A200976 -id:A200976 - cp's OEIS frontend

A328867 Heinz numbers of integer partitions in which no two distinct parts are relatively prime.

1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 21, 23, 25, 27, 29, 31, 32, 37, 39, 41, 43, 47, 49, 53, 57, 59, 61, 63, 64, 65, 67, 71, 73, 79, 81, 83, 87, 89, 91, 97, 101, 103, 107, 109, 111, 113, 115, 117, 121, 125, 127, 128, 129, 131, 133, 137, 139, 147, 149

Offset: 1

Gus Wiseman, Oct 30 2019

nonn

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

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

			The sequence of terms together with their prime indices begins:
    1: {}
    2: {1}
    3: {2}
    4: {1,1}
    5: {3}
    7: {4}
    8: {1,1,1}
    9: {2,2}
   11: {5}
   13: {6}
   16: {1,1,1,1}
   17: {7}
   19: {8}
   21: {2,4}
   23: {9}
   25: {3,3}
   27: {2,2,2}
   29: {10}
   31: {11}
   32: {1,1,1,1,1}

These are the Heinz numbers of the partitions counted by A328673.
The strict case is A318719.
The relatively prime version is A328868.
A ranking using binary indices is A326910.
The version for non-isomorphic multiset partitions is A319752.
The version for divisibility (instead of relative primality) is A316476.
Cf. A000837, A056239, A112798, A200976, A289509, A303283, A305843, A318715, A318716, A328336.

Select[Range[100],And@@(GCD[##]>1&)@@@Subsets[PrimePi/@First/@FactorInteger[#],{2}]&]

A337667 Number of compositions of n where any two parts have a common divisor > 1.

Original entry on oeis.org

1, 0, 1, 1, 2, 1, 5, 1, 8, 4, 17, 1, 38, 1, 65, 19, 128, 1, 284, 1, 518, 67, 1025, 1, 2168, 16, 4097, 256, 8198, 1, 16907, 7, 32768, 1027, 65537, 79, 133088, 19, 262145, 4099, 524408, 25, 1056731, 51, 2097158, 16636, 4194317, 79, 8421248, 196, 16777712

Offset: 0

Gus Wiseman, Oct 05 2020

nonn

First differs from A178472 at a(31) = 7, a(31) = 1.

			The a(2) = 1 through a(10) = 17 compositions (A = 10):
   2   3   4    5   6     7   8      9     A
           22       24        26     36    28
                    33        44     63    46
                    42        62     333   55
                    222       224          64
                              242          82
                              422          226
                              2222         244
                                           262
                                           424
                                           442
                                           622
                                           2224
                                           2242
                                           2422
                                           4222
                                           22222

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..290

A101268 = 1 + A337462 is the pairwise coprime version.
A328673 = A200976 + 1 is the unordered version.
A337604 counts these compositions of length 3.
A337666 ranks these compositions.
A337694 gives Heinz numbers of the unordered version.
A337983 is the strict case.
A051185 counts intersecting set-systems, with spanning case A305843.
A318717 is the unordered strict case.
A319786 is the version for factorizations, with strict case A318749.
A327516 counts pairwise coprime partitions.
A333227 ranks pairwise coprime compositions.
A333228 ranks compositions whose distinct parts are pairwise coprime.
Cf. A051424, A082024, A178472, A284825, A319752, A327039, A337599, A337605, A337665.

stabQ[u_,Q_]:=And@@Not/@Q@@@Tuples[u,2];
Table[Length[Join@@Permutations/@Select[IntegerPartitions[n],stabQ[#,CoprimeQ]&]],{n,0,15}]

A302568 Odd numbers that are either prime or whose prime indices are pairwise coprime.

Original entry on oeis.org

3, 5, 7, 11, 13, 15, 17, 19, 23, 29, 31, 33, 35, 37, 41, 43, 47, 51, 53, 55, 59, 61, 67, 69, 71, 73, 77, 79, 83, 85, 89, 93, 95, 97, 101, 103, 107, 109, 113, 119, 123, 127, 131, 137, 139, 141, 143, 145, 149, 151, 155, 157, 161, 163, 165, 167, 173, 177, 179

