A209122 Numbers a(n) for which there exists k>1 such that the number of partitions of a(n) into k parts is k.

4, 5, 6, 9, 12, 17, 22, 30, 39, 52, 67, 89, 114, 149, 191, 247, 314, 403, 509, 647, 813, 1024, 1278, 1599, 1983, 2462, 3037, 3746, 4594, 5634, 6873, 8381, 10176, 12344, 14918, 18013, 21674, 26053, 31224, 37378, 44624, 53216, 63304, 75219

Offset: 1

Author

Clark Kimberling, Mar 05 2012

Keywords

• nonn

Comments

For n>2, k=a(n)-n.

Examples

			The partitions of a(4)=9 into k=9-4 parts are
5+1+1+1+1, 4+2+1+1+1, 3+3+1+1+1, 3+2+3+1+1, 2+2+2+2+1.
		

Crossrefs

Cf. A000041, A133041.

Programs

Mathematica

f[n_, k_] := Length[IntegerPartitions[n, {k}]]
t[n_] := Table[f[n, k] - k, {k, 1, n}]
b[n_] := Position[t[n], 0]
c = Flatten[Table[Last[b[n]], {n, 1, 60}]]
x = Flatten[Position[c, 1]]
y = Complement[Range[Length[x]], x]

Formula

a(1)=4, a(2)=5, and a(n)=n+A000041(n) if n>2; i.e., k(n)=A000041(n) for n>2.