A335238 -id:A335238 - cp's OEIS frontend

A337602 Number of ordered triples of positive integers summing to n whose set of distinct parts is pairwise coprime, where a singleton is always considered coprime.

0, 0, 0, 1, 3, 6, 10, 9, 18, 16, 24, 21, 43, 24, 51, 31, 54, 42, 94, 45, 102, 55, 99, 69, 163, 66, 150, 88, 168, 96, 265, 93, 228, 121, 246, 126, 337, 132, 315, 169, 342, 162, 487, 165, 420, 217, 411, 213, 619, 207, 558, 259, 540, 258, 784, 264, 654, 325, 660

Offset: 0

Gus Wiseman, Sep 20 2020

nonn

			The a(3) = 1 through a(8) = 18 triples:
  (1,1,1)  (1,1,2)  (1,1,3)  (1,1,4)  (1,1,5)  (1,1,6)
           (1,2,1)  (1,2,2)  (1,2,3)  (1,3,3)  (1,2,5)
           (2,1,1)  (1,3,1)  (1,3,2)  (1,5,1)  (1,3,4)
                    (2,1,2)  (1,4,1)  (2,2,3)  (1,4,3)
                    (2,2,1)  (2,1,3)  (2,3,2)  (1,5,2)
                    (3,1,1)  (2,2,2)  (3,1,3)  (1,6,1)
                             (2,3,1)  (3,2,2)  (2,1,5)
                             (3,1,2)  (3,3,1)  (2,3,3)
                             (3,2,1)  (5,1,1)  (2,5,1)
                             (4,1,1)           (3,1,4)
                                               (3,2,3)
                                               (3,3,2)
                                               (3,4,1)
                                               (4,1,3)
                                               (4,3,1)
                                               (5,1,2)
                                               (5,2,1)
                                               (6,1,1)

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..10000

The complement in A014311 of A337695 ranks these compositions.
A220377*6 is the strict case.
A337600 is the unordered version.
A337603 does not consider a singleton to be coprime unless it is (1).
A337664 counts these compositions of any length.
A000740 counts relatively prime compositions.
A337561 counts pairwise coprime strict compositions.
A000217 counts 3-part compositions.
A001399/A069905/A211540 count 3-part partitions.
A023023 counts relatively prime 3-part partitions.
A051424 counts pairwise coprime or singleton partitions.
A101268 counts pairwise coprime or singleton compositions.
A305713 counts pairwise coprime strict partitions.
A327516 counts pairwise coprime partitions.
A333227 ranks pairwise coprime compositions.
A333228 ranks compositions whose distinct parts are pairwise coprime.
A337461 counts pairwise coprime 3-part compositions.
Cf. A000212, A007359, A087087, A284825, A302696, A304709, A304712, A307719, A328673, A335235, A335238, A337483, A337562, A337601.

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n,{3}],SameQ@@#||CoprimeQ@@Union[#]&]],{n,0,100}]

A337664 Number of compositions of n whose set of distinct parts is pairwise coprime, where a singleton is always considered coprime.

Original entry on oeis.org

1, 1, 2, 4, 8, 16, 30, 58, 111, 210, 396, 750, 1420, 2688, 5079, 9586, 18092, 34157, 64516, 121899, 230373, 435463, 823379, 1557421, 2946938, 5578111, 10561990, 20005129, 37902514, 71832373, 136173273, 258211603, 489738627, 929074448, 1762899110, 3345713034

Offset: 0

Gus Wiseman, Sep 21 2020

nonn

			The a(0) = 1 through a(5) = 16 compositions:
  ()  (1)  (2)   (3)    (4)     (5)
           (11)  (12)   (13)    (14)
                 (21)   (22)    (23)
                 (111)  (31)    (32)
                        (112)   (41)
                        (121)   (113)
                        (211)   (122)
                        (1111)  (131)
                                (212)
                                (221)
                                (311)
                                (1112)
                                (1121)
                                (1211)
                                (2111)
                                (11111)

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..220

A304712 is the unordered version.
A337562 is the strict case.
A337602 is the length-3 case.
A337665 does not consider a singleton to be coprime unless it is (1).
A337695 ranks the complement of these compositions.
A000740 counts relatively prime compositions.
A051424 counts pairwise coprime or singleton partitions.
A101268 counts pairwise coprime or singleton compositions.
A305713 counts pairwise coprime strict partitions.
A327516 counts pairwise coprime partitions.
A333227 ranks pairwise coprime compositions.
A333228 ranks compositions whose distinct parts are pairwise coprime.
A337461 counts pairwise coprime length-3 compositions.
A337561 counts pairwise coprime strict compositions.
Cf. A007359, A007360, A087087, A302569, A304709, A307719, A335235, A335238, A335239, A337562, A337603, A337667.

