A212187 Number of distinct sums of reciprocals of parts of partitions of n.

1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 41, 55, 75, 97, 129, 166, 213, 272, 343, 430, 536, 664, 822, 1008, 1230, 1495, 1808, 2178, 2616, 3122, 3720, 4416, 5221, 6164, 7249, 8497, 9941, 11593, 13481, 15665, 18150, 20971, 24184, 27827, 31974, 36650, 41944, 47930, 54670

Offset

0,3

Comments

This is also the number of distinct spring constants you can make with n Belleville washers.

Links

Alois P. Heinz, Table of n, a(n) for n = 0..73

Wikipedia, Belleville washer

Example

For n = 4, the partitions are [4], which gives 1/4, [3,1] which gives 1/3+1/1 = 4/3, [2,2] which gives 1/2+1/2 = 1, [2,1,1] which gives 1/2+1/1+1/1 = 5/2 and [1,1,1,1] which gives 1/1+1/1+1/1+1/1 = 4.  These five sums are all distinct, so a(4) = 5.

Maple

b:= proc(n, i) option remember; `if`(n=0, {0}, `if`(i<1, {},
      {b(n, i-1)[], `if`(i>n, {}, map(x-> x+1/i, b(n-i, i)))[]}))
    end:
a:= n-> nops(b(n, n)):
seq(a(n), n=0..40);  # Alois P. Heinz, Feb 18 2013

Mathematica

a[n_] := Length@Union[Plus @@@ (1/IntegerPartitions[n])]; a/@Range[30] (* Giovanni Resta, Feb 14 2013  *)
b[n_, i_] := b[n, i] = If[n == 0, {0}, If[i < 1, {}, Union @ Flatten @ {b[n, i-1], If[i > n, {}, Map[Function[x, x + 1/i], b[n-i, i]]]}]];
a[n_] := Length[b[n, n]];
Table[a[n], {n, 0, 40}] (* Jean-François Alcover, May 28 2018, after Alois P. Heinz *)

Crossrefs

Sequence in context: A325857 A023030 A246580 * A332280 A035998 A137792

Adjacent sequences: A212184 A212185 A212186 * A212188 A212189 A212190

Keyword

nonn

Author

Sachi Hashimoto, Feb 13 2013

Extensions

a(31)-a(48) from Giovanni Resta, Feb 18 2013

Status

approved