François Alcover, Sep 30 2016 *)

T(n,1) = 1. T(n,k+1) = Sum_{d|n, d1} T(d-1,k).

A367613 Numbers with exactly one comma-child.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 15, 16, 17, 19, 20, 21, 22, 23, 24, 25, 26, 28, 29, 30, 31, 32, 34, 35, 37, 38, 39, 40, 41, 42, 43, 44, 46, 47, 48, 49, 50, 51, 53, 55, 56, 57, 58, 59, 60, 61, 62, 64, 65, 66, 67, 68, 69, 70, 73, 74, 75, 76, 77, 78, 79, 80, 82, 83

Offset: 1

N. J. A. Sloane, Dec 15 2023

Complement of union of A367341 and A367346.

See A367338 for definition of comma-child.

Michael S. Branicky, Table of n, a(n) for n = 1..10000
Eric Angelini, Michael S. Branicky, Giovanni Resta, N. J. A. Sloane, and David W. Wilson, The Comma Sequence: A Simple Sequence With Bizarre Properties, arXiv:2401.14346, Youtube

Cf. A121895, A367341 (numbers with no comma-children), A367346 (numbers with two comma-children).

fQ[n_]:=Module[{k=n+10*Last[IntegerDigits[n]]+Range[9]},Length[Select[k,#-n==FromDigits[{Last[IntegerDigits[n]],First[IntegerDigits[#]]}]&]]]==1;
Select[Range[83],fQ[#]&] (* Ivan N. Ianakiev, Dec 16 2023 *)

def ok(n):
    m = n + 10*(n%10)
    return len([m+y for y in range(1, 10) if int(str(m+y)[0]) == y]) == 1
print([k for k in range(1, 100) if ok(k)]) # Michael S. Branicky, Dec 28 2023

cp's OEIS Frontend

A122934 Triangle T(n,k) = number of partitions of n into k parts, with each part size divisible by the next.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Formula