Offset: 1

Gus Wiseman, Apr 10 2018

nonn

Also Heinz numbers of partitions with pairwise coprime parts all greater than 1 (A007359), where singletons are considered coprime. 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:
      3: {2}       43: {14}      89: {24}      141: {2,15}
      5: {3}       47: {15}      93: {2,11}    143: {5,6}
      7: {4}       51: {2,7}     95: {3,8}     145: {3,10}
     11: {5}       53: {16}      97: {25}      149: {35}
     13: {6}       55: {3,5}    101: {26}      151: {36}
     15: {2,3}     59: {17}     103: {27}      155: {3,11}
     17: {7}       61: {18}     107: {28}      157: {37}
     19: {8}       67: {19}     109: {29}      161: {4,9}
     23: {9}       69: {2,9}    113: {30}      163: {38}
     29: {10}      71: {20}     119: {4,7}     165: {2,3,5}
     31: {11}      73: {21}     123: {2,13}    167: {39}
     33: {2,5}     77: {4,5}    127: {31}      173: {40}
     35: {3,4}     79: {22}     131: {32}      177: {2,17}
     37: {12}      83: {23}     137: {33}      179: {41}
     41: {13}      85: {3,7}    139: {34}      181: {42}
Entry A302242 describes a correspondence between positive integers and multiset multisystems. In this case it gives the following sequence of multiset systems.
03: {{1}}
05: {{2}}
07: {{1,1}}
11: {{3}}
13: {{1,2}}
15: {{1},{2}}
17: {{4}}
19: {{1,1,1}}
23: {{2,2}}
29: {{1,3}}
31: {{5}}
33: {{1},{3}}
35: {{2},{1,1}}
37: {{1,1,2}}
41: {{6}}
43: {{1,4}}
47: {{2,3}}
51: {{1},{4}}
53: {{1,1,1,1}}

A005117 is a superset.
A007359 counts partitions with these Heinz numbers.
A302569 allows evens, with squarefree version A302798.
A337694 is the pairwise non-coprime instead of pairwise coprime version.
A337984 does not include the primes.
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.
A337667 counts pairwise non-coprime compositions, ranked by A337666.
A337697 counts pairwise coprime compositions with no 1's.
Cf. A005408, A051424, A056239, A087087, A112798, A200976, A302797, A303282, A304711, A335235, A338468.

primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[1,400,2],Or[PrimeQ[#],CoprimeQ@@primeMS[#]]&]

Equals A065091 \/ A337984.

Equals A302569 /\ A005408.

Extended by Gus Wiseman, Oct 29 2020

A337694 Numbers with no two relatively prime prime indices.

Original entry on oeis.org

1, 2, 3, 5, 7, 9, 11, 13, 17, 19, 21, 23, 25, 27, 29, 31, 37, 39, 41, 43, 47, 49, 53, 57, 59, 61, 63, 65, 67, 71, 73, 79, 81, 83, 87, 89, 91, 97, 101, 103, 107, 109, 111, 113, 115, 117, 121, 125, 127, 129, 131, 133, 137, 139, 147, 149, 151, 157, 159, 163, 167, 169, 171, 173, 179, 181, 183, 185, 189, 191, 193, 197, 199

Offset: 1

Gus Wiseman, Sep 23 2020

nonn

First differs from A305078 in having 1 and lacking 195.
First differs from A305103 in having 1 and 169 and lacking 195.
First differs from A328336 in lacking 897, with prime indices (2,6,9).
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 integer partitions in which no two parts are relatively prime. The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

			The sequence of terms together with their prime indices begins:
   1: {}      37: {12}     79: {22}      121: {5,5}
   3: {2}     39: {2,6}    81: {2,2,2,2} 125: {3,3,3}
   5: {3}     41: {13}     83: {23}      127: {31}
   7: {4}     43: {14}     87: {2,10}    129: {2,14}
   9: {2,2}   47: {15}     89: {24}      131: {32}
  11: {5}     49: {4,4}    91: {4,6}     133: {4,8}
  13: {6}     53: {16}     97: {25}      137: {33}
  17: {7}     57: {2,8}   101: {26}      139: {34}
  19: {8}     59: {17}    103: {27}      147: {2,4,4}
  21: {2,4}   61: {18}    107: {28}      149: {35}
  23: {9}     63: {2,2,4} 109: {29}      151: {36}
  25: {3,3}   65: {3,6}   111: {2,12}    157: {37}
  27: {2,2,2} 67: {19}    113: {30}      159: {2,16}
  29: {10}    71: {20}    115: {3,9}     163: {38}
  31: {11}    73: {21}    117: {2,2,6}   167: {39}

