Displaying 1-10 of 16 results found.
Numbers k such that the k-th composition in standard order is a co-necklace.
+10
25
1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 15, 16, 17, 18, 19, 21, 23, 31, 32, 33, 34, 35, 36, 37, 38, 39, 42, 43, 45, 47, 63, 64, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 77, 78, 79, 85, 87, 91, 95, 127, 128, 129, 130, 131, 132, 133, 134, 135, 136, 137, 138, 139, 140
COMMENTS
A co-necklace is a finite sequence that is lexicographically greater than or equal to any cyclic rotation.
A composition of n is a finite sequence of positive integers summing to n. The k-th composition in standard order (row k of 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
EXAMPLE
The sequence together with the corresponding co-necklaces begins:
1: (1) 32: (6) 69: (4,2,1)
2: (2) 33: (5,1) 70: (4,1,2)
3: (1,1) 34: (4,2) 71: (4,1,1,1)
4: (3) 35: (4,1,1) 73: (3,3,1)
5: (2,1) 36: (3,3) 74: (3,2,2)
7: (1,1,1) 37: (3,2,1) 75: (3,2,1,1)
8: (4) 38: (3,1,2) 77: (3,1,2,1)
9: (3,1) 39: (3,1,1,1) 78: (3,1,1,2)
10: (2,2) 42: (2,2,2) 79: (3,1,1,1,1)
11: (2,1,1) 43: (2,2,1,1) 85: (2,2,2,1)
15: (1,1,1,1) 45: (2,1,2,1) 87: (2,2,1,1,1)
16: (5) 47: (2,1,1,1,1) 91: (2,1,2,1,1)
17: (4,1) 63: (1,1,1,1,1,1) 95: (2,1,1,1,1,1)
18: (3,2) 64: (7) 127: (1,1,1,1,1,1,1)
19: (3,1,1) 65: (6,1) 128: (8)
21: (2,2,1) 66: (5,2) 129: (7,1)
23: (2,1,1,1) 67: (5,1,1) 130: (6,2)
31: (1,1,1,1,1) 68: (4,3) 131: (6,1,1)
MATHEMATICA
stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n, 2]], 1], 0]]//Reverse;
coneckQ[q_]:=Array[OrderedQ[{RotateRight[q, #], q}]&, Length[q]-1, 1, And];
Select[Range[100], coneckQ[stc[#]]&]
CROSSREFS
Necklaces covering an initial interval are A019536. Numbers whose prime signature is a necklace are A329138. Length of co-Lyndon factorization of binary expansion is A329312. Length of Lyndon factorization of reversed binary expansion is A329313. All of the following pertain to compositions in standard order ( A066099): - Rotational symmetries are counted by A138904. - Constant compositions are A272919. - Co-Lyndon compositions are A326774. - Aperiodic compositions are A328594. - Length of Lyndon factorization is A329312. Cf. A000740, A001037, A027375, A059966, A211100, A302291, A328596, A329142, A333765, A333939, A333941.
Numbers whose prime signature is a Lyndon word.
+10
22
2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 18, 19, 23, 25, 27, 29, 31, 32, 37, 41, 43, 47, 49, 50, 53, 54, 59, 61, 64, 67, 71, 73, 75, 79, 81, 83, 89, 97, 98, 101, 103, 107, 108, 109, 113, 121, 125, 127, 128, 131, 137, 139, 147, 149, 150, 151, 157, 162, 163, 167
COMMENTS
First differs from A133811 in having 50. A Lyndon word is a finite sequence that is lexicographically strictly less than all of its cyclic rotations.
A number's prime signature is the sequence of positive exponents in its prime factorization.
EXAMPLE
The prime signature of 30870 is (1,2,1,3), which is a Lyndon word, so 30870 is in the sequence.
The sequence of terms together with their prime indices begins:
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}
18: {1,2,2}
19: {8}
23: {9}
25: {3,3}
27: {2,2,2}
29: {10}
31: {11}
32: {1,1,1,1,1}
MATHEMATICA
lynQ[q_]:=Array[Union[{q, RotateRight[q, #]}]=={q, RotateRight[q, #]}&, Length[q]-1, 1, And];
Select[Range[2, 100], lynQ[Last/@FactorInteger[#]]&]
CROSSREFS
Numbers whose reversed binary expansion is Lyndon are A328596. Numbers whose prime signature is a necklace are A329138. Numbers whose prime signature is aperiodic are A329139. Cf. A000031, A000740, A000961, A001037, A025487, A027375, A097318, A112798, A118914, A304678, A318731, A329140, A329142.
