A328673 -id:A328673 - cp's OEIS frontend

A337984 Heinz numbers of pairwise coprime integer partitions with no 1's, where a singleton is not considered coprime.

15, 33, 35, 51, 55, 69, 77, 85, 93, 95, 119, 123, 141, 143, 145, 155, 161, 165, 177, 187, 201, 205, 209, 215, 217, 219, 221, 249, 253, 255, 265, 287, 291, 295, 309, 323, 327, 329, 335, 341, 355, 381, 385, 391, 395, 403, 407, 411, 413, 415, 437, 447, 451, 465

Offset: 1

Gus Wiseman, Oct 22 2020

nonn

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.

			The sequence of terms together with their prime indices begins:
     15: {2,3}     155: {3,11}     265: {3,16}
     33: {2,5}     161: {4,9}      287: {4,13}
     35: {3,4}     165: {2,3,5}    291: {2,25}
     51: {2,7}     177: {2,17}     295: {3,17}
     55: {3,5}     187: {5,7}      309: {2,27}
     69: {2,9}     201: {2,19}     323: {7,8}
     77: {4,5}     205: {3,13}     327: {2,29}
     85: {3,7}     209: {5,8}      329: {4,15}
     93: {2,11}    215: {3,14}     335: {3,19}
     95: {3,8}     217: {4,11}     341: {5,11}
    119: {4,7}     219: {2,21}     355: {3,20}
    123: {2,13}    221: {6,7}      381: {2,31}
    141: {2,15}    249: {2,23}     385: {3,4,5}
    143: {5,6}     253: {5,9}      391: {7,9}
    145: {3,10}    255: {2,3,7}    395: {3,22}

A005117 is a superset.
A337485 counts these partitions.
A302568 considers singletons to be coprime.
A304711 allows 1's, with squarefree version A302797.
A337694 is the pairwise non-coprime instead of pairwise coprime version.
A007359 counts partitions into singleton or pairwise coprime parts with no 1's
A101268 counts pairwise coprime or singleton compositions, ranked by A335235.
A305713 counts pairwise coprime strict partitions.
A327516 counts pairwise coprime partitions, ranked by A302696.
A337462 counts pairwise coprime compositions, ranked by A333227.
A337561 counts pairwise coprime strict compositions.
A337665 counts compositions whose distinct parts are pairwise coprime, ranked by A333228.
A337667 counts pairwise non-coprime compositions, ranked by A337666.
A337697 counts pairwise coprime compositions with no 1's.
A337983 counts pairwise non-coprime strict compositions, with unordered version A318717 ranked by A318719.
Cf. A051424, A056239, A087087, A112798, A200976, A220377, A302569, A303140, A303282, A328673, A328867.

Select[Range[1,100,2],SquareFreeQ[#]&&CoprimeQ@@PrimePi/@First/@FactorInteger[#]&]

Equals A302568\A000040.

A202425 Number of partitions of n into parts having pairwise common factors but no overall common factor.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 2, 0, 0, 0, 3, 0, 3, 0, 0, 1, 6, 0, 5, 0, 2, 2, 9, 0, 8, 2, 4, 3, 16, 0, 22, 5, 6, 5, 19, 2, 35, 8, 14, 6, 44, 4, 55, 13, 16, 19, 64, 6, 82, 17, 39, 31, 108, 10, 105, 40, 66, 46, 161, 14, 182, 61, 97, 72, 207, 37, 287, 85, 144, 93, 357, 59

Offset: 31

Alois P. Heinz, Dec 19 2011

nonn

			a(31) = 1: [6,10,15] = [2*3,2*5,3*5].
a(37) = 2: [6,6,10,15], [10,12,15].
a(41) = 3: [6,10,10,15], [6,15,20], [6,14,21].
a(47) = 6: [6,6,10,10,15], [10,10,12,15], [6,6,15,20], [12,15,20], [6,6,14,21], [12,14,21].
a(49) = 5: [6,6,6,6,10,15], [6,6,10,12,15], [10,12,12,15], [6,10,15,18], [10,15,24].

