A303140 -id:A303140 - cp's OEIS frontend

A320423 Number of set partitions of {1,...,n} where each block's elements are pairwise coprime.

1, 1, 1, 2, 2, 8, 4, 28, 18, 120, 60, 888, 252, 5220, 1860, 22224, 9552, 311088, 59616, 2473056, 565920, 13627008, 4051872, 235039392, 33805440, 1932037632, 465239808, 20604487680, 4294865664, 386228795904, 35413136640

Offset: 0

Gus Wiseman, Jan 08 2019

Two or more numbers are pairwise coprime if no pair of them has a common divisor > 1. A single number is not considered to be pairwise coprime unless it is equal to 1.

			The a(5) = 8 set partitions:
  {{1},{2,3},{4,5}}
  {{1},{2,5},{3,4}}
   {{1,2},{3,4,5}}
   {{1,4},{2,3,5}}
   {{1,2,3},{4,5}}
   {{1,2,5},{3,4}}
   {{1,3,4},{2,5}}
   {{1,4,5},{2,3}}

Cf. A000110, A051424, A084422, A085945, A186974, A187106, A302569, A302696, A303139, A303140, A320424, A320426, A320430, A320768.

spsu[,{}]:={{}};spsu[foo,set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@spsu[Select[foo,Complement[#,Complement[set,s]]=={}&],Complement[set,s]]]/@Cases[foo,{i,_}];
Table[Length[spsu[Select[Subsets[Range[n]],CoprimeQ@@#&],Range[n]]],{n,10}]

a(17)-a(18) from Alois P. Heinz, Jan 17 2019

a(19)-a(30) from Christian Sievers, Nov 28 2024

A303138 Regular triangle where T(n,k) is the number of strict integer partitions of n with greatest common divisor k.

Original entry on oeis.org

1, 0, 1, 1, 0, 1, 1, 0, 0, 1, 2, 0, 0, 0, 1, 2, 1, 0, 0, 0, 1, 4, 0, 0, 0, 0, 0, 1, 4, 1, 0, 0, 0, 0, 0, 1, 6, 0, 1, 0, 0, 0, 0, 0, 1, 7, 2, 0, 0, 0, 0, 0, 0, 0, 1, 11, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 10, 2, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 17, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 17, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 23, 0, 2, 0, 1

Offset: 1

Gus Wiseman, Apr 19 2018

			Triangle begins:
01:   1
02:   0  1
03:   1  0  1
04:   1  0  0  1
05:   2  0  0  0  1
06:   2  1  0  0  0  1
07:   4  0  0  0  0  0  1
08:   4  1  0  0  0  0  0  1
09:   6  0  1  0  0  0  0  0  1
10:   7  2  0  0  0  0  0  0  0  1
11:  11  0  0  0  0  0  0  0  0  0  1
12:  10  2  1  1  0  0  0  0  0  0  0  1
13:  17  0  0  0  0  0  0  0  0  0  0  0  1
14:  17  4  0  0  0  0  0  0  0  0  0  0  0  1
15:  23  0  2  0  1  0  0  0  0  0  0  0  0  0  1
The strict partitions counted in row 12 are the following.
T(12,1) = 10: (11,1) (9,2,1) (8,3,1) (7,5) (7,4,1) (7,3,2) (6,5,1) (6,3,2,1) (5,4,3) (5,4,2,1)
T(12,2) = 2:  (10,2) (6,4,2)
T(12,3) = 1:  (9,3)
T(12,4) = 1:  (8,4)
T(12,12) = 1: (12)

First column is A078374. Second column at even indices is same as first column. Row sums are A000009. Row sums with first column removed are A303280.

Cf. A000009, A000837, A018783, A051424, A117408, A168532, A289508, A289509, A298748, A300486, A302698, A302796, A303139, A303140, A303280.

Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&GCD@@#===k&]],{n,15},{k,n}]

If k divides n, T(n,k) = A078374(n/k); otherwise T(n,k) = 0.

A328672 Number of integer partitions of n with relatively prime parts in which no two distinct parts are relatively prime.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 3, 1, 1, 1, 4, 1, 4, 1, 1, 2, 7, 1, 6, 1, 3, 3, 10, 1, 9, 3, 5, 4, 17, 1, 23, 6, 7, 6, 20, 3, 36, 9, 15, 7, 45, 5, 56, 14, 17, 20, 65, 7, 83, 18, 40

Offset: 0

Gus Wiseman, Oct 29 2019

nonn

Positions of terms greater than 1 are {31, 37, 41, 43, 46, 47, 49, ...}.

A partition with no two distinct parts relatively prime is said to be intersecting.

			Examples:
  a(31) = 2:         a(46) = 2:
    (15,10,6)          (15,15,10,6)
    (1^31)             (1^46)
  a(37) = 3:         a(47) = 7:
    (15,12,10)         (20,15,12)
    (15,10,6,6)        (21,14,12)
    (1^37)             (20,15,6,6)
  a(41) = 4:           (21,14,6,6)
    (20,15,6)          (15,12,10,10)
    (21,14,6)          (15,10,10,6,6)
    (15,10,10,6)       (1^47)
    (1^41)           a(49) = 6:
  a(43) = 4:           (24,15,10)
    (18,15,10)         (18,15,10,6)
    (15,12,10,6)       (15,12,12,10)
    (15,10,6,6,6)      (15,12,10,6,6)
    (1^43)             (15,10,6,6,6,6)
                       (1^39)

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..400

The Heinz numbers of these partitions are A328679.
The strict case is A318715.
The version for non-isomorphic multiset partitions is A319759.
Relatively prime partitions are A000837.
Intersecting partitions are A328673.
Cf. A078374, A285573, A289509, A291166, A303140, A305148, A316476, A326910, A326912.

