A273667 -id:A273667 - cp's OEIS frontend

A275837 Permutation of nonnegative integers: a(n) = A225901(A273667(n)).

0, 1, 2, 4, 18, 5, 6, 22, 12, 3, 96, 20, 72, 17, 14, 19, 114, 13, 94, 10, 11, 23, 108, 15, 24, 79, 84, 100, 101, 21, 48, 102, 8, 9, 600, 71, 480, 49, 78, 118, 119, 16, 65, 73, 98, 99, 696, 27, 360, 594, 62, 63, 70, 112, 54, 113, 74, 7, 45, 95, 552, 116, 50, 117, 672, 603, 454, 569, 37, 40, 29, 106, 444, 41, 36, 107, 85, 97, 38, 621, 86, 82, 545, 110, 528, 59, 56, 111

Offset: 0

Antti Karttunen, Aug 13 2016

Inverse: A275838.
Cf. A225901, A273667.
Cf. A275839 (fixed points).
Cf. also A275835, A275836.

(define (A275837 n) (A225901 (A273667 n)))

a(n) = A225901(A273667(n)).
Other identities. For all n >= 0:
a(n) = A266193(a(A153880(n))). [Restriction to A153880 induces the same permutation.]
A275841(n) = A273663(a(A273670(n))). [While restriction to A273670 induces another permutation.]

A275835 Permutation of nonnegative integers: a(n) = A273667(A225901(n)).

Original entry on oeis.org

0, 1, 6, 3, 4, 2, 56, 20, 36, 17, 21, 9, 48, 15, 30, 13, 16, 7, 18, 8, 24, 10, 12, 5, 495, 135, 74, 31, 132, 53, 582, 401, 147, 59, 157, 158, 361, 116, 216, 173, 117, 47, 136, 155, 380, 46, 78, 82, 420, 111, 61, 25, 108, 45, 490, 347, 123, 51, 133, 134, 312, 93, 192, 149, 94, 41, 112, 131, 327, 40, 64, 68, 360, 270, 80, 38, 88, 89, 416, 303, 99, 44, 109, 110, 288, 34

Offset: 0

Antti Karttunen, Aug 13 2016

Inverse: A275836.
Cf. A225901, A255411, A257684, A273667.
Cf. also A275837, A275838.

(define (A275835 n) (A273667 (A225901 n)))

a(n) = A273667(A225901(n)).
Other identities. For all n >= 0:
a(n) = A257684(a(A255411(n))). [Restriction to A255411 induces the same permutation.]

A273668 Permutation of nonnegative integers: a(0) = 0, a(A255411(n)) = A153880(a(n)), a(A256450(n)) = A273670(a(n)).

Original entry on oeis.org

0, 1, 3, 5, 2, 9, 4, 15, 7, 21, 11, 29, 8, 17, 41, 13, 14, 23, 6, 57, 19, 20, 32, 33, 10, 77, 27, 28, 44, 45, 16, 101, 39, 40, 61, 63, 22, 129, 53, 55, 83, 87, 31, 165, 71, 75, 107, 111, 12, 43, 213, 95, 56, 99, 137, 141, 18, 59, 269, 119, 26, 76, 125, 177, 80, 183, 38, 25, 81, 341, 134, 153, 30, 37, 100, 161, 62, 225, 104, 231, 52, 35

Offset: 0

Antti Karttunen, May 30 2016

Inverse: A273667.
Cf. A153880, A255411, A256450, A257680, A257684, A273662, A273670.
Similar or related permutations: A255565, A273666.

a(0) = 0; for n >= 1: if A257680(n) = 0 [when n is one of the terms of A255411] then a(n) = A153880(a(A257684(n))), otherwise [when n is one of the terms of A256450], a(n) = A273670(a(A273662(n))).
As a composition of other permutations:
a(n) = A273666(A255565(n)).

A275847 Permutation of natural numbers: a(0) = 0, a(A153880(n)) = A255411(a(n)), a(A273670(n)) = A256450(n).

Original entry on oeis.org

0, 1, 4, 2, 3, 5, 18, 6, 12, 7, 8, 9, 16, 10, 22, 11, 13, 14, 15, 17, 19, 20, 21, 23, 96, 24, 48, 25, 26, 27, 72, 28, 52, 29, 30, 31, 60, 32, 64, 33, 34, 35, 36, 37, 38, 39, 40, 41, 90, 42, 66, 43, 44, 45, 114, 46, 70, 47, 49, 50, 76, 51, 84, 53, 54, 55, 56, 57, 58, 59, 61, 62, 88, 63, 94, 65, 67, 68, 100, 69, 108, 71, 73, 74, 112, 75, 118, 77, 78

Offset: 0

Antti Karttunen, Aug 13 2016

Inverse: A275848.
Cf. A153880, A225901, A255411, A256450, A257680, A266193, A273663, A273670.
Similar permutations: A273667 (a more recursed variant), A275845, A275846.

a(0) = 0; for n >= 1: if A257680(A225901(n)) = 0 [when n is one of the terms of A153880] then a(n) = A255411(a(A266193(n))), otherwise [when n is one of the terms of A273670], a(n) = A256450(A273663(n)).