Fausto A. C. Cariboni, Table of n, a(n) for n = 31..400 (terms 31..251 from Alois P. Heinz)

Cf. A202385, A018783.
The version with only distinct parts compared is A328672.
The Heinz numbers of these partitions are A328868.
The strict case is A202385, which is essentially the same as A318715.
The version for non-isomorphic multiset partitions is A319759.
The version for set-systems is A326364.
Intersecting partitions are A200976.
Cf. A000837, A305148, A305843, A306006, A318717, A328673, A328867.

with(numtheory):
w:= (m, h)-> mul(`if`(j>=h, 1, j), j=factorset(m)):
b:= proc(n, i, g, s) option remember; local j, ok, si;
      if n<0 then 0
    elif n=0 then `if`(g>1, 0, 1)
    elif i<2 or member(1, s) then 0
    else ok:= evalb(i<=n);
         si:= map(x->w(x, i), s);
         for j in s while ok do ok:= igcd(i, j)>1 od;
         b(n, i-1, g, si) +`if`(ok, add(b(n-t*i, i-1, igcd(i, g),
                      si union {w(i,i)} ), t=1..iquo(n, i)), 0)
      fi
    end:
a:= n-> b(n, n, 0, {}):
seq(a(n), n=31..100);

w[m_, h_] := Product[If[j >= h, 1, j], {j, FactorInteger[m][[All, 1]]}]; b[n_, i_, g_, s_] := b[n, i, g, s] = Module[{j, ok, si}, Which[n<0, 0, n == 0, If[g>1, 0, 1], i<2 || MemberQ[s, 1], 0, True, ok = (i <= n); si = w[#, i]& /@ s; Do[If[ok, ok = (GCD[i, j]>1)], {j, s}]; b[n, i-1, g, si] + If[ok, Sum[b[n-t*i, i-1, GCD[i, g], si ~Union~ {w[i, i]}], {t, 1, Quotient[n, i]}], 0]]]; a[n_] := b[n, n, 0, {}]; Table[a[n], {n, 31, 100}] (* Jean-François Alcover, Feb 16 2017, translated from Maple *)
Table[Length[Select[IntegerPartitions[n],GCD@@#==1&&And@@(GCD[##]>1&)@@@Tuples[#,2]&]],{n,0,40}] (* Gus Wiseman, Nov 04 2019 *)

a(n > 0) = A328672(n) - 1. - Gus Wiseman, Nov 04 2019

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.

A337983 Number of compositions of n into distinct parts, any two of which have a common divisor > 1.

Original entry on oeis.org

1, 0, 1, 1, 1, 1, 3, 1, 3, 3, 5, 1, 13, 1, 13, 7, 19, 1, 35, 1, 59, 15, 65, 1, 117, 5, 133, 27, 195, 1, 411, 7, 435, 67, 617, 17, 941, 7, 1177, 135, 1571, 13, 2939, 31, 3299, 375, 4757, 13, 6709, 43, 8813, 643, 11307, 61, 16427, 123, 24331, 1203, 30461, 67

Offset: 0

Gus Wiseman, Oct 06 2020

nonn

Number of pairwise non-coprime strict compositions of n.

			The a(2) = 1 through a(15) = 7 compositions (A..F = 10..15):
  2  3  4  5  6   7  8   9   A   B  C    D  E    F
              24     26  36  28     2A      2C   3C
              42     62  63  46     39      4A   5A
                             64     48      68   69
                             82     84      86   96
                                    93      A4   A5
                                    A2      C2   C3
                                    246     248
                                    264     284
                                    426     428
                                    462     482
                                    624     824
                                    642     842

A318717 is the unordered version.
A318719 is the version for Heinz numbers of partitions.
A337561 is the pairwise coprime instead of pairwise non-coprime version, or A337562 if singletons are considered coprime.
A337605*6 counts these compositions of length 3.
A337667 is the non-strict version, ranked by A337666.
A337696 ranks these compositions.
A051185 and A305843 (covering) count pairwise intersecting set-systems.
A101268 counts pairwise coprime or singleton compositions.
A200976 and A328673 are the unordered version.
A233564 ranks strict compositions.
A318749 is the version for factorizations, with non-strict version A319786.
A333228 ranks compositions whose distinct parts are pairwise coprime.
A335236 ranks compositions neither a singleton nor pairwise coprime.
A337462 counts pairwise coprime compositions.
A337694 lists numbers with no two relatively prime prime indices.
Cf. A082024, A284825, A302797, A305713, A319752, A327516, A333227, A335235, A336737, A337604.

stabQ[u_,Q_]:=And@@Not/@Q@@@Tuples[u,2];
Table[Length[Join@@Permutations/@Select[IntegerPartitions[n],UnsameQ@@#&&stabQ[#,CoprimeQ]&]],{n,0,30}]

A338315 Number of integer partitions of n with no 1's whose distinct parts are pairwise coprime, where a singleton is not considered coprime unless it is (1).

