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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.
3
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
1,2
COMMENTS
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.
FORMULA
Intersection of A337666 and A233564.
EXAMPLE
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)
MATHEMATICA
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]&]
CROSSREFS
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.
Sequence in context: A331380 A115423 A221467 * A045603 A072011 A068535
KEYWORD
nonn
AUTHOR
Gus Wiseman, Oct 06 2020
STATUS
approved