A269388 -id:A269388 - cp's OEIS frontend

A252754 Inverse of "Tree of Eratosthenes" permutation: a(1) = 0, after which, a(2n) = 1 + 2a(n), a(2n+1) = 2 a(A268674(2n+1)).

0, 1, 2, 3, 4, 5, 8, 7, 6, 9, 16, 11, 32, 17, 10, 15, 64, 13, 128, 19, 14, 33, 256, 23, 12, 65, 18, 35, 512, 21, 1024, 31, 22, 129, 20, 27, 2048, 257, 34, 39, 4096, 29, 8192, 67, 30, 513, 16384, 47, 24, 25, 26, 131, 32768, 37, 28, 71, 38, 1025, 65536, 43, 131072, 2049, 66, 63, 36, 45, 262144, 259, 46, 41, 524288, 55, 1048576, 4097, 130, 515, 40

Offset: 1

Antti Karttunen, Jan 02 2015

nonn

Antti Karttunen, Table of n, a(n) for n = 1..1024
Index entries for sequences that are permutations of the natural numbers

Inverse: A252753.
Fixed points of a(n)+1: A253789.
Cf. A250470, A253556, A253557, A253558, A253559, A268674, A302041.
Similar permutations: A156552, A252756, A054429, A250246, A269388.
Differs from A156552 for the first time at n=21, where a(21) = 14, while A156552(21) = 18.

A252754(n) = if(1==n,0,if(!(n%2),1 + 2*A252754(n/2),2*A252754(A268674(n)))); \\ Antti Karttunen, Mar 31 2018

;; With memoization-macro definec.
(definec (A252754 n) (cond ((= 1 n) (- n 1)) ((even? n) (+ 1 (* 2 (A252754 (/ n 2))))) (else (* 2 (A252754 (A250470 n))))))

a(1) = 0, after which, a(2n) = 1 + 2*a(n), a(2n+1) = 2 * a(A268674(2n+1)).
As a composition of related permutations:
a(n) = A054429(A252756(n)).
a(n) = A156552(A250246(n)).
From Antti Karttunen, Mar 31 2018: (Start)
A000120(a(n)) = A253557(n).
A069010(a(n)) = A302041(n).
A132971(a(n)) = A302050(n).
A106737(a(n)) = A302051(n).
(End)

Name edited and formula corrected by Antti Karttunen, Mar 31 2018

A269380 a(1) = 1, after which, for odd numbers: a(n) = A260739(n)-th number k for which A260738(k) = A260738(n)-1, and for even numbers: a(n) = a(n/2).

Original entry on oeis.org

1, 1, 2, 1, 3, 2, 5, 1, 4, 3, 7, 2, 11, 5, 6, 1, 13, 4, 9, 3, 8, 7, 17, 2, 23, 11, 10, 5, 25, 6, 19, 1, 12, 13, 15, 4, 29, 9, 14, 3, 37, 8, 41, 7, 16, 17, 43, 2, 21, 23, 18, 11, 47, 10, 31, 5, 20, 25, 35, 6, 53, 19, 22, 1, 27, 12, 61, 13, 24, 15, 67, 4, 55, 29, 26, 9, 71, 14, 33, 3, 28, 37, 77, 8, 49, 41, 30, 7, 83, 16, 89, 17, 32, 43, 39, 2

Offset: 1

Antti Karttunen, Mar 01 2016

nonn

For odd numbers n > 1, a(n) tells which term is on the immediately preceding row of A255127 (square array generated by Ludic sieve), in the same column where n itself is.

Antti Karttunen, Table of n, a(n) for n = 1..33011

Cf. A255127, A260738, A260739.
Cf. A269172, A269355, A269357, A269382, A269386, A269388 (sequences that use this function).
Cf. also A268674, A269370.

a(1) = 1; after which, for even numbers a(n) = a(n/2), and for odd numbers a(n) = A255127(A260738(n)-1, A260739(n)).
Other identities. For all n >= 1:
a(A269379(n)) = n.