A276949 Index of row where n is located in array A276953 (equally: column in A276955).

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, Sep 22 2016

This is the smallest difference that occurs between any nonzero digit's radix (which is one more than its one-based position from the right) and that digit's value in the factorial base representation of n. See A225901 and the example.

a(0) = 0 by convention, as there are no nonzero digits present, and neither does 0 occur in arrays A276953 & A276955.

			For n=8, its factorial base representation (A007623) is "110", where the radix for each digit position 1, 2, 3 (from the right) is 2, 3, 4 (one larger than the position). For the 1 in the middle position the difference is 3-1 = 2, while for the 1 at the left we obtain 4-1 = 3. Of these two differences 2 is smaller, thus a(8)=2.

Antti Karttunen, Table of n, a(n) for n = 0..5040
Index entries for sequences related to factorial base representation

Cf. A007623, A257679, A266193, A276950.
Cf. A276951 (for the other index).
Cf. arrays A276953 & A276955.
Cf. also A225901, A273667, A275847.

a(0) = 0, and for n >= 1: if A276950(n) = 1, then a(n) = 1, otherwise a(n) = 1 + a(A266193(n)).
Other identities. For all n >= 0:
a(n) = A257679(A225901(n)) = A257679(A275847(n)) = A257679(A273667(n)).

A255566 a(0) = 0; after which, a(2n) = A255411(a(n)), a(2n+1) = A256450(a(n)).

Original entry on oeis.org

0, 1, 4, 2, 18, 6, 12, 3, 96, 24, 48, 8, 72, 15, 16, 5, 600, 120, 240, 30, 360, 56, 60, 10, 480, 87, 88, 20, 90, 21, 22, 7, 4320, 720, 1440, 144, 2160, 270, 288, 36, 2880, 416, 420, 67, 432, 73, 66, 13, 3600, 567, 568, 107, 570, 109, 108, 26, 576, 111, 112, 27, 114, 28, 52, 9, 35280, 5040, 10080, 840, 15120, 1584, 1680, 168

Offset: 0

Antti Karttunen, May 05 2015

This sequence can be represented as a binary tree. Each left hand child is produced as A255411(n), and each right hand child as A256450(n), when parent contains n >= 1:
                                      0
                                      |
                   ...................1...................
                  4                                       2
       18......../ \........6                  12......../ \........3
       / \                 / \                 / \                 / \
      /   \               /   \               /   \               /   \
     /     \             /     \             /     \             /     \
   96       24         48       8          72       15         16       5
600  120 240  30    360  56   60 10     480  87   88  20     90  21   22 7
etc.
Because all terms of A255411 are even it means that odd terms can occur only in odd positions (together with some even terms, for each one of which there is a separate infinite cycle), while terms in even positions are all even.
After its initial 1, A255567 seems to give all the terms like 2, 3, 12, ... where the left hand child of the right hand child is one more than the right hand child of the left hand child (as for 2: 16 = 15+1, as for 3: 22 = 21+1, as for 12: 88 = 87+1).

Inverse: A255565.
Cf. A000142, A001511, A001563, A083318, A255411, A256450, A257679.
Cf. also A255567 and arrays A257503, A257505.
Related or similar permutations: A273666, A273667.

a(0) = 0; after which, a(2n) = A255411(a(n)), a(2n+1) = A256450(a(n)).
Other identities:
For all n >= 0, a(2^n) = A001563(n+1). [The leftmost branch of the binary tree is given by n*n!]
For all n >= 0, a(A083318(n)) = A000142(n+1). [And the next innermost vertices by (n+1)! This follows because A256450(n*n! - 1) = (n+1)! - 1.]
For all n >= 1, A257679(a(n)) = A001511(n).

Formula changed because of the changed starting offset of A256450 - Antti Karttunen, May 30 2016

A275845 Permutation of natural numbers: a(0) = 0, a(A153880(n)) = A255411(n), a(A273670(n)) = A256450(a(n)).

Original entry on oeis.org

0, 1, 4, 2, 6, 3, 12, 8, 16, 5, 15, 10, 18, 21, 22, 7, 20, 13, 24, 27, 28, 9, 26, 17, 48, 30, 52, 33, 34, 11, 60, 32, 64, 23, 56, 36, 66, 61, 70, 39, 40, 14, 73, 38, 78, 29, 67, 42, 72, 80, 76, 74, 85, 45, 84, 46, 88, 19, 89, 44, 90, 97, 94, 35, 81, 49, 87, 99, 93, 91, 105, 53, 96, 104, 100, 54, 109, 25, 108, 110, 112, 51, 111, 121, 114, 117, 118, 41

Offset: 0

Antti Karttunen, Aug 13 2016

Inverse: A275846.
Cf. A000142, A052849, A153880, A225901, A255411, A256450, A257680, A266193, A273663, A273670.
Similar permutations: A273667 (a more recursed variant), A275847, A275848.

a(0) = 0; for n >= 1: if A257680(A225901(n)) = 0 [when n is one of the terms of A153880] then a(n) = A255411(A266193(n)), otherwise [when n is one of the terms of A273670], a(n) = A256450(a(A273663(n))).
Other identities:
a(A000142(n)) = A052849(n) for all n >= 2.

