A209862 -id:A209862 - cp's OEIS frontend

A209863 a(n) = Number of fixed points of permutation A209861/A209862 in range [2^(n-1),(2^n)-1].

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

Offset: 0

Antti Karttunen, Mar 24 2012

nonn

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.

Antti Karttunen, Table of n, a(n) for n = 0..24

A209860, A209864, A209865, A209866, A209867.

A209864 a(n) = number of cycles in range [2^(n-1),(2^n)-1] of permutation A209861/A209862.

Original entry on oeis.org

1, 1, 2, 4, 7, 8, 11, 12, 14, 10, 21, 14, 20, 26, 22, 18, 18, 28, 23, 30, 32

Offset: 0

Antti Karttunen, Mar 24 2012

nonn

a(0) gives the number of cycles in range [0,0], i.e. 1.

Antti Karttunen, Listing showing a more detailed cycle-structure of permutations A209861/A209862 up to n=20.

Cf. A209863, A209865, A209866, A209867.

A209866 a(n) = least common multiple of all cycle sizes in range [2^(n-1),(2^n)-1] of permutation A209861/A209862.

Original entry on oeis.org

1, 1, 1, 1, 2, 5, 24, 26, 672, 246, 3755388, 13827240, 1768910220, 99034598880, 1463488641762840, 612823600, 171768365608799778, 16338317307187487976, 27491145139913884194480, 14794457633180140325810400, 2084886621890359572790082258379649440

Offset: 0

Antti Karttunen, Mar 24 2012

nonn

a(0) gives the LCM of cycle sizes in range [0,0], i.e., 1.

			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.

Antti Karttunen, Listing showing a more detailed cycle-structure of permutations A209861/A209862 up to n=20.

Cf. A209863, A209864, A209865, A209867.

A209865 a(n) = maximal cycle size in range [2^(n-1),(2^n)-1] of permutation A209861/A209862.

Original entry on oeis.org

1, 1, 1, 1, 2, 5, 8, 26, 96, 246, 181, 540, 868, 724, 6038, 4405, 23302, 39514, 34480, 83424, 270884

Offset: 0

Antti Karttunen, Mar 24 2012

nonn

a(0) gives the maximum cycle size in range [0,0], i.e. 1.

Cf. A209863, A209864, A209866, A209867.

A209867 a(n) = number of integers in range [2^(n-1),(2^n)-1] which permutation A209861/A209862 sends to odd-sized orbits.

Original entry on oeis.org

1, 1, 2, 4, 6, 16, 12, 8, 14, 8, 406, 8, 56, 80, 1686, 8866, 8272, 15178, 9462, 938, 41128

Offset: 0

Antti Karttunen, Mar 24 2012

nonn

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.

			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.

a(n) = A000079(n-1) - A209868(n) for all n>0. Cf. A209860, A209863, A209864, A209865, A209866.

A209860 Fixed points of permutation A209861/A209862.

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, Mar 24 2012

nonn

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).

Antti Karttunen, Table of n, a(n) for n = 0..171

A209863 gives the number of these fixed points in each range [2^(n-1),(2^n)-1].

Those i, for which A209861(i)=i, or equally A209862(i)=i.

A209868 a(n) = number of integers in range [2^(n-1),(2^n)-1] which permutation A209861/A209862 sends to even-sized orbits.

Original entry on oeis.org

0, 0, 0, 0, 2, 0, 20, 56, 114, 248, 106, 1016, 1992, 4016, 6506, 7518, 24496, 50358, 121610, 261206, 483160

Offset: 0

Antti Karttunen, Mar 24 2012

nonn

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.

			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.

a(n) = A000079(n-1) - A209867(n) for all n>0. Cf. A209860, A209863, A209864, A209865, A209866.

A339195 Triangle of squarefree numbers grouped by greatest prime factor, read by rows.

Original entry on oeis.org

1, 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

Offset: 0

Gus Wiseman, Dec 02 2020

Also Heinz numbers of subsets of {1..n} that contain n if n>0, where the Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
A019565 in its triangle form, with each row's terms in increasing order. - Peter Munn, Feb 26 2021
From David James Sycamore, Jan 09 2025: (Start)
Alternative definition, with offset = 1: a(1) = 1. For n>1 if a(n-1) = A002110(k), a(n) = prime(k+1). Otherwise a(n) is the smallest novel squarefree number whose prime factors have already occurred as previous terms.
Permutation of A005117, Squarefree version A379746. (End)

			Triangle begins:
   1
   2
   3   6
   5  10  15  30
   7  14  21  35  42  70  105  210

Alois P. Heinz, Rows n = 0..14, flattened
Michael De Vlieger, Plot p | a(n) at (x,y) = (n,pi(p)), n = 0..2047, 12X vertical exaggeration.
Michael De Vlieger, Fan style binary tree showing a(n), n = 0..2047, with a color function related to the order of a(n) in A019565.
Michael De Vlieger, Fan style binary tree showing a(n), n = 0..2047, with a color function showing 1 in gray, primes in red, primorials in bright green, even squarefree semiprimes in yellow, odd squarefree semiprimes in light green, thereafter, progressively deeper green related to omega(a(n)) = m until m >= 6.