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 3, 2, 4, 4, 10, 6, 15, 13, 16, 21, 31, 29, 43, 41, 50, 63, 79, 81, 99, 113, 129, 145, 179, 197, 228, 249, 284, 328, 363, 418, 472, 522, 581, 655, 741, 828, 921, 1008, 1123, 1259, 1407, 1546, 1709, 1889, 2077, 2292, 2554, 2799, 3061, 3369

Offset: 0

Gus Wiseman, Oct 23 2020

nonn

The Heinz numbers of these partitions are given by A337987. 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.

			The a(5) = 1 through a(13) = 15 partitions (empty column indicated by dot, A = 10, B = 11):
  32   .  43    53    54     73     65      75      76
          52    332   72     433    74      543     85
          322         522    532    83      552     94
                      3222   3322   92      732     A3
                                    443     5322    B2
                                    533     33222   544
                                    722             553
                                    3332            733
                                    5222            922
                                    32222           4333
                                                    5332
                                                    7222
                                                    33322
                                                    52222
                                                    322222

A200976 is a pairwise non-coprime instead of pairwise coprime version.
A304709 allows 1's, with strict case A305713 and Heinz numbers A304711.
A318717 counts pairwise non-coprime strict partitions.
A337485 is the strict version, with Heinz numbers A337984.
A337987 gives the Heinz numbers of these partitions.
A338317 considers singletons coprime, with Heinz numbers A338316.
A007359 counts singleton or pairwise coprime partitions with no 1's.
A327516 counts pairwise coprime partitions, ranked by A302696.
A328673 counts partitions with no two distinct parts relatively prime.
A337462 counts pairwise coprime compositions, ranked by A333227.
A337561 counts pairwise coprime strict compositions.
A337665 counts compositions whose distinct parts are pairwise coprime.
A337667 counts pairwise non-coprime compositions, ranked by A337666.
A337697 counts pairwise coprime compositions with no 1's.
Cf. A051424, A101268, A220377, A302568, A328867, A333228, A337563.

Table[Length[Select[IntegerPartitions[n],!MemberQ[#,1]&&CoprimeQ@@Union[#]&]],{n,0,30}]

A337696 Numbers k such that the k-th composition in standard order (A066099) is strict and pairwise non-coprime, meaning the parts are distinct and any two of them have a common divisor > 1.

Original entry on oeis.org

0, 2, 4, 8, 16, 32, 34, 40, 64, 128, 130, 160, 256, 260, 288, 512, 514, 520, 544, 640, 1024, 2048, 2050, 2052, 2056, 2082, 2088, 2176, 2178, 2208, 2304, 2560, 2568, 2592, 4096, 8192, 8194, 8200, 8224, 8226, 8232, 8320, 8704, 8706, 8832, 10240, 10248, 10368

Offset: 1

Gus Wiseman, Oct 06 2020

nonn

Differs from A291165 in having 1090535424, corresponding to the composition (6,10,15).

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:
       0: ()        512: (10)       2304: (3,9)
       2: (2)       514: (8,2)      2560: (2,10)
       4: (3)       520: (6,4)      2568: (2,6,4)
       8: (4)       544: (4,6)      2592: (2,4,6)
      16: (5)       640: (2,8)      4096: (13)
      32: (6)      1024: (11)       8192: (14)
      34: (4,2)    2048: (12)       8194: (12,2)
      40: (2,4)    2050: (10,2)     8200: (10,4)
      64: (7)      2052: (9,3)      8224: (8,6)
     128: (8)      2056: (8,4)      8226: (8,4,2)
     130: (6,2)    2082: (6,4,2)    8232: (8,2,4)
     160: (2,6)    2088: (6,2,4)    8320: (6,8)
     256: (9)      2176: (4,8)      8704: (4,10)
     260: (6,3)    2178: (4,6,2)    8706: (4,8,2)
     288: (3,6)    2208: (4,2,6)    8832: (4,2,8)