Numbers with more than one prime factor and, in the ordered factorization, the exponent never increases when read from left to right.
+10
20
6, 10, 12, 14, 15, 20, 21, 22, 24, 26, 28, 30, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45, 46, 48, 51, 52, 55, 56, 57, 58, 60, 62, 63, 65, 66, 68, 69, 70, 72, 74, 76, 77, 78, 80, 82, 84, 85, 86, 87, 88, 91, 92, 93, 94, 95, 96, 99, 100, 102, 104, 105, 106, 110, 111, 112, 114
COMMENTS
If n = Product_{k=1..m} p(k)^e(k), then m > 1, e(1) >= e(2) >= ... >= e(m).
These are numbers whose ordered prime signature is weakly decreasing. Weakly increasing is A304678. Ordered prime signature is A124010. - Gus Wiseman, Nov 10 2019
EXAMPLE
60 is 2^2*3^1*5^1, A001221(60)=3 and 2>=1>=1, so 60 is in sequence.
MAPLE
q:= n-> (l-> (t-> t>1 and andmap(i-> l[i, 2]>=l[i+1, 2],
[$1..t-1]))(nops(l)))(sort(ifactors(n)[2])):
MATHEMATICA
fQ[n_] := Module[{f = Transpose[FactorInteger[n]][[2]]}, Length[f] > 1 && Max[Differences[f]] <= 0]; Select[Range[2, 200], fQ] (* T. D. Noe, Nov 04 2013 *)
PROG
(PARI) for(n=1, 130, F=factor(n); t=0; s=matsize(F)[1]; if(s>1, for(k=1, s-1, if(F[k, 2]<F[k+1, 2], t=1; break)); if(!t, print1(n", "))))
Numbers whose prime signature is an aperiodic word.
+10
20
1, 2, 3, 4, 5, 7, 8, 9, 11, 12, 13, 16, 17, 18, 19, 20, 23, 24, 25, 27, 28, 29, 31, 32, 37, 40, 41, 43, 44, 45, 47, 48, 49, 50, 52, 53, 54, 56, 59, 60, 61, 63, 64, 67, 68, 71, 72, 73, 75, 76, 79, 80, 81, 83, 84, 88, 89, 90, 92, 96, 97, 98, 99, 101, 103, 104
COMMENTS
First differs from A319161 in having 1260 = 2*2 * 3^2 * 5^1 * 7^1. First differs from A325370 in having 420 = 2^2 * 3^1 * 5^1 * 7^1. A number's prime signature ( A124010) is the sequence of positive exponents in its prime factorization. A sequence is aperiodic if its cyclic rotations are all different.
EXAMPLE
The sequence of terms together with their prime signatures begins:
1: ()
2: (1)
3: (1)
4: (2)
5: (1)
7: (1)
8: (3)
9: (2)
11: (1)
12: (2,1)
13: (1)
16: (4)
17: (1)
18: (1,2)
19: (1)
20: (2,1)
23: (1)
24: (3,1)
25: (2)
27: (3)
MATHEMATICA
aperQ[q_]:=Array[RotateRight[q, #1]&, Length[q], 1, UnsameQ];
Select[Range[100], aperQ[Last/@FactorInteger[#]]&]
CROSSREFS
Aperiodic compositions are A000740. Aperiodic binary words are A027375. Numbers whose binary expansion is aperiodic are A328594. Numbers whose prime signature is a Lyndon word are A329131. Numbers whose prime signature is a necklace are A329138. Cf. A025487, A097318, A112798, A124010, A178472, A181819, A304678, A329133, A329135, A329136, A329137, A329142.
Numbers k such that the k-th composition in standard order is a reversed necklace.
+10
19
1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 15, 16, 17, 18, 19, 21, 23, 31, 32, 33, 34, 35, 36, 37, 39, 41, 42, 43, 45, 47, 63, 64, 65, 66, 67, 68, 69, 71, 73, 74, 75, 77, 79, 81, 83, 85, 87, 91, 95, 127, 128, 129, 130, 131, 132, 133, 135, 136, 137, 138, 139, 141, 143
COMMENTS
A necklace is a finite sequence that is lexicographically minimal among all of its cyclic rotations. Reversed necklaces are different from co-necklaces ( A333764). A composition of n is a finite sequence of positive integers summing to n. The k-th composition in standard order (row k of 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.
