A318749 -id:A318749 - cp's OEIS frontend

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

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}]

A319786 Number of factorizations of n where no two factors are relatively prime.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 2, 1, 1, 1, 5, 1, 2, 1, 2, 1, 1, 1, 4, 2, 1, 3, 2, 1, 1, 1, 7, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 2, 2, 1, 1, 7, 2, 2, 1, 2, 1, 4, 1, 4, 1, 1, 1, 3, 1, 1, 2, 11, 1, 1, 1, 2, 1, 1, 1, 7, 1, 1, 2, 2, 1, 1, 1, 7, 5, 1, 1, 3, 1, 1, 1, 4, 1, 3, 1, 2, 1, 1, 1, 12, 1, 2, 2, 4, 1, 1, 1, 4, 1

Offset: 1

Gus Wiseman, Sep 27 2018

nonn

First differs from A305193 at a(36) = 4, A305193(36) = 5.

a(n) depends only on prime signature of n (cf. A025487). - Antti Karttunen, Nov 07 2018

			The a(48) = 7 factorizations are (2*2*2*6), (2*2*12), (2*4*6), (2*24), (4*12), (6*8), (48).

Antti Karttunen, Table of n, a(n) for n = 1..11520
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..100000
Index entries for sequences computed from exponents in factorization of n

Cf. A001055, A007716, A045778, A049311, A162247, A281116, A283877, A305193, A305854, A306006, A316980, A318749.

Cf. A319755, A319759, A319760, A319765, A319779, A319787, A319782, A319789.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Table[Length[Select[facs[n],!Or@@CoprimeQ@@@Subsets[#,{2}]&]],{n,100}]

A319786(n, m=n, facs=List([])) = if(1==n, (1!=gcd(Vec(facs))), my(s=0, newfacs); fordiv(n, d, if((d>1)&&(d<=m), newfacs = List(facs); listput(newfacs,d); s += A319786(n/d, d, newfacs))); (s)); \\ Antti Karttunen, Nov 07 2018

More terms from Antti Karttunen, Nov 07 2018

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}]

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.

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

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A319786 Number of factorizations of n where no two factors are relatively prime.

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

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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.

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