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],SameQ@@#||CoprimeQ@@Union[#]&]],{n,0,15}]

A335240 Number of integer partitions of n that are not pairwise coprime, where a singleton is not coprime unless it is (1).

Original entry on oeis.org

1, 0, 1, 1, 2, 2, 5, 6, 11, 16, 25, 34, 51, 69, 98, 134, 181, 238, 316, 410, 536, 691, 887, 1122, 1423, 1788, 2246, 2800, 3483, 4300, 5304, 6508, 7983, 9745, 11869, 14399, 17436, 21040, 25367, 30482, 36568, 43735, 52239, 62239, 74073, 87950, 104277, 123348

Offset: 0

Gus Wiseman, May 30 2020

nonn

We use the Mathematica definition for CoprimeQ, so a singleton is not considered coprime unless it is (1).

These are also partitions that are a singleton or whose product is strictly greater than the LCM of their parts.

			The a(2) = 1 through a(9) = 16 partitions:
  (2)  (3)  (4)   (5)    (6)     (7)      (8)       (9)
            (22)  (221)  (33)    (322)    (44)      (63)
                         (42)    (331)    (62)      (333)
                         (222)   (421)    (332)     (432)
                         (2211)  (2221)   (422)     (441)
                                 (22111)  (2222)    (522)
                                          (3221)    (621)
                                          (3311)    (3222)
                                          (4211)    (3321)
                                          (22211)   (4221)
                                          (221111)  (22221)
                                                    (32211)
                                                    (33111)
                                                    (42111)
                                                    (222111)
                                                    (2211111)

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..750

The version for relatively prime instead of coprime is A018783.
The Heinz numbers of these partitions are the complement of A302696.
The complement is counted by A327516.
Singleton or pairwise coprime partitions are counted by A051424.
Singleton or pairwise coprime sets are ranked by A087087.
Numbers whose binary indices are pairwise coprime are A326675.
All of the following pertain to compositions in standard order (A066099):
- GCD is A326674.
- LCM is A333226.
- Coprime compositions are A333227.
- Compositions whose distinct parts are coprime are A333228.
- Non-coprime compositions are A335239.
Cf. A007360, A101268, A302569, A335235, A335236, A335237, A335238.

Table[Length[Select[IntegerPartitions[n],!CoprimeQ@@#&]],{n,0,30}]

A337695 Numbers k such that the distinct parts of the k-th composition in standard order (A066099) are not pairwise coprime, where a singleton is always considered coprime.

Original entry on oeis.org

34, 40, 69, 70, 81, 88, 98, 104, 130, 138, 139, 141, 142, 160, 162, 163, 168, 177, 184, 197, 198, 209, 216, 226, 232, 260, 261, 262, 274, 276, 277, 278, 279, 282, 283, 285, 286, 288, 290, 296, 321, 324, 325, 326, 327, 328, 337, 344, 352, 354, 355, 360, 369

Offset: 1

Gus Wiseman, Sep 22 2020

nonn

The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.

			The sequence together with the corresponding compositions begins:
     34: (4,2)        163: (2,4,1,1)    277: (4,2,2,1)
     40: (2,4)        168: (2,2,4)      278: (4,2,1,2)
     69: (4,2,1)      177: (2,1,4,1)    279: (4,2,1,1,1)
     70: (4,1,2)      184: (2,1,1,4)    282: (4,1,2,2)
     81: (2,4,1)      197: (1,4,2,1)    283: (4,1,2,1,1)
     88: (2,1,4)      198: (1,4,1,2)    285: (4,1,1,2,1)
     98: (1,4,2)      209: (1,2,4,1)    286: (4,1,1,1,2)
    104: (1,2,4)      216: (1,2,1,4)    288: (3,6)
    130: (6,2)        226: (1,1,4,2)    290: (3,4,2)
    138: (4,2,2)      232: (1,1,2,4)    296: (3,2,4)
    139: (4,2,1,1)    260: (6,3)        321: (2,6,1)
    141: (4,1,2,1)    261: (6,2,1)      324: (2,4,3)
    142: (4,1,1,2)    262: (6,1,2)      325: (2,4,2,1)
    160: (2,6)        274: (4,3,2)      326: (2,4,1,2)
    162: (2,4,2)      276: (4,2,3)      327: (2,4,1,1,1)

