A327780 Number of integer partitions of n whose LCM is 2 * n.

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 2, 12, 0, 0, 6, 0, 10, 32, 6, 0, 8, 0, 9, 0, 32, 0, 505, 0, 0, 108, 16, 147, 258, 0, 20, 170, 134, 0, 2030, 0, 140, 1865, 30, 0, 80, 0, 105, 350, 236, 0, 419, 500, 617, 474, 49, 0, 40966, 0, 56, 8225, 0, 785

Offset

0,15

Links

Andrew Howroyd, Table of n, a(n) for n = 0..1000

Formula

a(n) = Sum_{d|2*n} mu(d)*([x^n] B(2*n/d, x)) for n > 0, where B(m,x) = 1/(Product_{d|m} 1 - x^d). - Andrew Howroyd, Feb 12 2022

Example

The a(10) = 1 through a(20) = 10 partitions (A = 10) (empty columns not shown):
  (541)  (831)  (7421)   (A32)       (9432)     (A82)
                (74111)  (5532)      (9441)     (8552)
                         (6522)      (94221)    (A811)
                         (6531)      (94311)    (85421)
                         (A311)      (942111)   (85511)
                         (53322)     (9411111)  (852221)
                         (65211)                (854111)
                         (532221)               (8522111)
                         (533211)               (85211111)
                         (651111)               (851111111)
                         (5322111)
                         (53211111)

Mathematica

Table[Length[Select[IntegerPartitions[n], LCM@@#==2*n&]], {n, 30}]

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)={if(n<1, 0, sumdiv(2*n, d, moebius(d)*b(2*n/d, n)))} \\ Andrew Howroyd, Oct 09 2019

Crossrefs

The Heinz numbers of these partitions are given by A327775.

Partitions whose LCM is a multiple of their sum are A327778.

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. A018818, A290103, A316413, A326842.

Sequence in context: A110982 A079331 A265490 * A350258 A364781 A166489

Adjacent sequences: A327777 A327778 A327779 * A327781 A327782 A327783

Keyword

nonn

Author

Gus Wiseman, Sep 25 2019

Status

approved