EXAMPLE
The sequence together with the corresponding reversed necklaces begins:
1: (1) 32: (6) 69: (4,2,1)
2: (2) 33: (5,1) 71: (4,1,1,1)
3: (1,1) 34: (4,2) 73: (3,3,1)
4: (3) 35: (4,1,1) 74: (3,2,2)
5: (2,1) 36: (3,3) 75: (3,2,1,1)
7: (1,1,1) 37: (3,2,1) 77: (3,1,2,1)
8: (4) 39: (3,1,1,1) 79: (3,1,1,1,1)
9: (3,1) 41: (2,3,1) 81: (2,4,1)
10: (2,2) 42: (2,2,2) 83: (2,3,1,1)
11: (2,1,1) 43: (2,2,1,1) 85: (2,2,2,1)
15: (1,1,1,1) 45: (2,1,2,1) 87: (2,2,1,1,1)
16: (5) 47: (2,1,1,1,1) 91: (2,1,2,1,1)
17: (4,1) 63: (1,1,1,1,1,1) 95: (2,1,1,1,1,1)
18: (3,2) 64: (7) 127: (1,1,1,1,1,1,1)
19: (3,1,1) 65: (6,1) 128: (8)
21: (2,2,1) 66: (5,2) 129: (7,1)
23: (2,1,1,1) 67: (5,1,1) 130: (6,2)
31: (1,1,1,1,1) 68: (4,3) 131: (6,1,1)
MATHEMATICA
stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n, 2]], 1], 0]]//Reverse;
neckQ[q_]:=Array[OrderedQ[{q, RotateRight[q, #1]}]&, Length[q]-1, 1, And];
Select[Range[100], neckQ[Reverse[stc[#]]]&]
CROSSREFS
The non-reversed version is A065609. Necklaces covering an initial interval are A019536. Numbers whose prime signature is a necklace are A329138. Length of co-Lyndon factorization of binary expansion is A329312. Length of Lyndon factorization of reversed binary expansion is A329313. All of the following pertain to compositions in standard order ( A066099): - Rotational symmetries are counted by A138904. - Constant compositions are A272919. - Co-Lyndon compositions are A326774. - Aperiodic compositions are A328594. - Length of Lyndon factorization is A329312. - Length of co-Lyndon factorization is A334029. Cf. A000740, A001037, A027375, A059966, A211100, A302291, A328596, A329142, A333765, A333939, A333941.
Number of Lyndon factorizations of the k-th composition in standard order.
+10
15
1, 1, 1, 2, 1, 2, 1, 3, 1, 2, 2, 4, 1, 2, 1, 5, 1, 2, 2, 4, 1, 4, 2, 7, 1, 2, 1, 4, 1, 2, 1, 7, 1, 2, 2, 4, 2, 5, 2, 7, 1, 2, 3, 9, 2, 5, 2, 12, 1, 2, 1, 4, 1, 2, 2, 7, 1, 2, 1, 4, 1, 2, 1, 11, 1, 2, 2, 4, 2, 5, 2, 7, 1, 4, 4, 11, 2, 5, 2, 12, 1, 2, 2, 4, 1, 7
COMMENTS
We define the Lyndon product of two or more finite sequences to be the lexicographically maximal sequence obtainable by shuffling the sequences together. For example, the Lyndon product of (231) with (213) is (232131), the product of (221) with (213) is (222131), and the product of (122) with (2121) is (2122121). A Lyndon factorization of a composition c is a multiset of compositions whose Lyndon product is c.
A composition of n is a finite sequence of positive integers summing to n. 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. Also the number of multiset partitions of the Lyndon-word factorization of the n-th composition in standard order.
FORMULA
For n > 0, Sum_{k = 2^(n-1)..2^n-1} a(k) = A034691(n).