Gus Wiseman, Statistics, classes, and transformations of standard compositions

A304712 counts the complement, with ordered version A337664.
A333228 ranks compositions whose distinct parts are pairwise coprime.
A335238 does not consider a singleton coprime unless it is (1).
A337600 counts 3-part partitions in the complement.
A000740 counts relatively prime compositions.
A051424 counts pairwise coprime or singleton partitions.
A101268 counts pairwise coprime or singleton compositions.
A327516 counts pairwise coprime partitions.
A333227 ranks pairwise coprime compositions.
A337461 counts pairwise coprime 3-part compositions.
A337561 counts pairwise coprime strict compositions.
A337665 counts compositions whose distinct parts are pairwise coprime.
A337666 ranks pairwise non-coprime compositions.
Cf. A007359, A007360, A087087, A302569, A302696, A304709, A305713, A335235, A337562, A337601, A337602, A337603.

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Select[Range[0,100],!(SameQ@@stc[#]||CoprimeQ@@Union[stc[#]])&]

A335241 Numbers whose prime indices are not pairwise coprime, where a singleton is not coprime unless it is {1}.

Original entry on oeis.org

1, 3, 5, 7, 9, 11, 13, 17, 18, 19, 21, 23, 25, 27, 29, 31, 36, 37, 39, 41, 42, 43, 45, 47, 49, 50, 53, 54, 57, 59, 61, 63, 65, 67, 71, 72, 73, 75, 78, 79, 81, 83, 84, 87, 89, 90, 91, 97, 98, 99, 100, 101, 103, 105, 107, 108, 109, 111, 113, 114, 115, 117, 121

Offset: 1

Gus Wiseman, May 30 2020

nonn

We use the Mathematica definition for CoprimeQ, so a singleton is not considered coprime unless it is (1).

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

			The sequence of terms together with their prime indices begins:
    1: {}          31: {11}          61: {18}
    3: {2}         36: {1,1,2,2}     63: {2,2,4}
    5: {3}         37: {12}          65: {3,6}
    7: {4}         39: {2,6}         67: {19}
    9: {2,2}       41: {13}          71: {20}
   11: {5}         42: {1,2,4}       72: {1,1,1,2,2}
   13: {6}         43: {14}          73: {21}
   17: {7}         45: {2,2,3}       75: {2,3,3}
   18: {1,2,2}     47: {15}          78: {1,2,6}
   19: {8}         49: {4,4}         79: {22}
   21: {2,4}       50: {1,3,3}       81: {2,2,2,2}
   23: {9}         53: {16}          83: {23}
   25: {3,3}       54: {1,2,2,2}     84: {1,1,2,4}
   27: {2,2,2}     57: {2,8}         87: {2,10}
   29: {10}        59: {17}          89: {24}

The complement is A302696.
The version for relatively prime instead of coprime is A318978.
The version for standard compositions is A335239.
These are the Heinz numbers of the partitions counted by A335240.
Singleton or pairwise coprime partitions are counted by A051424.
Singleton or pairwise coprime sets are ranked by A087087.
Primes and numbers with pairwise coprime prime indices are A302569.
Numbers whose binary indices are pairwise coprime are A326675.
Coprime standard composition numbers are A333227.
Cf. A007360, A101268, A326674, A327516, A333228, A335236, A335237, A335238.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],!CoprimeQ@@primeMS[#]&]

A336620 Numbers that are not a product of elements of A304711.

Original entry on oeis.org

3, 5, 7, 9, 11, 13, 17, 19, 21, 23, 25, 27, 29, 31, 37, 39, 41, 42, 43, 47, 49, 53, 57, 59, 61, 63, 65, 67, 71, 73, 78, 79, 81, 83, 87, 89, 91, 97, 101, 103, 105, 107, 109, 111, 113, 114, 115, 117, 121, 125, 126, 127, 129, 130, 131, 133, 137, 139, 147, 149

Offset: 1

Gus Wiseman, Aug 02 2020

nonn

A304711 lists numbers whose distinct prime indices are pairwise coprime.

The first term divisible by 4 is a(421) = 1092.

			The sequence of terms together with their prime indices begins:
      3: {2}         39: {2,6}       78: {1,2,6}
      5: {3}         41: {13}        79: {22}
      7: {4}         42: {1,2,4}     81: {2,2,2,2}
      9: {2,2}       43: {14}        83: {23}
     11: {5}         47: {15}        87: {2,10}
     13: {6}         49: {4,4}       89: {24}
     17: {7}         53: {16}        91: {4,6}
     19: {8}         57: {2,8}       97: {25}
     21: {2,4}       59: {17}       101: {26}
     23: {9}         61: {18}       103: {27}
     25: {3,3}       63: {2,2,4}    105: {2,3,4}
     27: {2,2,2}     65: {3,6}      107: {28}
     29: {10}        67: {19}       109: {29}
     31: {11}        71: {20}       111: {2,12}
     37: {12}        73: {21}       113: {30}