A269172 Permutation of natural numbers: a(1) = 1, a(2n) = 2*a(n), a(2n+1) = A250469(a(A269380(2n+1))).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 25, 20, 21, 22, 19, 24, 23, 26, 27, 28, 29, 30, 49, 32, 33, 34, 35, 36, 31, 50, 39, 40, 37, 42, 41, 44, 45, 38, 43, 48, 55, 46, 51, 52, 47, 54, 121, 56, 57, 58, 77, 60, 53, 98, 63, 64, 65, 66, 59, 68, 69, 70, 61, 72, 169, 62, 75, 100, 67, 78, 85, 80, 81

Offset: 1

Antti Karttunen, Mar 03 2016

nonn

Antti Karttunen, Table of n, a(n) for n = 1..2244
Index entries for sequences that are permutations of the natural numbers

Inverse: A269171.
Cf. A000035, A003309, A008578, A083221, A250469, A260738, A260739, A269380.
Related or similar permutations: A260741, A260742, A269356, A269358, A255422.
Cf. also A269394 (a(3n)/3) and A269396.
Differs from A255408 for the first time at n=38, where a(38) = 50, while A255408(38) = 38.

a(1) = 1, then after for even n, a(n) = 2*a(n/2), and for odd n, A250469(a(A269380(n))).
a(1) = 1, for n > 1, a(n) = A083221(A260738(n), a(A260739(n))).
As a composition of other permutations:
a(n) = A252755(A269386(n)).
a(n) = A252753(A269388(n)).
Other identities. For all n >= 1:
A000035(a(n)) = A000035(n). [This permutation preserves the parity of n.]
a(A003309(n)) = A008578(n). [Maps Ludic numbers to noncomposites.]

A302026 Permutation of natural numbers mapping "Ludic factorization" to ordinary factorization: a(1) = 1, a(2n) = 2*a(n), a(A269379(n)) = A003961(a(n)).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 25, 20, 27, 22, 19, 24, 23, 26, 21, 28, 29, 30, 49, 32, 45, 34, 35, 36, 31, 50, 33, 40, 37, 54, 41, 44, 81, 38, 43, 48, 125, 46, 75, 52, 47, 42, 121, 56, 63, 58, 77, 60, 53, 98, 39, 64, 55, 90, 59, 68, 135, 70, 61, 72, 169, 62, 51, 100, 67, 66, 175, 80, 99, 74, 71, 108, 343, 82, 105, 88

Offset: 1

Antti Karttunen, Apr 03 2018

nonn

See comments and examples in A302032 to see how Ludic factorization proceeds.

Cf. A302025 (inverse permutation).
Cf. A005940, A250246, A269172, A269388 (similar or related permutations).
Cf. A003961, A064989, A269379, A269380, A302031, A302032, A302034, A302037.

\\ With A269380 precomputed:
A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ From A003961
A302026(n) = if(1==n,n,if(!(n%2),2*A302026(n/2),A003961(A302026(A269380(n)))));
(Scheme, with memoization-macro definec)
(definec (A302026 n) (cond ((= 1 n) n) ((even? n) (* 2 (A302026 (/ n 2)))) (else (A003961 (A302026 (A269380 n))))))

a(1) = 1, a(2n) = 2*a(n), a(2n+1) = A003961(a(A269380(2n+1))).
a(n) = A250246(A269172(n)).
a(n) = A005940(1+A269388(n)).
Other identities. For all n >= 1:
A001221(a(n)) = A302031(n).
A001222(a(n)) = A302037(n).

A269387 Tree of Ludic sieve: a(0) = 1, a(1) = 2; after which, a(2n) = A269379(a(n)), a(2n+1) = 2*a(n).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 9, 8, 7, 10, 15, 12, 19, 18, 21, 16, 11, 14, 27, 20, 35, 30, 33, 24, 31, 38, 51, 36, 49, 42, 45, 32, 13, 22, 39, 28, 65, 54, 57, 40, 59, 70, 87, 60, 79, 66, 69, 48, 55, 62, 111, 76, 125, 102, 105, 72, 85, 98, 123, 84, 109, 90, 93, 64, 17, 26, 63, 44, 95, 78, 81, 56, 113, 130, 159, 108, 139, 114, 117, 80

Offset: 0

Antti Karttunen, Mar 01 2016

Permutation of natural numbers obtained from the Ludic sieve. Note the indexing: Domain starts from 0, range from 1.
This sequence can be represented as a binary tree. Each left hand child is obtained by applying A269379 to the parent's contents, and each right hand child is obtained by doubling the parent's contents:
                                    1
                                    |
                 ...................2...................
                3                                       4
      5......../ \........6                   9......../ \........8
     / \                 / \                 / \                 / \
    /   \               /   \               /   \               /   \
   /     \             /     \             /     \             /     \
  7       10         15       12         19       18         21       16
11 14   27  20     35  30   33  24     31  38   51  36     49  42   45  32
etc.
Sequence A269385 is obtained from the mirror image of the same tree.

