Displaying 1-10 of 21 results found.
a(n) = Number of fixed points of permutation A209861/ A209862 in range [2^(n-1),(2^n)-1].
+20
9
1, 1, 2, 4, 6, 6, 6, 8, 8, 8, 12, 8, 10, 8, 10, 6, 6, 10, 8, 6, 6, 10, 8, 8, 6
COMMENTS
See the conjecture given in A209860. If true, then all the terms from a(2) onward are even. a(0) gives the number of fixed points in range [0,0], i.e. 1.
a(n) = number of cycles in range [2^(n-1),(2^n)-1] of permutation A209861/ A209862.
+20
8
1, 1, 2, 4, 7, 8, 11, 12, 14, 10, 21, 14, 20, 26, 22, 18, 18, 28, 23, 30, 32
COMMENTS
a(0) gives the number of cycles in range [0,0], i.e. 1.
a(n) = least common multiple of all cycle sizes in range [2^(n-1),(2^n)-1] of permutation A209861/ A209862.
+20
8
1, 1, 1, 1, 2, 5, 24, 26, 672, 246, 3755388, 13827240, 1768910220, 99034598880, 1463488641762840, 612823600, 171768365608799778, 16338317307187487976, 27491145139913884194480, 14794457633180140325810400, 2084886621890359572790082258379649440
COMMENTS
a(0) gives the LCM of cycle sizes in range [0,0], i.e., 1.
EXAMPLE
In range [2^(6-1),(2^6)-1] ([32,63]) of permutations A209861 & A209862, there are 6 cycles of size 1 (fixed points), 2 cycles of size 3, one cycle of size 4, and 2 cycles of size 8 (6 + 2*3 + 4 + 2*8 = 32), thus a(6) = lcm(1,3,4,8) = 24.
a(n) = maximal cycle size in range [2^(n-1),(2^n)-1] of permutation A209861/ A209862.
+20
7
1, 1, 1, 1, 2, 5, 8, 26, 96, 246, 181, 540, 868, 724, 6038, 4405, 23302, 39514, 34480, 83424, 270884
COMMENTS
a(0) gives the maximum cycle size in range [0,0], i.e. 1.
a(n) = number of integers in range [2^(n-1),(2^n)-1] which permutation A209861/ A209862 sends to odd-sized orbits.
+20
7
1, 1, 2, 4, 6, 16, 12, 8, 14, 8, 406, 8, 56, 80, 1686, 8866, 8272, 15178, 9462, 938, 41128
COMMENTS
a(0) gives the number of odd sized cycles in range [0,0], i.e. 1, as there is just one fixed point in that range.
EXAMPLE
In range [2^(6-1),(2^6)-1] ([32,63]) of permutations A209861 & A209862, there are 6 cycles of size 1 (six fixed points), 2 cycles of size 3, one cycle of size 4, and 2 cycles of size 8, 6*1 + 2*3 + 1*4 + 2*8 = 32 in total, of which 6*1 + 2*3 elements are in odd-sized cycles, thus a(6)=12.
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 13, 14, 15, 16, 17, 18, 29, 30, 31, 32, 33, 34, 61, 62, 63, 64, 65, 66, 73, 118, 125, 126, 127, 128, 129, 130, 148, 235, 253, 254, 255, 256, 257, 258, 274, 493, 509, 510, 511, 512, 513, 514, 651, 689, 710, 825, 846, 884, 1021, 1022, 1023, 1024, 1025, 1026, 1097, 1974, 2045, 2046, 2047, 2048, 2049
COMMENTS
Conjecture: for every a(n), also A054429(a(n)) is in the sequence. Conversely, if i is not in the sequence, then neither is A054429(i).
CROSSREFS
A209863 gives the number of these fixed points in each range [2^(n-1),(2^n)-1].
a(n) = number of integers in range [2^(n-1),(2^n)-1] which permutation A209861/ A209862 sends to even-sized orbits.
+20
3
0, 0, 0, 0, 2, 0, 20, 56, 114, 248, 106, 1016, 1992, 4016, 6506, 7518, 24496, 50358, 121610, 261206, 483160
COMMENTS
a(0) gives the number of even-sized cycles in range [0,0], i.e. 0, as there is only one fixed point in that range.
EXAMPLE
In range [2^(6-1),(2^6)-1] ([32,63]) of permutations A209861 & A209862, there are 6 cycles of size 1 (six fixed points), 2 cycles of size 3, one cycle of size 4, and 2 cycles of size 8, i.e. 6*1 + 2*3 + 1*4 + 2*8 = 32 in total, of which 4 + 2*8 elements are in even-sized cycles, thus a(6)=20.