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

A318719 gives the Heinz numbers of the unordered version, with non-strict version A337694.
A337667 counts the non-strict version.
A337983 counts these compositions, with unordered version A318717.
A051185 counts intersecting set-systems, with spanning case A305843.
A200976 and A328673 count the unordered non-strict version.
A337462 counts pairwise coprime compositions.
A318749 counts pairwise non-coprime factorizations, with strict case A319786.
All of the following pertain to compositions in standard order (A066099):
- A000120 is length.
- A070939 is sum.
- A124767 counts runs.
- A233564 ranks strict compositions.
- A272919 ranks constant compositions.
- A333219 is Heinz number.
- A333227 ranks pairwise coprime compositions, or A335235 if singletons are considered coprime.
- A333228 ranks compositions whose distinct parts are pairwise coprime.
- A335236 ranks compositions neither a singleton nor pairwise coprime.
- A337561 is the pairwise coprime instead of pairwise non-coprime version, or A337562 if singletons are considered coprime.
- A337666 ranks the non-strict version.
Cf. A082024, A101268, A302797, A305713, A319752, A327040, A327516, A336737, A337599, A337604, A337605.

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
stabQ[u_,Q_]:=And@@Not/@Q@@@Tuples[u,2];
Select[Range[0,1000],UnsameQ@@stc[#]&&stabQ[stc[#],CoprimeQ]&]

Intersection of A337666 and A233564.

A338316 Odd numbers whose distinct prime indices are pairwise coprime, where a singleton is always considered coprime.

Original entry on oeis.org

1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 23, 25, 27, 29, 31, 33, 35, 37, 41, 43, 45, 47, 49, 51, 53, 55, 59, 61, 67, 69, 71, 73, 75, 77, 79, 81, 83, 85, 89, 93, 95, 97, 99, 101, 103, 107, 109, 113, 119, 121, 123, 125, 127, 131, 135, 137, 139, 141, 143, 145, 149, 151

Offset: 1

Gus Wiseman, Oct 24 2020

nonn

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 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. a(n) gives the n-th Heinz number of an integer partition with no 1's and pairwise coprime distinct parts, where a singleton is always considered coprime (A338317).

			The sequence of terms together with their prime indices begins:
      1: {}          33: {2,5}       71: {20}
      3: {2}         35: {3,4}       73: {21}
      5: {3}         37: {12}        75: {2,3,3}
      7: {4}         41: {13}        77: {4,5}
      9: {2,2}       43: {14}        79: {22}
     11: {5}         45: {2,2,3}     81: {2,2,2,2}
     13: {6}         47: {15}        83: {23}
     15: {2,3}       49: {4,4}       85: {3,7}
     17: {7}         51: {2,7}       89: {24}
     19: {8}         53: {16}        93: {2,11}
     23: {9}         55: {3,5}       95: {3,8}
     25: {3,3}       59: {17}        97: {25}
     27: {2,2,2}     61: {18}        99: {2,2,5}
     29: {10}        67: {19}       101: {26}
     31: {11}        69: {2,9}      103: {27}

A338315 does not consider singletons coprime, with Heinz numbers A337987.
A338317 counts the partitions with these Heinz numbers.
A337694 is a pairwise non-coprime instead of pairwise coprime version.
A007359 counts singleton or pairwise coprime partitions with no 1's, with Heinz numbers A302568.
A101268 counts pairwise coprime or singleton compositions, ranked by A335235.
A302797 lists squarefree numbers whose distinct parts are pairwise coprime.
A304709 counts partitions whose distinct parts are pairwise coprime, with Heinz numbers A304711.
A327516 counts pairwise coprime partitions, ranked by A302696.
A337485 counts pairwise coprime partitions with no 1's, with Heinz numbers A337984.
A337561 counts pairwise coprime strict compositions.
A337665 counts compositions whose distinct parts are pairwise coprime, ranked by A333228.
A337697 counts pairwise coprime compositions with no 1's.
Cf. A051424, A056239, A112798, A220377, A289509, A302569, A303282, A318719, A328673, A328867.

Select[Range[1,100,2],#==1||PrimePowerQ[#]||CoprimeQ@@Union[PrimePi/@First/@FactorInteger[#]]&]

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.

A328871 Number of integer partitions of n whose distinct parts are pairwise indivisible (stable) and pairwise non-relatively prime (intersecting).

