A327778 Number of integer partitions of n whose LCM is a multiple of n.

0, 1, 1, 1, 1, 1, 2, 1, 1, 1, 5, 1, 11, 1, 11, 23, 1, 1, 23, 1, 85, 85, 45, 1, 152, 1, 84, 1, 451, 1, 1787, 1, 1, 735, 260, 1925, 1908, 1, 437, 1877, 4623, 1, 14630, 1, 6934, 10519, 1152, 1, 6791, 1, 1817, 10159, 22556, 1, 2819, 47927, 69333, 22010, 4310, 1

Offset

0,7

Links

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Formula

a(n) = 1 <=> n in { A000961 }. - Alois P. Heinz, Sep 26 2019

Example

The partitions of n = 6, 10, 12, and 15 whose LCM is a multiple of n:
  (6)      (10)         (12)             (15)
  (3,2,1)  (5,3,2)      (5,4,3)          (6,5,4)
           (5,4,1)      (6,4,2)          (7,5,3)
           (5,2,2,1)    (8,3,1)          (9,5,1)
           (5,2,1,1,1)  (4,3,3,2)        (10,3,2)
                        (4,4,3,1)        (5,4,3,3)
                        (6,4,1,1)        (5,5,3,2)
                        (4,3,2,2,1)      (6,5,2,2)
                        (4,3,3,1,1)      (6,5,3,1)
                        (4,3,2,1,1,1)    (10,3,1,1)
                        (4,3,1,1,1,1,1)  (5,3,3,2,2)
                                         (5,3,3,3,1)
                                         (5,4,3,2,1)
                                         (5,5,3,1,1)
                                         (6,5,2,1,1)
                                         (5,3,2,2,2,1)
                                         (5,3,3,2,1,1)
                                         (5,4,3,1,1,1)
                                         (6,5,1,1,1,1)
                                         (5,3,2,2,1,1,1)
                                         (5,3,3,1,1,1,1)
                                         (5,3,2,1,1,1,1,1)
                                         (5,3,1,1,1,1,1,1,1)

Maple

a:= proc(m) option remember; local b; b:=
      proc(n, i, l) option remember; `if`(n=0 or i=1,
        `if`(l=m, 1, 0), `if`(i<2, 0, b(n, i-1, l))+
         b(n-i, min(n-i, i), igcd(m, ilcm(l, i))))
      end; `if`(isprime(m), 1, b(m$2, 1))
    end:
seq(a(n), n=0..60);  # Alois P. Heinz, Sep 26 2019

Mathematica

Table[Length[Select[IntegerPartitions[n], Divisible[LCM@@#, n]&]], {n, 30}]
(* Second program: *)
a[m_] := a[m] = Module[{b}, b[n_, i_, l_] := b[n, i, l] = If[n == 0 || i == 1, If[l == m, 1, 0], If[i<2, 0, b[n, i - 1, l]] + b[n - i, Min[n - i, i], GCD[m, LCM[l, i]]]]; If[PrimeQ[m], 1, b[m, m, 1]]];
a /@ Range[0, 60] (* Jean-François Alcover, May 18 2021, after Alois P. Heinz *)

Crossrefs

The Heinz numbers of these partitions are given by A327783.

Partitions whose LCM is equal to their sum are A074761.

Partitions whose LCM is greater than their sum are A327779.

Partitions whose LCM is less than their sum are A327781.

Cf. A000961, A018818, A067538, A290103, A319333, A326842, A326843, A327780.

Sequence in context: A361781 A082063 A260148 * A099940 A157249 A351861

Adjacent sequences: A327775 A327776 A327777 * A327779 A327780 A327781

Keyword

nonn

Author

Gus Wiseman, Sep 25 2019

Status

approved