Antti Karttunen, Table of n, a(n) for n = 0..8191
Index entries for sequences that are permutations of the natural numbers

Inverse: A269388.
Cf. A003309 (left edge of the tree).
Cf. A269379.
Related permutations: A260741, A269171, A269385.
Cf. also A252753, A269377.

a(0) = 1, a(1) = 2; after which, a(2n) = A269379(a(n)), a(2n+1) = 2*a(n).
As a composition of other permutations:
a(n) = A269171(A252753(n)).
a(n) = A260741(A269377(n)).

A269386 Permutation of nonnegative integers: a(1) = 0, a(2) = 1, a(2n) = 2a(n), a(2n+1) = 1 + 2a(A269380(2n+1)).

Original entry on oeis.org

0, 1, 3, 2, 7, 6, 15, 4, 5, 14, 31, 12, 63, 30, 13, 8, 127, 10, 11, 28, 9, 62, 255, 24, 511, 126, 29, 60, 1023, 26, 23, 16, 25, 254, 27, 20, 2047, 22, 61, 56, 4095, 18, 8191, 124, 17, 510, 16383, 48, 19, 1022, 21, 252, 32767, 58, 47, 120, 57, 2046, 55, 52, 65535, 46, 125, 32, 59, 50, 131071, 508, 49, 54, 262143, 40, 95, 4094, 253, 44

Offset: 1

Antti Karttunen, Mar 01 2016

nonn

Note the indexing: Domain starts from 1, range from 0.

Antti Karttunen, Table of n, a(n) for n = 1..1024
Index entries for sequences that are permutations of the natural numbers

Inverse: A269385.
Cf. A269380.
Related permutations: A260742, A269172, A269388.
Cf. also A252756, A269376.
Differs from A243071 and A252756 for the first time at n=19, which here a(19) = 11, instead of 255.

a(1) = 0, a(2) = 1, a(2n) = 2*a(n), a(2n+1) = 1 + 2*a(A269380(2n+1)).
As a composition of related permutations:
a(n) = A252756(A269172(n)).
a(n) = A269376(A260742(n)).

A302031 An omega (A001221) analog based on the Ludic sieve (A255127): a(1) = 0; for n > 1, a(n) = 1 + a(A302034(n)).

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 2, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 2, 2, 1, 3, 1, 1, 2, 2, 2, 2, 1, 2, 2, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 2, 2, 1, 3, 1, 2, 2, 2, 2, 3, 1, 2, 2, 1, 2, 3, 1, 2, 2, 3, 1, 2, 1, 2, 2, 2, 1, 3, 2, 2, 2, 2, 1, 2, 1, 2, 3, 2, 1, 2, 1, 2, 1, 2, 2, 2, 1, 2, 2, 2, 1, 3, 2, 2, 2

Offset: 1

Antti Karttunen, Apr 02 2018

nonn

Antti Karttunen, Table of n, a(n) for n = 1..32768
Index entries for sequences generated by sieves

Cf. A001221, A302026, A302037, A302041,
Cf. A302036 (positions of terms < 2).
Differs from similar A302041 for the first time at n=59, where a(59) = 2, while A302041(59) = 1.

\\ Assuming that A269379 and A269380 have been precomputed:
A302034(n) = if(1==n,n,my(k=0); while((n%2), n = A269380(n); k++); n = (n/2^valuation(n, 2)); while(k>0, n = A269379(n); k--); (n));
A302031(n) = if(1==n,0,1+A302031(A302034(n)));

a(1) = 0; for n > 1, a(n) = 1 + a(A302034(n)).
a(n) = A001221(A302026(n)).
a(n) = A069010(A269388(n)).

A302037 A bigomega (A001222) analog based on the Ludic sieve (A255127): a(1) = 0; for n > 1, a(n) = 1 + a(A302032(n)).

Original entry on oeis.org

0, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 3, 1, 2, 2, 4, 1, 3, 2, 3, 3, 2, 1, 4, 1, 2, 2, 3, 1, 3, 2, 5, 3, 2, 2, 4, 1, 3, 2, 4, 1, 4, 1, 3, 4, 2, 1, 5, 3, 2, 3, 3, 1, 3, 2, 4, 3, 2, 2, 4, 1, 3, 2, 6, 2, 4, 1, 3, 4, 3, 1, 5, 2, 2, 2, 4, 1, 3, 3, 5, 3, 2, 1, 5, 3, 2, 3, 4, 1, 5, 1, 3, 5, 2, 2, 6, 1, 4, 2, 3, 2, 4, 2, 4, 4

Offset: 1

Antti Karttunen, Apr 01 2018

nonn

Antti Karttunen, Table of n, a(n) for n = 1..32768
Index entries for sequences generated by sieves

Cf. A302026, A302032.
Cf. A003309 (gives the positions of terms <= 1), A302038 (gives the positions of 2's).
Cf. A302031 (an omega-analog), A253557.