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 11, 13, 14, 15, 16, 17, 18, 21, 19, 22, 24, 27, 20, 23, 25, 28, 26, 29, 30, 31, 32, 33, 34, 38, 35, 39, 42, 48, 36, 40, 43, 49, 45, 51, 54, 58, 37, 41, 44, 50, 46, 52, 55, 59, 47, 53, 56, 60, 57, 61, 62, 63, 64, 65, 66, 71, 67, 72, 76, 86, 68, 73, 77, 87, 80, 90, 96, 106, 69, 74, 78, 88, 81, 91, 97, 107, 83
COMMENTS
Conjecture: For all n, A209861( A054429(n)) = A054429( A209861(n)), i.e. A054429 acts as an homomorphism (automorphism) of the cyclic group generated by this permutation. This implies also a weaker conjecture given in A209860. The scatterplot graph of the sequence has a nice texture and interesting pattern.
Irregular triangle of numbers that are squarefree and smooth (row n contains squarefree p-smooth numbers, where p is the n-th prime).
+10
13
1, 2, 1, 2, 3, 6, 1, 2, 3, 5, 6, 10, 15, 30, 1, 2, 3, 5, 6, 7, 10, 14, 15, 21, 30, 35, 42, 70, 105, 210, 1, 2, 3, 5, 6, 7, 10, 11, 14, 15, 21, 22, 30, 33, 35, 42, 55, 66, 70, 77, 105, 110, 154, 165, 210, 231, 330, 385, 462, 770, 1155, 2310, 1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 21, 22, 26, 30, 33, 35, 39, 42
COMMENTS
If we define a triangle whose n-th row consists of all squarefree numbers whose prime factors are all less than prime(k), we get this same triangle except starting with a row {1}, with offset 1. - Gus Wiseman, Aug 24 2021
EXAMPLE
Triangle begins:
1, 2; squarefree and 2-smooth
1, 2, 3, 6; squarefree and 3-smooth
1, 2, 3, 5, 6, 10, 15, 30;
1, 2, 3, 5, 6, 7, 10, 14, 15, 21, 30, 35, 42, 70, 105, 210;
...
MAPLE
b:= proc(n) option remember; `if`(n=0, [1],
sort(map(x-> [x, x*ithprime(n)][], b(n-1))))
end:
T:= n-> b(n)[]:
MATHEMATICA
primorial[n_] := Times @@ Prime[Range[n]]; row[n_] := Select[ Divisors[ primorial[n]], SquareFreeQ]; Table[row[n], {n, 1, 10}] // Flatten
CROSSREFS
Cf. A000079 (2-smooth), A003586 (3-smooth), A051037 (5-smooth), A002473 (7-smooth), A018336 (7-smooth & squarefree), A051038 (11-smooth), A087005 (11-smooth & squarefree), A080197 (13-smooth), A087006 (13-smooth & squarefree), A087007 (17-smooth & squarefree), A087008 (19-smooth & squarefree). Rightmost terms (or column k = 2^n) are A002110. Rows are partial unions of rows of A019565. A072047 counts prime factors of squarefree numbers.
Triangle of squarefree numbers grouped by greatest prime factor, read by rows.
+10
12
2, 3, 6, 5, 10, 15, 30, 7, 14, 21, 35, 42, 70, 105, 210, 11, 22, 33, 55, 66, 77, 110, 154, 165, 231, 330, 385, 462, 770, 1155, 2310, 13, 26, 39, 65, 78, 91, 130, 143, 182, 195, 273, 286, 390, 429, 455, 546, 715, 858, 910, 1001, 1365, 1430, 2002, 2145, 2730, 3003, 4290, 5005, 6006, 10010, 15015, 30030
COMMENTS
Also Heinz numbers of subsets of {1..n} that contain n, where the Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
A019565 in its triangle form, but omitting its initial row and with each row's terms in increasing order. - Peter Munn, Feb 26 2021
FORMULA
For n > 1, T(n,k) = prime(n) * A261144(n-1,k).
EXAMPLE
Triangle begins:
2
3 6
5 10 15 30
7 14 21 35 42 70 105 210
MATHEMATICA
Table[Prime[n]*Sort[Times@@Prime/@#&/@Subsets[Range[n-1]]], {n, 5}]
CROSSREFS
A000079 (shifted left) gives row lengths.
A209862 takes prime indices to binary indices in these terms.
A246867 groups squarefree numbers by Heinz weight, with row sums A147655.
A005117 lists squarefree numbers, ordered lexicographically by prime factors: A019565.
A006881 lists squarefree semiprimes.
A072047 counts prime factors of squarefree numbers.
A319246 is the sum of prime indices of the n-th squarefree number.
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