A011782 gives row lengths.
A339360 gives row sums.
A008578 (shifted) is column k = 1.
A100484 is column k = 2.
A001748 is column k = 3.
A002110 is column k = 2^(n-1).
A070826 is column k = 2^(n-1) - 1.
A209862 takes prime indices to binary indices in these terms.
A246867 groups squarefree numbers by Heinz weight, with row sums A147655.
A261144 divides the n-th row by prime(n), with row sums A054640.
A339116 is the restriction to semiprimes, with row sums A339194.
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.
A329631 lists prime indices of squarefree numbers, reversed: A319247.
A338899/A270650/A270652 give the prime indices of squarefree semiprimes.
Cf. A001221, A014342, A014466, A019565, A098350, A112798, A320656, A326882, A338901, A379770.
Cf. A379746.

T:= proc(n) option remember; `if`(n=0, 1, (p-> map(
      x-> x*p, {seq(T(i), i=0..n-1)})[])(ithprime(n)))
    end:
seq(T(n), n=0..6);  # Alois P. Heinz, Jan 08 2025

Table[Prime[n]*Sort[Times@@Prime/@#&/@Subsets[Range[n-1]]],{n,5}]

For n > 1, T(n,k) = prime(n) * A261144(n-1,k).

a(n) = A019565(A379770(n)). - Michael De Vlieger, Jan 08 2025

Row n=0 (term 1) prepended by Alois P. Heinz, Jan 08 2025

A209861 Permutation of nonnegative integers which maps A209641 to A209642.

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, Mar 24 2012

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.

Antti Karttunen, Table of n, a(n) for n = 0..32767
Index entries for sequences that are permutations of the natural numbers

Inverse permutation: A209862. Cf. A209860, A209863, A209864, A209865, A209866, A209867, A209868.

a(n) = A209640(A209642(n)).

A261144 Irregular triangle of numbers that are squarefree and smooth (row n contains squarefree p-smooth numbers, where p is the n-th prime).

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Nov 26 2015

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

			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;
...

Jean-François Alcover, Table of n, a(n) for n = 1..2046 (first 10 rows)
A. Hildebrand and G. Tenenbaum, Integers without large prime factors, Journal de théorie des nombres de Bordeaux (1993) Volume:5, Issue:2, p. 411-484.
Eric Weisstein's MathWorld, Smooth number.
Wikipedia, Smooth number

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).
Row lengths are A000079.
Rightmost terms (or column k = 2^n) are A002110.
Rows are partial unions of rows of A019565.
Row n is A027750(A002110(n)), i.e., divisors of primorials.
Row sums are A054640.
Column k = 2^n-1 is A070826.
Multiplying row n by prime(n+1) gives A339195, row sums A339360.
A005117 lists squarefree numbers.
A056239 adds up prime indices, row sums of A112798.
A072047 counts prime factors of squarefree numbers.
A246867 groups squarefree numbers by Heinz weight, row sums A147655.
A329631 lists prime indices of squarefree numbers, sums A319246.
A339116 groups squarefree semiprimes by greater factor, sums A339194.
Cf. A000040, A001221, A006881, A071403, A209862, A319247.

b:= proc(n) option remember; `if`(n=0, [1],
      sort(map(x-> [x, x*ithprime(n)][], b(n-1))))
    end:
T:= n-> b(n)[]:
seq(T(n), n=1..7);  # Alois P. Heinz, Nov 28 2015

primorial[n_] := Times @@ Prime[Range[n]]; row[n_] := Select[ Divisors[ primorial[n]], SquareFreeQ]; Table[row[n], {n, 1, 10}] // Flatten

T(n-1,k) = A339195(n,k)/prime(n). - Gus Wiseman, Aug 24 2021

cp's OEIS Frontend

A209863 a(n) = Number of fixed points of permutation A209861/A209862 in range [2^(n-1),(2^n)-1].

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A209864 a(n) = number of cycles in range [2^(n-1),(2^n)-1] of permutation A209861/A209862.

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A209866 a(n) = least common multiple of all cycle sizes in range [2^(n-1),(2^n)-1] of permutation A209861/A209862.

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A209865 a(n) = maximal cycle size in range [2^(n-1),(2^n)-1] of permutation A209861/A209862.

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A209867 a(n) = number of integers in range [2^(n-1),(2^n)-1] which permutation A209861/A209862 sends to odd-sized orbits.

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A209860 Fixed points of permutation A209861/A209862.

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A209868 a(n) = number of integers in range [2^(n-1),(2^n)-1] which permutation A209861/A209862 sends to even-sized orbits.

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A339195 Triangle of squarefree numbers grouped by greatest prime factor, read by rows.

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A209861 Permutation of nonnegative integers which maps A209641 to A209642.

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A261144 Irregular triangle of numbers that are squarefree and smooth (row n contains squarefree p-smooth numbers, where p is the n-th prime).

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