A337462 -id:A337462 - cp's OEIS frontend

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

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]&]

A337450 Number of relatively prime compositions of n with no 1's.

Original entry on oeis.org

0, 0, 0, 0, 0, 2, 0, 7, 5, 17, 17, 54, 51, 143, 168, 358, 482, 986, 1313, 2583, 3663, 6698, 9921, 17710, 26489, 46352, 70928, 121137, 188220, 317810, 497322, 832039, 1313501, 2177282, 3459041, 5702808, 9094377, 14930351, 23895672, 39084070, 62721578

Offset: 0

Gus Wiseman, Aug 31 2020

nonn

A composition of n is a finite sequence of positive integers summing to n.

			The a(5) = 2 through a(10) = 17 compositions (empty column indicated by dot):
  (2,3)  .  (2,5)    (3,5)    (2,7)      (3,7)
  (3,2)     (3,4)    (5,3)    (4,5)      (7,3)
            (4,3)    (2,3,3)  (5,4)      (2,3,5)
            (5,2)    (3,2,3)  (7,2)      (2,5,3)
            (2,2,3)  (3,3,2)  (2,2,5)    (3,2,5)
            (2,3,2)           (2,3,4)    (3,3,4)
            (3,2,2)           (2,4,3)    (3,4,3)
                              (2,5,2)    (3,5,2)
                              (3,2,4)    (4,3,3)
                              (3,4,2)    (5,2,3)
                              (4,2,3)    (5,3,2)
                              (4,3,2)    (2,2,3,3)
                              (5,2,2)    (2,3,2,3)
                              (2,2,2,3)  (2,3,3,2)
                              (2,2,3,2)  (3,2,2,3)
                              (2,3,2,2)  (3,2,3,2)
                              (3,2,2,2)  (3,3,2,2)

Alois P. Heinz, Table of n, a(n) for n = 0..2000 (first 171 terms from Fausto A. C. Cariboni)

A000740 is the version allowing 1's.
2*A055684(n) is the case of length 2.
A302697 ranks the unordered case.
A302698 is the unordered version.
A337451 is the strict version.
A337452 is the unordered strict version.
A000837 counts relatively prime partitions.
A002865 counts partitions with no 1's.
A101268 counts singleton or pairwise coprime compositions.
A212804 counts compositions with no 1's.
A291166 appears to rank relatively prime compositions.
A337462 counts pairwise coprime compositions.
Cf. A000010, A007359, A023023, A101268, A178472, A289509, A302568, A337485.

b:= proc(n, g) option remember; `if`(n=0,
     `if`(g=1, 1, 0), add(b(n-j, igcd(g, j)), j=2..n))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..42);

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],!MemberQ[#,1]&&GCD@@#==1&]],{n,0,15}]

A337451 Number of relatively prime strict compositions of n with no 1's.

Original entry on oeis.org

0, 0, 0, 0, 0, 2, 0, 4, 2, 10, 8, 20, 14, 34, 52, 72, 90, 146, 172, 244, 390, 502, 680, 956, 1218, 1686, 2104, 3436, 4078, 5786, 7200, 10108, 12626, 17346, 20876, 32836, 38686, 53674, 67144, 91528, 113426, 152810, 189124, 245884, 343350, 428494, 552548, 719156

Offset: 0

Gus Wiseman, Aug 31 2020

nonn

A strict composition of n is a finite sequence of distinct positive integers summing to n.

			The a(5) = 2 through a(10) = 8 compositions (empty column indicated by dot):
  (2,3)  .  (2,5)  (3,5)  (2,7)    (3,7)
  (3,2)     (3,4)  (5,3)  (4,5)    (7,3)
            (4,3)         (5,4)    (2,3,5)
            (5,2)         (7,2)    (2,5,3)
                          (2,3,4)  (3,2,5)
                          (2,4,3)  (3,5,2)
                          (3,2,4)  (5,2,3)
                          (3,4,2)  (5,3,2)
                          (4,2,3)
                          (4,3,2)

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..350

A032022 does not require relative primality.
A302698 is the unordered non-strict version.
A332004 is the version allowing 1's.
A337450 is the non-strict version.
A337452 is the unordered version.
A000837 counts relatively prime partitions.
A032020 counts strict compositions.
A078374 counts strict relatively prime partitions.
A002865 counts partitions with no 1's.
A212804 counts compositions with no 1's.
A291166 appears to rank relatively prime compositions.
A337462 counts pairwise coprime compositions.
A337561 counts strict pairwise coprime compositions.
Cf. A000010, A007359, A101268, A178472, A216652, A289509, A337562, A337563.

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],UnsameQ@@#&&!MemberQ[#,1]&&GCD@@#==1&]],{n,0,15}]

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.

A337697 Number of pairwise coprime compositions of n with no 1's, where a singleton is not considered coprime.

Original entry on oeis.org

0, 0, 0, 0, 0, 2, 0, 4, 2, 4, 8, 8, 14, 10, 16, 12, 30, 38, 46, 46, 48, 52, 62, 152, 96, 156, 112, 190, 256, 338, 420, 394, 326, 402, 734, 622, 1150, 802, 946, 898, 1730, 1946, 2524, 2200, 2328, 2308, 3356, 5816, 4772, 5350, 4890, 6282, 6316, 12092, 8902

Offset: 0

Gus Wiseman, Oct 06 2020

nonn

A composition of n is a finite sequence of positive integers summing to n. These compositions must be strict.

