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A023023 Number of partitions of n into 3 unordered relatively prime parts. 22
1, 1, 2, 2, 4, 4, 6, 6, 10, 8, 14, 12, 16, 16, 24, 18, 30, 24, 32, 30, 44, 32, 50, 42, 54, 48, 70, 48, 80, 64, 80, 72, 96, 72, 114, 90, 112, 96, 140, 96, 154, 120, 144, 132, 184, 128, 196, 150, 192, 168, 234, 162, 240, 192, 240, 210, 290, 192, 310, 240, 288, 256, 336, 240, 374 (list; graph; refs; listen; history; text; internal format)
OFFSET

3,3

LINKS

Fausto A. C. Cariboni, Table of n, a(n) for n = 3..10000

Mohamed El Bachraoui, Partitions with relatively prime parts [From Jonathan Sondow, May 27 2009]

FORMULA

G.f. for the number of partitions of n into m unordered relatively prime parts is Sum(moebius(k)*x^(m*k)/Product(1-x^(i*k), i=1..m), k=1..infinity). - Vladeta Jovovic, Dec 21 2004

a(n) = (n^2/12)*Product_{prime p|n} (1 - 1/p^2) for n > 3 (proved by Mohamed El Bachraoui). [Jonathan Sondow, May 27 2009]

a(n) = Sum_{k=1..floor(n/3)} Sum_{i=k..floor((n-k)/2)} floor(1/gcd(i,k,n-i-k)). - Wesley Ivan Hurt, Jan 02 2021

EXAMPLE

From Gus Wiseman, Oct 08 2020: (Start)

The a(3) = 1 through a(13) = 14 triples (A = 10, B = 11):

  111   211   221   321   322   332   432   433   443   543   544

              311   411   331   431   441   532   533   552   553

                          421   521   522   541   542   651   643

                          511   611   531   631   551   732   652

                                      621   721   632   741   661

                                      711   811   641   831   733

                                                  722   921   742

                                                  731   A11   751

                                                  821         832

                                                  911         841

                                                              922

                                                              931

                                                              A21

                                                              B11

(End)

MATHEMATICA

Table[Length[Select[IntegerPartitions[n, {3}], GCD@@#==1&]], {n, 3, 50}] (* Gus Wiseman, Oct 08 2020 *)

CROSSREFS

Cf. A023024-A023030, A000742-A000743, A023032-A023035.

A000741 is the ordered version.

A000837 counts these partitions of any length.

A001399(n-3) does not require relative primality.

A023022 is the 2-part version.

A101271 is the strict case.

A284825 counts the case that is also pairwise non-coprime.

A289509 intersected with A014612 gives the Heinz numbers.

A307719 is the pairwise coprime instead of relatively prime version.

A337599 is the pairwise non-coprime instead of relative prime version.

A008284 counts partitions by sum and length.

A078374 counts relatively prime strict partitions.

A337601 counts 3-part partitions whose distinct parts are pairwise coprime.

Cf. A000010, A000217, A055684, A078374, A200976, A220377, A302698, A327516, A337563, A337600, A337605.

Sequence in context: A340280 A340279 A343100 * A184157 A008643 A008644

Adjacent sequences:  A023020 A023021 A023022 * A023024 A023025 A023026

KEYWORD

nonn

AUTHOR

David W. Wilson

STATUS

approved

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Last modified June 14 21:27 EDT 2021. Contains 345041 sequences. (Running on oeis4.)