Robert Israel, Table of n, a(n) for n = 1..10000

A200976 and A328673 count these partitions.
A302696 and A302569 are pairwise coprime instead of pairwise non-coprime.
A318719 is the squarefree case.
A328867 looks at distinct prime indices.
A337666 is the version for standard compositions.
A101268 counts pairwise coprime or singleton compositions.
A318717 counts strict pairwise non-coprime partitions.
A327516 counts pairwise coprime partitions.
A333227 ranks pairwise coprime compositions.
A333228 ranks compositions whose distinct parts are pairwise coprime.
A335236 ranks compositions neither a singleton nor pairwise coprime.
A337462 counts pairwise coprime compositions.
A337667 counts pairwise non-coprime compositions.
Cf. A051185, A051424, A056239, A112798, A220377, A284825, A302797, A303282, A305843, A319752, A336737, A337599, A337604, A337605.

filter:= proc(n) local F,i,j,np;
  if n::even and n>2 then return false fi;
  F:= map(t -> numtheory:-pi(t[1]), ifactors(n)[2]);
  np:= nops(F);
  for i from 1 to np-1 do
    for j from i+1 to np do
      if igcd(F[i],F[j])=1 then return false fi
  od od;
  true
end proc:
select(filter, [$1..300]); # Robert Israel, Oct 06 2020

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
stabQ[u_,Q_]:=Array[#1==#2||!Q[u[[#1]],u[[#2]]]&,{Length[u],Length[u]},1,And];
Select[Range[100],stabQ[primeMS[#],CoprimeQ]&]

A337666 Numbers k such that any two parts of the k-th composition in standard order (A066099) have a common divisor > 1.

Original entry on oeis.org

0, 2, 4, 8, 10, 16, 32, 34, 36, 40, 42, 64, 128, 130, 136, 138, 160, 162, 168, 170, 256, 260, 288, 292, 512, 514, 520, 522, 528, 544, 546, 552, 554, 640, 642, 648, 650, 672, 674, 680, 682, 1024, 2048, 2050, 2052, 2056, 2058, 2080, 2082, 2084, 2088, 2090, 2176

Offset: 1

Gus Wiseman, Oct 05 2020

nonn

Differs from A291165 in having 1090535424, corresponding to the composition (6,10,15).
This is a ranking sequence for pairwise non-coprime compositions.
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: ()          138: (4,2,2)       546: (4,4,2)
       2: (2)         160: (2,6)         552: (4,2,4)
       4: (3)         162: (2,4,2)       554: (4,2,2,2)
       8: (4)         168: (2,2,4)       640: (2,8)
      10: (2,2)       170: (2,2,2,2)     642: (2,6,2)
      16: (5)         256: (9)           648: (2,4,4)
      32: (6)         260: (6,3)         650: (2,4,2,2)
      34: (4,2)       288: (3,6)         672: (2,2,6)
      36: (3,3)       292: (3,3,3)       674: (2,2,4,2)
      40: (2,4)       512: (10)          680: (2,2,2,4)
      42: (2,2,2)     514: (8,2)         682: (2,2,2,2,2)
      64: (7)         520: (6,4)        1024: (11)
     128: (8)         522: (6,2,2)      2048: (12)
     130: (6,2)       528: (5,5)        2050: (10,2)
     136: (4,4)       544: (4,6)        2052: (9,3)

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

A337604 counts these compositions of length 3.
A337667 counts these compositions.
A337694 is the version for Heinz numbers of partitions.
A337696 is the strict case.
A051185 and A305843 (covering) count pairwise intersecting set-systems.
A101268 counts pairwise coprime or singleton compositions.
A200976 and A328673 count pairwise non-coprime partitions.
A318717 counts strict pairwise non-coprime partitions.
A327516 counts pairwise coprime partitions.
A335236 ranks compositions neither a singleton nor pairwise coprime.
A337462 counts pairwise coprime compositions.
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.
- A291166 appears to rank relatively prime compositions.
- A326674 is greatest common divisor.
- A333219 is Heinz number.
- A333227 ranks coprime (Mathematica definition) compositions.
- A333228 ranks compositions with distinct parts coprime.
- A335235 ranks singleton or coprime compositions.
Cf. A082024, A284825, A305713, A319752, A319786, A327039, A327040, A336737, A337599, 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],stabQ[stc[#],CoprimeQ]&]

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

Original entry on oeis.org

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

A328868 Heinz numbers of integer partitions with no two (not necessarily distinct) parts relatively prime, but with no divisor in common to all of the parts.