A273665 a(0) = 0; for n >= 1: if n = A153880(k) for some k, then a(n) = 2a(k), otherwise, n = A273670(h) for some h, and a(n) = 1 + 2a(h).

Original entry on oeis.org

0, 1, 2, 3, 5, 7, 4, 11, 6, 15, 9, 23, 10, 13, 14, 31, 19, 47, 21, 27, 29, 63, 39, 95, 8, 43, 22, 55, 59, 127, 12, 79, 30, 191, 17, 87, 18, 45, 46, 111, 119, 255, 25, 159, 61, 383, 35, 175, 20, 37, 26, 91, 93, 223, 28, 239, 62, 511, 51, 319, 38, 123, 94, 767, 71, 351, 41, 75, 53, 183, 187, 447, 42, 57, 54, 479, 125, 1023, 58

Offset: 0

Antti Karttunen, May 30 2016

Inverse: A273666.
Cf. A153880, A225901, A257680, A266193, A273663, A273670.
Related or similar permutations: A255565, A273667.

a(0) = 0; for n >= 1: if A257680(A225901(n)) = 0 [i.e., n is one of the terms of A153880], then a(n) = 2*a(A266193(n)), otherwise [when n is one of the terms of A273670], a(n) = 1 + 2*a(A273663(n)).

A275839 Fixed points of permutations A275837 & A275838.

Original entry on oeis.org

0, 1, 2, 5, 6, 14, 24, 54, 120, 145, 264, 411, 464, 720, 842, 1560, 2042, 2408, 2670, 5040, 5766, 10800, 13686, 16590, 18144, 40320, 45384, 85680, 105864, 106153, 131184, 141960, 145728

Offset: 0

Antti Karttunen, Aug 13 2016

Numbers n for which A225901(n) = A273667(n).

If n is a member, then A153880(n) is also a member, thus sequence is infinite and all factorial numbers (A000142) are present.

Index entries for sequences related to factorial base representation

Cf. A000142 (a subsequence).

Cf. A153880, A225901, A273667, A275837, A275838.

A276957 Permutation of natural numbers: a(A153880(n)) = A255411(n), a(A273670(n)) = A256450(n).

Original entry on oeis.org

1, 4, 2, 3, 5, 12, 6, 16, 7, 8, 9, 18, 10, 22, 11, 13, 14, 15, 17, 19, 20, 21, 23, 48, 24, 52, 25, 26, 27, 60, 28, 64, 29, 30, 31, 66, 32, 70, 33, 34, 35, 36, 37, 38, 39, 40, 41, 72, 42, 76, 43, 44, 45, 84, 46, 88, 47, 49, 50, 90, 51, 94, 53, 54, 55, 56, 57, 58, 59, 61, 62, 96, 63, 100, 65, 67, 68, 108, 69, 112, 71, 73

Offset: 1

Antti Karttunen, Sep 22 2016

Inverse: A276958.
Cf. A276950, A255411, A256450, A266193, A273663.
For more recursed variants see: A275845, A275847 & A273667.

(define (A276957 n) (cond ((zero? n) n) ((zero? (A276950 n)) (A255411 (A266193 n))) (else (A256450 (A273663 n)))))

If A276950(n) = 0, then a(n) = A255411(A266193(n)), otherwise a(n) = A256450(A273663(n)).

cp's OEIS Frontend

A275837 Permutation of nonnegative integers: a(n) = A225901(A273667(n)).

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A275835 Permutation of nonnegative integers: a(n) = A273667(A225901(n)).

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A273668 Permutation of nonnegative integers: a(0) = 0, a(A255411(n)) = A153880(a(n)), a(A256450(n)) = A273670(a(n)).

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A275847 Permutation of natural numbers: a(0) = 0, a(A153880(n)) = A255411(a(n)), a(A273670(n)) = A256450(n).

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A276949 Index of row where n is located in array A276953 (equally: column in A276955).

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A255566 a(0) = 0; after which, a(2n) = A255411(a(n)), a(2n+1) = A256450(a(n)).

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A275845 Permutation of natural numbers: a(0) = 0, a(A153880(n)) = A255411(n), a(A273670(n)) = A256450(a(n)).

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A273665 a(0) = 0; for n >= 1: if n = A153880(k) for some k, then a(n) = 2*a(k), otherwise, n = A273670(h) for some h, and a(n) = 1 + 2*a(h).

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A275839 Fixed points of permutations A275837 & A275838.

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A276957 Permutation of natural numbers: a(A153880(n)) = A255411(n), a(A273670(n)) = A256450(n).

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A273665 a(0) = 0; for n >= 1: if n = A153880(k) for some k, then a(n) = 2a(k), otherwise, n = A273670(h) for some h, and a(n) = 1 + 2a(h).