A343347 Number of strict integer partitions of n with a part divisible by all the others.

1, 1, 1, 2, 2, 2, 3, 3, 3, 4, 4, 4, 6, 5, 4, 6, 6, 6, 8, 7, 7, 10, 9, 9, 12, 10, 8, 11, 11, 10, 14, 13, 11, 13, 12, 15, 20, 17, 15, 19, 19, 19, 22, 18, 17, 23, 22, 22, 28, 25, 24, 31, 28, 26, 32, 32, 30, 34, 32, 29, 37, 33, 27, 36, 33, 34, 44, 38, 36, 45, 45

Offset

0,4

Comments

Alternative name: Number of strict integer partitions of n that are empty or have greatest part divisible by all the others.

Links

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

Formula

G.f.: 1 + Sum_{k>0} (x^k/(1 + x^k))*Product_{d|k} (1 + x^d). - Andrew Howroyd, Apr 17 2021

Example

The a(1) = 1 through a(15) = 6 partitions (A..F = 10..15):
  1  2  3   4   5   6   7    8   9    A    B    C     D    E    F
        21  31  41  42  61   62  63   82   A1   84    C1   C2   A5
                    51  421  71  81   91   632  93    841  D1   C3
                                 621  631  821  A2    931  842  E1
                                                B1    A21       C21
                                                6321            8421

Mathematica

Table[Length[Select[IntegerPartitions[n], #=={}||UnsameQ@@#&&And@@IntegerQ/@(Max@@#/#)&]], {n, 0, 30}]

Prog

(PARI) seq(n)={Vec(1 + sum(m=1, n, my(u=divisors(m)); x^m*prod(i=1, #u-1, 1 + x^u[i] + O(x^(n-m+1)))))} \\ Andrew Howroyd, Apr 17 2021

Crossrefs

The dual version is A097986 (non-strict: A083710).

The non-strict version is A130689 (Heinz numbers: complement of A343337).

The strict complement is counted by A343377.

The case with smallest part divisible by all the others is A343378.

The case with smallest part not divisible by all the others is A343380.

A000005 counts divisors.

A000009 counts strict partitions.

A000070 counts partitions with a selected part.

A015723 counts strict partitions with a selected part.

A018818 counts partitions into divisors (strict: A033630).

A167865 counts strict chains of divisors > 1 summing to n.

A339564 counts factorizations with a selected factor.

Cf. A064410, A098743, A200745, A264401, A339563, A341450, A343341, A343344, A343379, A343382.

Sequence in context: A062276 A350327 A053264 * A079440 A192262 A026414

Adjacent sequences: A343344 A343345 A343346 * A343348 A343349 A343350

Keyword

nonn

Author

Gus Wiseman, Apr 16 2021

Status

approved