Gus Wiseman, Sequences counting and encoding certain classes of multisets

A336426 is the version for superprimorials, with complement A181818.
A336497 is the version for superfactorials, with complement A336496.
A336735 is the complement.
A000837 counts relatively prime partitions, with strict case A007360.
A001055 counts factorizations.
A302696 lists numbers with coprime prime indices.
A304711 lists numbers with coprime distinct prime indices.
Cf. A007916, A112798, A302569, A327516, A333228, A335238, A335239, A335240, A335241, A336424, A336568, A336736.

nn=100;
dat=Select[Range[nn],CoprimeQ@@PrimePi/@First/@FactorInteger[#]&];
facsusing[s_,n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facsusing[Select[s,Divisible[n/d,#]&],n/d],Min@@#>=d&]],{d,Select[s,Divisible[n,#]&]}]];
Select[Range[nn],facsusing[dat,#]=={}&]

A336735 Products of elements of A304711.

Original entry on oeis.org

1, 2, 4, 6, 8, 10, 12, 14, 15, 16, 18, 20, 22, 24, 26, 28, 30, 32, 33, 34, 35, 36, 38, 40, 44, 45, 46, 48, 50, 51, 52, 54, 55, 56, 58, 60, 62, 64, 66, 68, 69, 70, 72, 74, 75, 76, 77, 80, 82, 84, 85, 86, 88, 90, 92, 93, 94, 95, 96, 98, 99, 100, 102, 104, 106

Offset: 1

Gus Wiseman, Aug 02 2020

nonn

A304711 lists numbers whose distinct prime indices are pairwise coprime.

First differs from A304711 in having 84.

			The sequence of terms together with their prime indices begins:
      1: {}            28: {1,1,4}         52: {1,1,6}
      2: {1}           30: {1,2,3}         54: {1,2,2,2}
      4: {1,1}         32: {1,1,1,1,1}     55: {3,5}
      6: {1,2}         33: {2,5}           56: {1,1,1,4}
      8: {1,1,1}       34: {1,7}           58: {1,10}
     10: {1,3}         35: {3,4}           60: {1,1,2,3}
     12: {1,1,2}       36: {1,1,2,2}       62: {1,11}
     14: {1,4}         38: {1,8}           64: {1,1,1,1,1,1}
     15: {2,3}         40: {1,1,1,3}       66: {1,2,5}
     16: {1,1,1,1}     44: {1,1,5}         68: {1,1,7}
     18: {1,2,2}       45: {2,2,3}         69: {2,9}
     20: {1,1,3}       46: {1,9}           70: {1,3,4}
     22: {1,5}         48: {1,1,1,1,2}     72: {1,1,1,2,2}
     24: {1,1,1,2}     50: {1,3,3}         74: {1,12}
     26: {1,6}         51: {2,7}           75: {2,3,3}

Gus Wiseman, Sequences counting and encoding certain classes of multisets

A181818 is the version for superprimorials, with complement A336426.
A336496 is the version for superfactorials, with complement A336497.
A336620 is the complement.
A000837 counts relatively prime partitions, with strict case A007360.
A001055 counts factorizations.
A302696 lists numbers with coprime prime indices.
A304711 lists numbers with coprime distinct prime indices.
Cf. A001221, A007360, A007916, A056239, A112798, A302569, A327516, A333228, A335238, A335239, A335240, A336424, A336497, A336736.

nn=100;
dat=Select[Range[nn],CoprimeQ@@PrimePi/@First/@FactorInteger[#]&];
facsusing[s_,n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facsusing[Select[s,Divisible[n/d,#]&],n/d],Min@@#>=d&]],{d,Select[s,Divisible[n,#]&]}]];
Select[Range[nn],facsusing[dat,#]!={}&]

cp's OEIS Frontend

A337602 Number of ordered triples of positive integers summing to n whose set of distinct parts is pairwise coprime, where a singleton is always considered coprime.

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A337664 Number of compositions of n whose set of distinct parts is pairwise coprime, where a singleton is always considered coprime.

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A335240 Number of integer partitions of n that are not pairwise coprime, where a singleton is not coprime unless it is (1).

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A337695 Numbers k such that the distinct parts of the k-th composition in standard order (A066099) are not pairwise coprime, where a singleton is always considered coprime.

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A335241 Numbers whose prime indices are not pairwise coprime, where a singleton is not coprime unless it is {1}.

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A336620 Numbers that are not a product of elements of A304711.

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A336735 Products of elements of A304711.

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