cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-7 of 7 results.

A249814 "Mountains of Eratosthenes" permutation: a(1) = 1, a(n) = A249741(A001511(n), a(A003602(n))).

Original entry on oeis.org

1, 2, 3, 4, 5, 8, 7, 6, 9, 14, 15, 24, 13, 20, 11, 10, 17, 26, 27, 34, 29, 44, 47, 48, 25, 38, 39, 54, 21, 32, 19, 12, 33, 50, 51, 64, 53, 80, 67, 76, 57, 86, 87, 114, 93, 140, 95, 120, 49, 74, 75, 94, 77, 116, 107, 90, 41, 62, 63, 84, 37, 56, 23, 16, 65, 98, 99, 124, 101, 152, 127, 118, 105, 158, 159, 204, 133, 200, 151, 142
Offset: 1

Views

Author

Antti Karttunen, Nov 06 2014

Keywords

Comments

This sequence is a "recursed variant" of A249811.
From Antti Karttunen, Jan 18 2015: (Start)
This can be viewed as an entanglement or encoding permutation where the complementary pairs of sequences to be interwoven together are even and odd numbers (A005843/A005408) which are entangled with another complementary pair: even numbers in the order they appear in A253886 and odd numbers in their usual order: (A253886/A005408).
From the above follows also that this sequence can be represented as a binary tree. Each child to the left is obtained by doubling the parent and subtracting one, and each child to the right is obtained by applying A253886 to the parent:
1
|
...................2...................
3 4
5......../ \........8 7......../ \........6
/ \ / \ / \ / \
/ \ / \ / \ / \
/ \ / \ / \ / \
9 14 15 24 13 20 11 10
17 26 27 34 29 44 47 48 25 38 39 54 21 32 19 12
(End)
For listening I recommend some (mostly) percussive MIDI-instrument and the pitch offset set to at least 29 and the tempo (rate) to about 60. - Antti Karttunen, Feb 17 2015

Links

Crossrefs

Inverse: A249813.
Similar or related permutations: A246684, A249811, A250244, A252755.
Cf. A000079, A000265, A001511, A003602, A006093, A083140, A083221, A249741, A250469, A253886.
Compare also the scatterplot of this sequence to the graphs of A252755 and A246684.
Differs from A246684 for the first time at n=14, where a(14) = 20, while A246684(14) = 26.

Formula

In the following formulas, A083221 and A249741 are interpreted as bivariate functions:
a(1) = 1, for n>1: a(n) = A083221(A001511(n), a(A003602(n))) - 1 = A249741(A001511(n), a(A003602(n))).
a(1) = 1, a(2n) = A253886(a(n)), a(2n+1) = (2*a(n+1))-1. - Antti Karttunen, Jan 18 2015
As a composition of other permutations:
a(n) = A250244(A246684(n)).
Other identities. For all n >= 1, the following holds:
a(n) = (1+a((2*n)-1))/2. [The odd bisection from a(1) onward with one added and then halved gives the sequence back.]
a(A000079(n-1)) = A006093(n).

A246683 Permutation of natural numbers: a(1) = 1, a(n) = A000079(A055396(n+1)-1) * ((2*a(A246277(n+1))) - 1).

Original entry on oeis.org

1, 2, 3, 4, 5, 8, 7, 6, 9, 16, 15, 32, 13, 10, 11, 64, 17, 128, 31, 18, 29, 256, 63, 12, 25, 14, 19, 512, 21, 1024, 127, 26, 33, 20, 255, 2048, 61, 58, 35, 4096, 57, 8192, 511, 30, 125, 16384, 23, 24, 49, 50, 27, 32768, 37, 36, 1023, 66, 41, 65536, 2047, 131072, 253, 62, 51, 52, 65, 262144, 39, 122, 509, 524288, 4095, 1048576, 121
Offset: 1

Views

Author

Antti Karttunen, Sep 06 2014

Keywords

Comments

See the comments in A246675. This is otherwise similar permutation, except that after having reached an even number 2m when we have shifted the prime factorization of n+1 k steps towards smaller primes, here, in contrary to A246675, we don't shift the binary representation of the odd number 2m-1, but instead of an odd number (2*a(m))-1 the same number (k) of bit-positions leftward, i.e. multiply it with 2^k.
See also the comments at the inverse permutation A246684.

