login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A327781 Number of integer partitions of n whose LCM is less than n. 6
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 (list; graph; refs; listen; history; text; internal format)
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}]

(* Second program: *)

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-Fran├žois 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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified July 25 03:36 EDT 2021. Contains 346282 sequences. (Running on oeis4.)