EXAMPLE
We have a(300) = 5, because the 300th composition (3,2,1,3) has the following Lyndon factorizations:
((3,2,1,3))
((1,3),(3,2))
((2),(3,1,3))
((3),(2,1,3))
((2),(3),(1,3))
MATHEMATICA
stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n, 2]], 1], 0]]//Reverse;
lynprod[]:={}; lynprod[{}, b_List]:=b; lynprod[a_List, {}]:=a; lynprod[a_List]:=a;
lynprod[{x_, a___}, {y_, b___}]:=Switch[Ordering[If[x=!=y, {x, y}, {lynprod[{a}, {x, b}], lynprod[{x, a}, {b}]}]], {2, 1}, Prepend[lynprod[{a}, {y, b}], x], {1, 2}, Prepend[lynprod[{x, a}, {b}], y]];
lynprod[a_List, b_List, c__List]:=lynprod[a, lynprod[b, c]];
sps[{}]:={{}}; sps[set:{i_, ___}]:=Join@@Function[s, Prepend[#, s]&/@sps[Complement[set, s]]]/@Cases[Subsets[set], {i, ___}];
dealings[q_]:=Union[Function[ptn, Sort[q[[#]]&/@ptn]]/@sps[Range[Length[q]]]];
Table[Length[Select[dealings[stc[n]], lynprod@@#==stc[n]&]], {n, 0, 100}]
CROSSREFS
Binary necklaces are counted by A000031. Necklace compositions are counted by A008965. Necklaces covering an initial interval are counted by A019536. Lyndon compositions are counted by A059966. Numbers whose reversed binary expansion is a necklace are A328595. Numbers whose prime signature is a necklace are A329138. Length of Lyndon factorization of binary expansion is A211100. Length of co-Lyndon factorization of binary expansion is A329312. Length of co-Lyndon factorization of reversed binary expansion is A329326. Length of Lyndon factorization of reversed binary expansion is A329313. All of the following pertain to compositions in standard order ( A066099): - Rotational symmetries are counted by A138904. - Constant compositions are A272919. - Co-Lyndon compositions are A326774. - Aperiodic compositions are A328594. - Reversed co-necklaces are A328595. - Length of Lyndon factorization is A329312. - Length of co-Lyndon factorization is A334029. - Combinatory separations are A334030.
Numbers whose prime signature is a periodic word.
+10
14
6, 10, 14, 15, 21, 22, 26, 30, 33, 34, 35, 36, 38, 39, 42, 46, 51, 55, 57, 58, 62, 65, 66, 69, 70, 74, 77, 78, 82, 85, 86, 87, 91, 93, 94, 95, 100, 102, 105, 106, 110, 111, 114, 115, 118, 119, 122, 123, 129, 130, 133, 134, 138, 141, 142, 143, 145, 146, 154
COMMENTS
First differs from A182853 in having 2100 = 2^2 * 3^1 * 5^2 * 7^1. A number's prime signature ( A124010) is the sequence of positive exponents in its prime factorization. A sequence is aperiodic if its cyclic rotations are all different.
EXAMPLE
The sequence of terms together with their prime signatures begins:
6: (1,1)
10: (1,1)
14: (1,1)
15: (1,1)
21: (1,1)
22: (1,1)
26: (1,1)
30: (1,1,1)
33: (1,1)
34: (1,1)
35: (1,1)
36: (2,2)
38: (1,1)
39: (1,1)
42: (1,1,1)
46: (1,1)
51: (1,1)
55: (1,1)
57: (1,1)
58: (1,1)
MATHEMATICA
aperQ[q_]:=Array[RotateRight[q, #1]&, Length[q], 1, UnsameQ];
Select[Range[100], !aperQ[Last/@FactorInteger[#]]&]
CROSSREFS
Numbers whose binary expansion is periodic are A121016. Numbers whose prime signature is a Lyndon word are A329131. Numbers whose prime signature is a necklace are A329138. Cf. A025487, A056239, A097318, A112798, A118914, A124010, A181819, A304678, A329132 A329134, A329143, A329144.
Number of co-Lyndon factorizations of the k-th composition in standard order.
+10
13
1, 1, 1, 2, 1, 1, 2, 3, 1, 1, 2, 1, 2, 2, 4, 5, 1, 1, 1, 1, 2, 1, 2, 1, 2, 2, 4, 2, 4, 4, 7, 7, 1, 1, 1, 1, 2, 1, 1, 1, 2, 2, 3, 1, 2, 2, 2, 1, 2, 2, 2, 2, 5, 2, 5, 2, 4, 4, 9, 4, 7, 7, 12, 11, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 2, 2, 2, 4, 1
COMMENTS
We define the co-Lyndon product of two or more finite sequences to be the lexicographically minimal sequence obtainable by shuffling the sequences together. For example, the co-Lyndon product of (2,3,1) with (2,1,3) is (2,1,2,3,1,3), the product of (2,2,1) with (2,1,3) is (2,1,2,2,1,3), and the product of (1,2,2) with (2,1,2,1) is (1,2,1,2,1,2,2). A co-Lyndon factorization of a composition c is a multiset of compositions whose co-Lyndon product is c.