\\ Assuming that A269379 and A269380 have been precomputed:
A302032(n) = if(1==n,n,my(k=0); while((n%2), n = A269380(n); k++); n = n/2; while(k>0, n = A269379(n); k--); (n));
A302037(n) = if(1==n,0,1+A302037(A302032(n)));

a(1) = 0; for n > 1, a(n) = 1 + a(A302032(n)).
a(n) = A000120(A269388(n)).
a(n) = A001222(A302026(n)).

A269378 Permutation of natural numbers: a(1) = 0, after which a(2n) = 1 + 2a(n), a(2n+1) = 2 a(A269370(n)).

Original entry on oeis.org

0, 1, 2, 3, 6, 5, 4, 7, 8, 13, 10, 11, 16, 9, 32, 15, 14, 17, 12, 27, 64, 21, 26, 23, 128, 33, 24, 19, 22, 65, 256, 31, 512, 29, 18, 35, 1024, 25, 20, 55, 30, 129, 2048, 43, 48, 53, 34, 47, 4096, 257, 8192, 67, 54, 49, 96, 39, 40, 45, 42, 131, 28, 513, 16384, 63, 46, 1025, 32768, 59, 65536, 37, 66, 71, 131072, 2049, 262144, 51, 38, 41

Offset: 1

Antti Karttunen, Mar 01 2016

nonn

Note the indexing: Domain starts from 1, range from 0.

Antti Karttunen, Table of n, a(n) for n = 1..1024
Index entries for sequences that are permutations of the natural numbers

Inverse: A269377.
Cf. A269370.
Related permutation: A269376.
Cf. also A252754, A269388.

a(1) = 0, after which, a(2n) = 1 + 2*a(n), a(2n+1) = 2 * a(A269370(n)).
As a composition of related permutations:
a(n) = A269388(A260741(n)).

cp's OEIS Frontend

A252754 Inverse of "Tree of Eratosthenes" permutation: a(1) = 0, after which, a(2n) = 1 + 2*a(n), a(2n+1) = 2 * a(A268674(2n+1)).

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A269380 a(1) = 1, after which, for odd numbers: a(n) = A260739(n)-th number k for which A260738(k) = A260738(n)-1, and for even numbers: a(n) = a(n/2).

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A269172 Permutation of natural numbers: a(1) = 1, a(2n) = 2*a(n), a(2n+1) = A250469(a(A269380(2n+1))).

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A302026 Permutation of natural numbers mapping "Ludic factorization" to ordinary factorization: a(1) = 1, a(2n) = 2*a(n), a(A269379(n)) = A003961(a(n)).

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A269387 Tree of Ludic sieve: a(0) = 1, a(1) = 2; after which, a(2n) = A269379(a(n)), a(2n+1) = 2*a(n).

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A269386 Permutation of nonnegative integers: a(1) = 0, a(2) = 1, a(2n) = 2*a(n), a(2n+1) = 1 + 2*a(A269380(2n+1)).

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A302031 An omega (A001221) analog based on the Ludic sieve (A255127): a(1) = 0; for n > 1, a(n) = 1 + a(A302034(n)).

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A302037 A bigomega (A001222) analog based on the Ludic sieve (A255127): a(1) = 0; for n > 1, a(n) = 1 + a(A302032(n)).

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A269378 Permutation of natural numbers: a(1) = 0, after which a(2n) = 1 + 2*a(n), a(2n+1) = 2 * a(A269370(n)).

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A252754 Inverse of "Tree of Eratosthenes" permutation: a(1) = 0, after which, a(2n) = 1 + 2a(n), a(2n+1) = 2 a(A268674(2n+1)).

A269386 Permutation of nonnegative integers: a(1) = 0, a(2) = 1, a(2n) = 2a(n), a(2n+1) = 1 + 2a(A269380(2n+1)).

A269378 Permutation of natural numbers: a(1) = 0, after which a(2n) = 1 + 2a(n), a(2n+1) = 2 a(A269370(n)).