Examples

			Consider 44 = 45-1. To find 45's position in array A246278, we start shifting its prime factorization 45 = 3 * 3 * 5 = p_2 * p_2 * p_3, step by step, until we get an even number, which in this case happens immediately after the first step, as p_1 * p_1 * p_2 = 2*2*3 = 12. 12 is in the 6th column of A246278, thus we take [here a(6) is computed recursively in the same way:] (2*a(6))-1 = (2*8)-1 = 15, "1111" in binary, and shift it one bit left (that is, multiply by 2), to give 2*15 = 30, thus a(44) = 30.

Links

Crossrefs

Inverse: A246684.
Variants: A246675, A246677.
Cf. A000051, A000079, A005408, A055396, A246277, A246278,
A249813, A250244.
Differs from A249813 for the first time at n=20, where a(20) = 18, while A249813(20) = 14.

Formula

a(1) = 1, a(n) = A000079(A055396(n+1)-1) * ((2*a(A246277(n+1))) - 1).
As a composition of other permutations:
a(n) = A249813(A250244(n)).
Other identities. For all n >= 1, the following holds:
a(n) = (1+a((2*n)-1))/2. [The odd bisection from a(1) onward with one added and then halved gives the sequence back].
For all n >= 0, the following holds: a(A000051(n)) = A000051(n). [Numbers of the form 2^n + 1 are among the fixed points].

A249812 Permutation of natural numbers: a(n) = A000079(A055396(n+1)-1) * ((2*A078898(n+1))-1).

Original entry on oeis.org

1, 2, 3, 4, 5, 8, 7, 6, 9, 16, 11, 32, 13, 10, 15, 64, 17, 128, 19, 14, 21, 256, 23, 12, 25, 18, 27, 512, 29, 1024, 31, 22, 33, 20, 35, 2048, 37, 26, 39, 4096, 41, 8192, 43, 30, 45, 16384, 47, 24, 49, 34, 51, 32768, 53, 28, 55, 38, 57, 65536, 59, 131072, 61, 42, 63, 36, 65, 262144, 67, 46, 69, 524288, 71, 1048576, 73, 50, 75, 40, 77, 2097152, 79, 54, 81, 4194304, 83, 44
Offset: 1

Views

Author

Antti Karttunen, Nov 06 2014

Keywords

Comments

In the essence, a(n) tells which number in the array A135764 is at the same position where n is in the array A249741, the sieve of Eratosthenes minus 1. As the topmost row in both arrays is A005408 (odd numbers), they are fixed, i.e., a(2n+1) = 2n+1 for all n.
Equally: a(n) tells which number in array A054582 is at the same position where n is in the array A114881, as they are the transposes of above two arrays.

Links

Crossrefs

Inverse: A249811.
Cf. A000079, A005408, A006093, A055396, A078898.
Similar or related permutations: A249813 ("deep variant"), A246675, A249816, A054582, A114881, A250252, A135764, A249741, A249742.
Differs from A246675 for the first time at n=20, where a(20)=14, while A246675(20)=18.

Programs

Formula

a(n) = A000079(A055396(n+1)-1) * ((2*A078898(n+1))-1).
As a composition of related permutations:
a(n) = A054582(A250252(n)-1).
a(n) = A135764(A249742(n)).
a(n) = A246675(A249816(n)).
Other identities. For all n >= 1 the following holds:
a(A006093(n)) = A000079(n-1).

