A209489 Number of partitions of 1 into rational parts a/b satisfying 0<a<b<=n.

0, 1, 3, 6, 12, 19, 33, 50, 77, 111, 166, 229, 329, 448, 615, 824, 1120, 1467, 1956, 2538, 3313, 4258, 5512, 6993, 8944, 11278, 14258, 17838, 22402, 27788, 34629, 42747, 52832, 64844, 79706, 97232, 118868, 144392, 175474, 212168, 256750

Offset

1,3

Links

Table of n, a(n) for n=1..41.

Formula

a(n)=a(n-1)-1+(number of partitions of n into relatively prime parts).

Example

a(1)=0.
a(2)=1 counts 1/2 + 1/2.
a(3)=3 counts 1/2 + 1/2, 1/3 + 1/3 + 1/3, and 1/3 + 2/3.

Mathematica

p[n_] := p[n] = IntegerPartitions[n];
l[n_] := Length[p[n]];
p[n_, j_] := p[n, j] = Part[p[n], j]
g[n_, j_] := g[n, j] = Apply[GCD, p[n, j]]
h[n_] := h[n] = Table[g[n, j], {j, 1, l[n]}]
c[n_] := c[n] = Count[h[n], 1]
Table[c[n], {n, 0, 45}]      (* A000837 *)
s[n_] := Sum[c[k], {k, 1, n}]
Table[s[n] - 1, {n, 1, 45}]  (* A209489 *)

Crossrefs

Cf. A000837.

Sequence in context: A180622 A125851 A351621 * A181962 A226220 A218076

Adjacent sequences: A209486 A209487 A209488 * A209490 A209491 A209492

Keyword

nonn

Author

Clark Kimberling, Mar 09 2012

Status

approved