A340827 Number of strict integer partitions of n into divisors of n whose length also divides n.

1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 5, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 5, 1, 1, 1, 3, 1, 2, 1, 1, 1, 1, 1, 6, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 25, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 18, 1, 1, 1, 1, 1, 2, 1, 3, 1, 1, 1, 17, 1, 1, 1, 1, 1, 14, 1, 1, 1, 1, 1, 12, 1, 1, 1, 3, 1, 2, 1, 1, 1

Offset

1,6

Comments

The first element not in A326715 that is however a Heinz number of these partitions is 273.

Links

David A. Corneth, Table of n, a(n) for n = 1..10000 (first 2519 terms from Antti Karttunen)

Antti Karttunen, Scheme program for computing this sequence

Example

The a(n) partitions for n = 6, 12, 24, 90, 84:
  6       12        24            90                      84
  3,2,1   6,4,2     12,8,4        45,30,15                42,28,14
          6,3,2,1   12,6,4,2      45,30,9,5,1             42,21,14,7
                    12,8,3,1      45,18,15,9,3            42,28,12,2
                    8,6,4,3,2,1   45,30,10,3,2            42,28,6,4,3,1
                                  45,18,15,10,2           42,28,7,4,2,1
                                  45,30,6,5,3,1           42,14,12,7,6,3
                                  45,30,9,3,2,1           42,21,12,4,3,2
                                  45,15,10,9,6,5          42,21,12,6,2,1
                                  45,18,10,9,5,3          42,21,14,4,2,1
                                  45,18,10,9,6,2          28,21,14,12,6,3
                                  45,18,15,6,5,1          28,21,14,12,7,2
                                  45,18,15,9,2,1          42,21,7,6,4,3,1
                                  30,18,15,10,6,5,3,2,1   42,14,12,7,4,3,2
                                                          42,14,12,7,6,2,1
                                                          28,21,14,12,4,3,2
                                                          28,21,14,12,6,2,1

Mathematica

Table[Length[Select[IntegerPartitions[n, All, Divisors[n]], UnsameQ@@#&&Divisible[n, Length[#]]&]], {n, 30}]

Prog

(PARI) A340827(n, divsleft=List(divisors(n)), rest=n, len=0) = if(rest<=0, !rest && !(n%len), my(s=0, d); forstep(i=#divsleft, 1, -1, d = divsleft[i]; listpop(divsleft, i); if(rest>=d, s += A340827(n, divsleft, rest-d, 1+len))); (s)); \\ Antti Karttunen, Feb 22 2023
(Scheme) ;; See the Links-section. - Antti Karttunen, Feb 22 2023

Crossrefs

Note: A-numbers of Heinz-number sequences are in parentheses below.

The non-strict case is A326842 (A326847).

A018818 = partitions using divisors (A326841).

A047993 = balanced partitions (A106529).

A067538 = partitions whose length/maximum divides sum (A316413/A326836).

A072233 = partitions by sum and length, with strict case A008289.

A102627 = strict partitions whose length divides sum.

A326850 = strict partitions whose maximum part divides sum.

A326851 = strict partitions w/ length and max dividing sum.

A340828 = strict partitions w/ length divisible by max.

A340829 = strict partitions w/ Heinz number divisible by sum.

A340830 = strict partitions w/ parts divisible by length.

Cf. A064173, A143773 (A316428), A168659, A326843 (A326837), A326852, A330950 (A324851), A340851, A340853.

Sequence in context: A319430 A285337 A328457 * A360119 A033630 A308608

Adjacent sequences: A340824 A340825 A340826 * A340828 A340829 A340830

Keyword

nonn

Author

Gus Wiseman, Feb 01 2021

Extensions

Data section extended up to a(105) by Antti Karttunen, Feb 22 2023

Status

approved