Original entry on oeis.org

17719, 40807, 43381, 50431, 74269, 83143, 101543, 105703, 116143, 121307, 123469, 139919, 140699, 142883, 171613, 181831, 185803, 191479, 203557, 205813, 211381, 213239, 215267, 219271, 230347, 246703, 249587, 249899, 279371, 286897, 289007, 296993, 300847

Offset: 1

Gus Wiseman, Oct 30 2019

nonn

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

			The sequence of terms together with their prime indices begins:
   17719: {6,10,15}
   40807: {6,14,21}
   43381: {6,15,20}
   50431: {10,12,15}
   74269: {6,10,45}
   83143: {10,15,18}
  101543: {6,21,28}
  105703: {6,15,40}
  116143: {12,14,21}
  121307: {10,15,24}
  123469: {12,15,20}
  139919: {6,15,50}
  140699: {6,22,33}
  142883: {6,10,75}
  171613: {6,14,63}
  181831: {6,20,45}
  185803: {10,14,35}
  191479: {14,18,21}
  203557: {15,18,20}
  205813: {10,15,36}
  211381: {10,12,45}
  213239: {6,15,70}
  215267: {6,10,105}
  219271: {6,26,39}
  230347: {6,6,10,15}

These are the Heinz numbers of the partitions counted by A202425.
Terms of A328679 that are not powers of 2.
The strict case is A318716 (preceded by 2).
A ranking using binary indices (instead of prime indices) is A326912.
Heinz numbers of relatively prime partitions are A289509.
Cf. A000837, A056239, A112798, A200976, A291166, A302796, A316476, A318715, A319752, A319759, A328336, A328672, A328677, A328867.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
dv=Select[Range[100000],GCD@@primeMS[#]==1&&And[And@@(GCD[##]>1&)@@@Tuples[Union[primeMS[#]],2]]&]

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}]

A337987 Odd numbers whose distinct prime indices are pairwise coprime, where a singleton is not considered coprime unless it is (1).

Original entry on oeis.org

15, 33, 35, 45, 51, 55, 69, 75, 77, 85, 93, 95, 99, 119, 123, 135, 141, 143, 145, 153, 155, 161, 165, 175, 177, 187, 201, 205, 207, 209, 215, 217, 219, 221, 225, 245, 249, 253, 255, 265, 275, 279, 287, 291, 295, 297, 309, 323, 327, 329, 335, 341, 355, 363, 369

Offset: 1

Gus Wiseman, Oct 23 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.

Also Heinz numbers of integer partitions with no 1's whose distinct parts are pairwise coprime (A338315). 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 sequence of terms together with their prime indices begins:
     15: {2,3}      135: {2,2,2,3}    215: {3,14}
     33: {2,5}      141: {2,15}       217: {4,11}
     35: {3,4}      143: {5,6}        219: {2,21}
     45: {2,2,3}    145: {3,10}       221: {6,7}
     51: {2,7}      153: {2,2,7}      225: {2,2,3,3}
     55: {3,5}      155: {3,11}       245: {3,4,4}
     69: {2,9}      161: {4,9}        249: {2,23}
     75: {2,3,3}    165: {2,3,5}      253: {5,9}
     77: {4,5}      175: {3,3,4}      255: {2,3,7}
     85: {3,7}      177: {2,17}       265: {3,16}
     93: {2,11}     187: {5,7}        275: {3,3,5}
     95: {3,8}      201: {2,19}       279: {2,2,11}
     99: {2,2,5}    205: {3,13}       287: {4,13}
    119: {4,7}      207: {2,2,9}      291: {2,25}
    123: {2,13}     209: {5,8}        295: {3,17}

A304711 is the not necessarily odd version, with squarefree case A302797.
A337694 is a pairwise non-coprime instead of pairwise coprime version.
A337984 is the squarefree case.
A338315 counts the partitions with these Heinz numbers.
A338316 considers singletons coprime.
A007359 counts partitions into singleton or pairwise coprime parts with no 1's, with Heinz numbers A302568.
A304709 counts partitions whose distinct parts are pairwise coprime.
A327516 counts pairwise coprime partitions, with Heinz numbers 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.
A318717 counts pairwise non-coprime strict partitions, with Heinz numbers A318719.
Cf. A005408, A051424, A056239, A087087, A112798, A200976, A289508, A289509, A302569, A303282, A328867, A337485.

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

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A328867 Heinz numbers of integer partitions in which no two distinct parts are relatively prime.

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A337667 Number of compositions of n where any two parts have a common divisor > 1.

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A302568 Odd numbers that are either prime or whose prime indices are pairwise coprime.

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A337694 Numbers with no two relatively prime prime indices.

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A337666 Numbers k such that any two parts of the k-th composition in standard order (A066099) have a common divisor > 1.

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

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

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A328868 Heinz numbers of integer partitions with no two (not necessarily distinct) parts relatively prime, but with no divisor in common to all of the parts.

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