A327781 Number of integer partitions of n whose LCM is less than n.

0, 0, 1, 2, 4, 5, 9, 12, 18, 22, 30, 37, 52, 69, 89, 110, 143, 163, 204, 243, 298, 374, 451, 516, 620, 790, 932, 1064, 1243, 1454, 1699, 2365, 2733, 3071, 3524, 3945, 4526, 5600, 6361, 7111, 8057, 9405, 10621, 12836, 14395, 16066, 18047, 19860, 22143, 25748

Offset

0,4

Links

Alois P. Heinz, Table of n, a(n) for n = 0..500

Example

The a(2) = 1 through a(8) = 18 partitions:
  (11)  (21)   (22)    (41)     (33)      (61)       (44)
        (111)  (31)    (221)    (42)      (322)      (62)
               (211)   (311)    (51)      (331)      (71)
               (1111)  (2111)   (222)     (421)      (332)
                       (11111)  (411)     (511)      (422)
                                (2211)    (2221)     (611)
                                (3111)    (3211)     (2222)
                                (21111)   (4111)     (3221)
                                (111111)  (22111)    (3311)
                                          (31111)    (4211)
                                          (211111)   (5111)
                                          (1111111)  (22211)
                                                     (32111)
                                                     (41111)
                                                     (221111)
                                                     (311111)
                                                     (2111111)
                                                     (11111111)

Maple

a:= proc(m) option remember; local b; b:=
      proc(n, i, l) option remember; `if`(n=0, 1,
       `if`(i>1, b(n, i-1, l), 0) +(h-> `if`(h<m,
        b(n-i, min(n-i, i), h), 0))(ilcm(l, i)))
      end: `if`(m>0, b(m$2, 1), 0)
    end:
seq(a(n), n=0..70);  # Alois P. Heinz, Oct 10 2019

Mathematica

Table[Length[Select[IntegerPartitions[n], LCM@@#<n&]], {n, 30}]
(* Alternative: *)
a[m_] := a[m] = Module[{b}, b[n_, i_, l_] := b[n, i, l] =
     If[n == 0, 1, If[i>1, b[n, i - 1, l], 0] + Function[h, If[h<m,
     b[n - i, Min[n - i, i], h], 0]][LCM[l, i]]]; If[m>0, b[m, m, 1], 0]];
a /@ Range[0, 70] (* Jean-François Alcover, May 18 2021, after Alois P. Heinz *)

Prog

(PARI)
b(m, n)={my(d=divisors(m)); polcoef(1/prod(i=1, #d, 1 - x^d[i] + O(x*x^n)), n)}
a(n)={sum(m=1, n-1, b(m, n)*sum(i=1, (n-1)\m, moebius(i)))} \\ Andrew Howroyd, Oct 09 2019

Crossrefs

The Heinz numbers of these partitions are given by A327776.

Partitions whose LCM is equal to their sum are A074761.

Partitions whose LCM is greater than their sum are A327779.

Cf. A018818, A290103, A316413, A319333, A326842, A327778, A327780.

Sequence in context: A241444 A082592 A241339 * A241411 A211373 A241734

Adjacent sequences: A327778 A327779 A327780 * A327782 A327783 A327784

Keyword

nonn

Author

Gus Wiseman, Sep 25 2019

Status

approved