			The a(5) = 2 through a(12) = 14 compositions (empty column indicated by dot):
  (2,3)  .  (2,5)  (3,5)  (2,7)  (3,7)    (2,9)  (5,7)
  (3,2)     (3,4)  (5,3)  (4,5)  (7,3)    (3,8)  (7,5)
            (4,3)         (5,4)  (2,3,5)  (4,7)  (2,3,7)
            (5,2)         (7,2)  (2,5,3)  (5,6)  (2,7,3)
                                 (3,2,5)  (6,5)  (3,2,7)
                                 (3,5,2)  (7,4)  (3,4,5)
                                 (5,2,3)  (8,3)  (3,5,4)
                                 (5,3,2)  (9,2)  (3,7,2)
                                                 (4,3,5)
                                                 (4,5,3)
                                                 (5,3,4)
                                                 (5,4,3)
                                                 (7,2,3)
                                                 (7,3,2)

A022340 intersected with A333227 is a ranking sequence (using standard compositions A066099) for these compositions.
A212804 does not require coprimality, with unordered version A002865.
A337450 is the relatively prime instead of pairwise coprime version, with strict case A337451 and unordered version A302698.
A337462 allows 1's, with strict case A337561 (or A101268 with singletons), unordered version A327516 with Heinz numbers A302696, and 3-part case A337461.
A337485 is the unordered version (or A007359 with singletons considered coprime), with Heinz numbers A337984.
A337563 is the case of unordered triples.
Cf. A078374, A178472, A302568, A302697, A305713, A307719, A332004, A337562.

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

For n > 1, the version where singletons are considered coprime is a(n) + 1.

A337482 Number of compositions of n that are neither strictly increasing nor weakly decreasing.

Original entry on oeis.org

0, 0, 0, 0, 2, 7, 18, 45, 101, 219, 461, 957, 1957, 3978, 8036, 16182, 32506, 65202, 130642, 261601, 523598, 1047709, 2096062, 4192946, 8386912, 16775117, 33551832, 67105663, 134213789, 268430636, 536865013, 1073734643, 2147474910, 4294956706, 8589921771

Offset: 0

Gus Wiseman, Sep 11 2020

nonn

A composition of n is a finite sequence of positive integers summing to n.

			The a(4) = 2 through a(4) = 18 compositions:
  (112)  (113)   (114)
  (121)  (122)   (132)
         (131)   (141)
         (212)   (213)
         (1112)  (231)
         (1121)  (312)
         (1211)  (1113)
                 (1122)
                 (1131)
                 (1212)
                 (1221)
                 (1311)
                 (2112)
                 (2121)
                 (11112)
                 (11121)
                 (11211)
                 (12111)

Ranked by the complement of the intersection of A114994 and A333255.
A128422 counts only the case of length 3.
A218004 counts the complement.
A332834 is the weak version.
A337481 is the strict version.
A001523 counts unimodal compositions, with complement counted by A115981.
A007318 and A097805 count compositions by length.
A032020 counts strict compositions, ranked by A233564.
A332745/A332835 count partitions/compositions with weakly increasing or weakly decreasing run-lengths.
Cf. A216652, A329398, A332831, A332833, A337462, A337483, A337484, A337605.

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],!Less@@#&&!GreaterEqual@@#&]],{n,0,15}]

a(n) = 2^(n-1) - A000009(n) - A000041(n) + 1, n > 0.

A337481 Number of compositions of n that are neither strictly increasing nor strictly decreasing.

Original entry on oeis.org

0, 0, 1, 1, 5, 11, 25, 55, 117, 241, 493, 1001, 2019, 4061, 8149, 16331, 32705, 65461, 130981, 262037, 524161, 1048425, 2096975, 4194097, 8388365, 16776933, 33554103, 67108481, 134217285, 268434945, 536870321, 1073741145, 2147482869, 4294966401, 8589933569

Offset: 0

Gus Wiseman, Sep 11 2020

nonn

A composition of n is a finite sequence of positive integers summing to n.

			The a(2) = 1 through a(5) = 11 compositions:
  (11)  (111)  (22)    (113)
               (112)   (122)
               (121)   (131)
               (211)   (212)
               (1111)  (221)
                       (311)
                       (1112)
                       (1121)
                       (1211)
                       (2111)
                       (11111)

Ranked by the complement of the intersection of A333255 and A333256.
A332834 is the weak version.
A337482 is the semi-strict version.
A337484 counts only compositions of length 3.
A007318 and A097805 count compositions by length.
A032020 counts strict compositions, ranked by A233564.
A218004 counts strictly increasing or weakly decreasing compositions.
Cf. A216652, A329398, A337462, A337483, A337605.

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],!Less@@#&&!Greater@@#&]],{n,0,15}]

a(n) = 2^(n-1) - 2*A000009(n) + 1, n > 0.

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}]

cp's OEIS Frontend

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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A337450 Number of relatively prime compositions of n with no 1's.

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A337451 Number of relatively prime strict compositions of n with no 1's.

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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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A337697 Number of pairwise coprime compositions of n with no 1's, where a singleton is not considered coprime.

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A337482 Number of compositions of n that are neither strictly increasing nor weakly decreasing.

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