A composition of n is a finite sequence of positive integers summing to n. 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. Also the number of multiset partitions of the co-Lyndon-word factorization of the n-th composition in standard order.
FORMULA
For n > 0, Sum_{k = 2^(n-1)..2^n-1} a(k) = A034691(n).
EXAMPLE
The a(54) = 5, a(61) = 7, and a(237) = 9 factorizations:
((1,2,1,2)) ((1,1,1,2,1)) ((1,1,2,1,2,1))
((1),(2,1,2)) ((1),(1,1,2,1)) ((1),(1,2,1,2,1))
((1,2),(2,1)) ((1,1),(1,2,1)) ((1,1),(2,1,2,1))
((2),(1,2,1)) ((2,1),(1,1,1)) ((1,2,1),(1,2,1))
((1),(2),(2,1)) ((1),(1),(1,2,1)) ((2,1),(1,1,2,1))
((1),(1,1),(2,1)) ((1),(1),(2,1,2,1))
((1),(1),(1),(2,1)) ((1,1),(2,1),(2,1))
((1),(2,1),(1,2,1))
((1),(1),(2,1),(2,1))
MATHEMATICA
stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n, 2]], 1], 0]]//Reverse;
colynprod[]:={}; colynprod[{}, b_List]:=b; colynprod[a_List, {}]:=a; colynprod[a_List]:=a;
colynprod[{x_, a___}, {y_, b___}]:=Switch[Ordering[If[x=!=y, {x, y}, {colynprod[{a}, {x, b}], colynprod[{x, a}, {b}]}]], {1, 2}, Prepend[colynprod[{a}, {y, b}], x], {2, 1}, Prepend[colynprod[{x, a}, {b}], y]];
colynprod[a_List, b_List, c__List]:=colynprod[a, colynprod[b, c]];
sps[{}]:={{}}; sps[set:{i_, ___}]:=Join@@Function[s, Prepend[#, s]&/@sps[Complement[set, s]]]/@Cases[Subsets[set], {i, ___}];
dealings[q_]:=Union[Function[ptn, Sort[q[[#]]&/@ptn]]/@sps[Range[Length[q]]]];
Table[Length[Select[dealings[stc[n]], colynprod@@#==stc[n]&]], {n, 0, 100}]
CROSSREFS
Binary necklaces are counted by A000031. Necklace compositions are counted by A008965. Necklaces covering an initial interval are counted by A019536. Lyndon compositions are counted by A059966. Numbers whose reversed binary expansion is a necklace are A328595. Numbers whose prime signature is a necklace are A329138. Length of Lyndon factorization of binary expansion is A211100. Length of co-Lyndon factorization of binary expansion is A329312. Length of co-Lyndon factorization of reversed binary expansion is A329326. Length of Lyndon factorization of reversed binary expansion is A329313. All of the following pertain to compositions in standard order ( A066099): - Rotational symmetries are counted by A138904. - Constant compositions are A272919. - Co-Lyndon compositions are A326774. - Aperiodic compositions are A328594. - Reversed co-necklaces are A328595. - Length of Lyndon factorization is A329312. - Length of co-Lyndon factorization is A334029. - Combinatory separations are A334030.
Length of the co-Lyndon factorization of the k-th composition in standard order.