Table[Length[Select[IntegerPartitions[n],GCD@@#==1&&And[And@@(GCD[##]>1&)@@@Subsets[Union[#],{2}]]&]],{n,0,32}]

a(n > 0) = A202425(n) + 1.

A338317 Number of integer partitions of n with no 1's and pairwise coprime distinct parts, where a singleton is always considered coprime.

Original entry on oeis.org

1, 0, 1, 1, 2, 2, 3, 4, 5, 6, 7, 11, 11, 16, 16, 19, 25, 32, 34, 44, 46, 53, 66, 80, 88, 101, 116, 132, 150, 180, 204, 229, 254, 287, 331, 366, 426, 473, 525, 584, 662, 742, 835, 922, 1013, 1128, 1262, 1408, 1555, 1711, 1894, 2080, 2297, 2555, 2806, 3064, 3376

Offset: 0

Gus Wiseman, Oct 24 2020

nonn

			The a(2) = 1 through a(12) = 11 partitions (A = 10, B = 11, C = 12):
  2   3   4    5    6     7     8      9      A       B       C
          22   32   33    43    44     54     55      65      66
                    222   52    53     72     73      74      75
                          322   332    333    433     83      444
                                2222   522    532     92      543
                                       3222   3322    443     552
                                              22222   533     732
                                                      722     3333
                                                      3332    5322
                                                      5222    33222
                                                      32222   222222

A007359 (A302568) gives the strict case.
A101268 (A335235) gives pairwise coprime or singleton compositions.
A200976 (A338318) gives the pairwise non-coprime instead of coprime version.
A304709 (A304711) gives partitions whose distinct parts are pairwise coprime, with strict case A305713 (A302797).
A304712 (A338331) allows 1's, with strict version A007360 (A302798).
A327516 (A302696) gives pairwise coprime partitions.
A328673 (A328867) gives partitions with no distinct relatively prime parts.
A338315 (A337987) does not consider singletons coprime.
A338317 (A338316) gives these partitions.
A337462 (A333227) gives pairwise coprime compositions.
A337485 (A337984) gives pairwise coprime integer partitions with no 1's.
A337665 (A333228) gives compositions with pairwise coprime distinct parts.
A337667 (A337666) gives pairwise non-coprime compositions.
A337697 (A022340 /\ A333227) = pairwise coprime compositions with no 1's.
A337983 (A337696) gives pairwise non-coprime strict compositions, with unordered version A318717 (A318719).
Cf. A051424, A289508, A302569, A303140, A337694.

Table[Length[Select[IntegerPartitions[n],!MemberQ[#,1]&&(SameQ@@#||CoprimeQ@@Union[#])&]],{n,0,15}]

The Heinz numbers of these partitions are given by A338316. The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), giving a bijective correspondence between positive integers and integer partitions.

A319300 Irregular triangle where T(n,k) is the number of strict integer partitions of n with GCD equal to the k-th divisor of n.

Original entry on oeis.org

1, 0, 1, 1, 1, 1, 0, 1, 2, 1, 2, 1, 0, 1, 4, 1, 4, 1, 0, 1, 6, 1, 1, 7, 2, 0, 1, 11, 1, 10, 2, 1, 1, 0, 1, 17, 1, 17, 4, 0, 1, 23, 2, 1, 1, 26, 4, 1, 0, 1, 37, 1, 36, 6, 2, 1, 0, 1, 53, 1, 53, 7, 2, 1, 0, 1, 70, 4, 1, 1, 77, 11, 0, 1, 103, 1, 103, 10, 4, 2, 1

Offset: 1

Gus Wiseman, Sep 16 2018

			Triangle begins:
   1
   0  1
   1  1
   1  0  1
   2  1
   2  1  0  1
   4  1
   4  1  0  1
   6  1  1
   7  2  0  1
  11  1
  10  2  1  1  0  1
  17  1
  17  4  0  1
  23  2  1  1
  26  4  1  0  1
  37  1
  36  6  2  1  0  1
  53  1
  53  7  2  1  0  1

A regular version is A303138. Row lengths are A000005. Row sums are A000009. First column is A078374.

Cf. A000837, A027750, A289508, A289509, A303140, A303280, A319299, A319301.

Table[Length[Select[IntegerPartitions[n],And[UnsameQ@@#,GCD@@#==k]&]],{n,20},{k,Divisors[n]}]

T(n,k) = A078374(n/A027750(n,k)).

cp's OEIS Frontend

A320423 Number of set partitions of {1,...,n} where each block's elements are pairwise coprime.

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A303138 Regular triangle where T(n,k) is the number of strict integer partitions of n with greatest common divisor k.

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A328672 Number of integer partitions of n with relatively prime parts in which no two distinct parts are relatively prime.

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A338317 Number of integer partitions of n with no 1's and pairwise coprime distinct parts, where a singleton is always considered coprime.

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A319300 Irregular triangle where T(n,k) is the number of strict integer partitions of n with GCD equal to the k-th divisor of n.

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