Original entry on oeis.org

1, 1, 2, 2, 3, 2, 4, 2, 4, 3, 5, 2, 6, 2, 7, 5, 7, 2, 10, 2, 11, 7, 14, 2, 16, 4, 19, 8, 22, 2, 30, 3, 29, 14, 37, 8, 48, 4, 50, 19, 59, 5, 82, 4, 81, 28, 93, 8, 128, 9, 128, 38, 147, 8, 199, 19, 196, 52, 223, 12, 308

Offset: 0

Gus Wiseman, Nov 12 2019

nonn

A partition with no two distinct parts divisible is said to be stable, and a partition with no two distinct parts relatively prime is said to be intersecting, so these are just stable intersecting partitions.

			The a(1) = 1 through a(10) = 5 partitions (A = 10):
  1  2   3    4     5      6       7        8         9          A
     11  111  22    11111  33      1111111  44        333        55
              1111         222              2222      111111111  64
                           111111           11111111             22222
                                                                 1111111111

The Heinz numbers of these partitions are A329366.
Replacing "intersecting" with "relatively prime" gives A328676.
Stable partitions are A305148.
Intersecting partitions are A328673.
Cf. A000837, A285573, A303362, A305148, A316476, A328671, A328677.

stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
Table[Length[Select[IntegerPartitions[n],stableQ[Union[#],Divisible]&&stableQ[Union[#],GCD[#1,#2]==1&]&]],{n,0,30}]

A338318 Composite numbers whose prime indices are pairwise intersecting (non-coprime).

Original entry on oeis.org

9, 21, 25, 27, 39, 49, 57, 63, 65, 81, 87, 91, 111, 115, 117, 121, 125, 129, 133, 147, 159, 169, 171, 183, 185, 189, 203, 213, 235, 237, 243, 247, 259, 261, 267, 273, 289, 299, 301, 303, 305, 319, 321, 325, 333, 339, 343, 351, 361, 365, 371, 377, 387, 393

Offset: 1

Gus Wiseman, Oct 31 2020

nonn

First differs from A322336 in lacking 2535, with prime indices {2,3,6,6}.
First differs from A327685 in having 17719, with prime indices {6,10,15}.
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.
Also Heinz numbers of pairwise intersecting (non-coprime) partitions with more than one part. The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), giving a bijective correspondence between positive integers and integer partitions.

			The sequence of terms together with their prime indices begins:
      9: {2,2}        121: {5,5}        243: {2,2,2,2,2}
     21: {2,4}        125: {3,3,3}      247: {6,8}
     25: {3,3}        129: {2,14}       259: {4,12}
     27: {2,2,2}      133: {4,8}        261: {2,2,10}
     39: {2,6}        147: {2,4,4}      267: {2,24}
     49: {4,4}        159: {2,16}       273: {2,4,6}
     57: {2,8}        169: {6,6}        289: {7,7}
     63: {2,2,4}      171: {2,2,8}      299: {6,9}
     65: {3,6}        183: {2,18}       301: {4,14}
     81: {2,2,2,2}    185: {3,12}       303: {2,26}
     87: {2,10}       189: {2,2,2,4}    305: {3,18}
     91: {4,6}        203: {4,10}       319: {5,10}
    111: {2,12}       213: {2,20}       321: {2,28}
    115: {3,9}        235: {3,15}       325: {3,3,6}
    117: {2,2,6}      237: {2,22}       333: {2,2,12}

A200976 counts the partitions with these Heinz numbers.
A302696 is the pairwise coprime instead of pairwise non-coprime version.
A337694 includes the primes.
A002808 lists composite numbers.
A318717 counts pairwise intersecting strict partitions.
A328673 counts partitions with pairwise intersecting distinct parts, with Heinz numbers A328867 and restriction to triples A337599 (except n = 3).
Cf. A008578, A051185, A056239, A101268, A112798, A284825, A302569, A305843, A319752, A327516, A335236, A337666, A337667.

stabstrQ[u_,Q_]:=And@@Not/@Q@@@Tuples[u,2];
Select[Range[2,100],!PrimeQ[#]&&stabstrQ[PrimePi/@First/@FactorInteger[#],CoprimeQ]&]

Equals A337694 \ A008578.

cp's OEIS Frontend

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A338315 Number of integer partitions of n with no 1's whose distinct parts are pairwise coprime, where a singleton is not considered coprime unless it is (1).

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A338316 Odd numbers whose distinct prime indices are pairwise coprime, where a singleton is always considered coprime.

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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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