+10
11
0, 1, 1, 2, 1, 1, 2, 3, 1, 1, 2, 1, 2, 2, 3, 4, 1, 1, 1, 1, 2, 1, 2, 1, 2, 2, 3, 2, 3, 3, 4, 5, 1, 1, 1, 1, 2, 1, 1, 1, 2, 2, 3, 1, 2, 2, 2, 1, 2, 2, 2, 2, 3, 2, 3, 2, 3, 3, 4, 3, 4, 4, 5, 6, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 2, 2, 2, 3, 1, 2
COMMENTS
We define the co-Lyndon product of two or more finite sequences to be the lexicographically minimal sequence obtainable by shuffling the sequences together. For example, the co-Lyndon product of (2,3,1) with (2,1,3) is (2,1,2,3,1,3), the product of (2,2,1) with (2,1,3) is (2,1,2,2,1,3), and the product of (1,2,2) with (2,1,2,1) is (1,2,1,2,1,2,2). A co-Lyndon word is a finite sequence that is prime with respect to the co-Lyndon product. Equivalently, a co-Lyndon word is a finite sequence that is lexicographically strictly greater than all of its cyclic rotations. Every finite sequence has a unique (orderless) factorization into co-Lyndon words, and if these factors are arranged in a certain order, their concatenation is equal to their co-Lyndon product. For example, (1,0,0,1) has co-Lyndon factorization {(1),(1,0,0)}.
A composition of n is a finite sequence of positive integers summing to n. The k-th composition in standard order (row k of 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.
EXAMPLE
The 441st composition in standard order is (1,2,1,1,3,1), with co-Lyndon factorization {(1),(3,1),(2,1,1)}, so a(441) = 3.
MATHEMATICA
stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n, 2]], 1], 0]]//Reverse;
colynQ[q_]:=Length[q]==0||Array[Union[{RotateRight[q, #1], q}]=={RotateRight[q, #1], q}&, Length[q]-1, 1, And];
colynfac[q_]:=If[Length[q]==0, {}, Function[i, Prepend[colynfac[Drop[q, i]], Take[q, i]]][Last[Select[Range[Length[q]], colynQ[Take[q, #1]]&]]]]
Table[Length[colynfac[stc[n]]], {n, 0, 100}]
CROSSREFS
The version for binary expansion is (also) A329312. The version for reversed binary expansion is A329326. Binary Lyndon/co-Lyndon words are counted by A001037. Necklaces covering an initial interval are A019536. Lyndon/co-Lyndon compositions are counted by A059966Length of Lyndon factorization of binomial expansion is A211100. Numbers whose prime signature is a necklace are A329138. Length of Lyndon factorization of reversed binary expansion is A329313. A list of all binary co-Lyndon words is A329318. All of the following pertain to compositions in standard order ( A066099): - Rotational symmetries are counted by A138904. - Constant compositions are A272919. - Co-Lyndon compositions are A326774. - Aperiodic compositions are A328594. - Reversed co-necklaces are A328595. - Co-Lyndon factorizations are counted by A333765. - Lyndon factorizations are counted by A333940. Cf. A034691, A060223, A102659, A211097, A292884, A296372, A328596, A329358, A329359, A329362, A329400, A329401, A333939.
Numbers whose prime signature is not a necklace.
+10
8
1, 12, 20, 24, 28, 40, 44, 45, 48, 52, 56, 60, 63, 68, 72, 76, 80, 84, 88, 90, 92, 96, 99, 104, 112, 116, 117, 120, 124, 126, 132, 135, 136, 140, 144, 148, 152, 153, 156, 160, 164, 168, 171, 172, 175, 176, 180, 184, 188, 189, 192, 198, 200, 204, 207, 208, 212
COMMENTS
After a(1) = 1, first differs from A112769 in lacking 1350. A number's prime signature ( A124010) is the sequence of positive exponents in its prime factorization. A necklace is a finite sequence that is lexicographically minimal among all of its cyclic rotations.
EXAMPLE
The sequence of terms together with their prime signatures begins:
1: ()
12: (2,1)
20: (2,1)
24: (3,1)
28: (2,1)
40: (3,1)
44: (2,1)
45: (2,1)
48: (4,1)
52: (2,1)
56: (3,1)
60: (2,1,1)
63: (2,1)
68: (2,1)
72: (3,2)
76: (2,1)
80: (4,1)
84: (2,1,1)
88: (3,1)
90: (1,2,1)
92: (2,1)
MATHEMATICA
neckQ[q_]:=Array[OrderedQ[{q, RotateRight[q, #]}]&, Length[q]-1, 1, And];
Select[Range[100], #==1||!neckQ[Last/@FactorInteger[#]]&]
CROSSREFS
Non-necklace compositions are A329145. Numbers whose reversed binary expansion is a necklace are A328595. Numbers whose prime signature is a Lyndon word are A329131. Numbers whose prime signature is periodic are A329140. Cf. A001037, A008965, A025487, A056239, A097318, A112798, A118914, A124010, A181819, A304678, A328596, A329139.
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