A250243 Permutation of natural numbers: a(n) = A246275(A055396(n+1), a(A078898(n+1))).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 26, 21, 22, 23, 24, 25, 20, 27, 28, 29, 30, 31, 44, 33, 34, 35, 36, 37, 32, 51, 40, 41, 42, 43, 80, 45, 46, 47, 48, 49, 74, 39, 52, 53, 124, 55, 62, 57, 58, 59, 60, 61, 38, 87, 54, 65, 66, 67, 134, 69, 70, 71, 72, 73, 50, 63, 76, 101, 78, 79, 98, 81
Offset: 1

Views

Author

Antti Karttunen, Nov 16 2014

Keywords

Comments

This is a "more recursed" variant of A249816. Preserves the parity of n.

Links

Crossrefs

Inverse: A250244.
Cf. A006093, A055396, A078898, A246275, A246278.
Similar or related permutations: A246684, A249813, A250246.
Differs from A249815 and A250244 for the first time at n=32, where a(32) = 44, while A249815(32) = A250244(32) = 38.
Differs from "shallow variant" A249816 for the first time at n=39, where a(39) = 51, while A249816(39) = 39.

Formula

a(n) = A246275(A055396(n+1), a(A078898(n+1))).
As a composition of other permutations:
a(n) = A246684(A249813(n)).
Other identities. For all n >= 1, the following holds:
a(n) = (1+a((2*n)-1))/2. [The odd bisection from a(1) onward with one added and then halved gives the sequence back.]
a(A006093(n)) = A006093(n). [Primes minus one are among the fixed points].

A269866 Permutation of natural numbers: a(1) = 1, a(2n) = 2*a(n), a(2n+1) = 1 + 2*a(A268674(2n+1)-1).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 9, 8, 7, 10, 13, 12, 21, 18, 11, 16, 25, 14, 33, 20, 19, 26, 29, 24, 17, 42, 15, 36, 53, 22, 73, 32, 27, 50, 37, 28, 45, 66, 43, 40, 57, 38, 81, 52, 23, 58, 77, 48, 49, 34, 51, 84, 117, 30, 41, 72, 67, 106, 169, 44, 213, 146, 39, 64, 85, 54, 89, 100, 59, 74, 109, 56, 149, 90, 35, 132, 101, 86, 113, 80, 31
Offset: 1

Views

Author

Antti Karttunen, Mar 12 2016

Keywords

Links

Crossrefs

Inverse: A269865.
Cf. A268674.
Related or similar permutations: A269867, A249813, A252756, A270196.
Differs from similarly constructed A246376 for the first time at n=21, where a(21) = 19, instead of 15.

Formula

a(1) = 1, a(2n) = 2*a(n), a(2n+1) = 1 + 2*a(A268674(2n+1)-1).

A269373 Permutation of natural numbers: a(1) = 1, a(n) = A000079(A260438(n+1)-1) * ((2 * a(A260439(n+1))) - 1).

Original entry on oeis.org

1, 2, 3, 6, 5, 4, 11, 8, 9, 10, 7, 16, 21, 32, 15, 22, 17, 12, 19, 64, 13, 18, 31, 128, 41, 24, 63, 14, 29, 256, 43, 512, 33, 42, 23, 1024, 37, 20, 127, 30, 25, 2048, 35, 48, 61, 34, 255, 4096, 81, 8192, 47, 38, 125, 96, 27, 40, 57, 26, 511, 44, 85, 16384, 1023, 62, 65, 32768, 83, 65536, 45, 82, 2047, 131072, 73, 262144, 39
Offset: 1

Views

Author

Antti Karttunen, Mar 01 2016

Keywords

Links

Crossrefs

Inverse: A269374.
Cf. A000035, A000079, A260438, A260439.
Cf. also A249813, A269383.

Formula

a(1) = 1, a(n) = A000079(A260438(n+1)-1) * ((2 * a(A260439(n+1))) - 1).
Other identities. For all n >= 0:
A000035(a(n)) = A000035(n). [This permutation preserves the parity of n.]

A269383 Permutation of natural numbers: a(1) = 1, a(n) = A000079(A260738(n+1)-1) * ((2 * a(A260739(n+1))) - 1).

Original entry on oeis.org

1, 2, 3, 4, 5, 8, 7, 6, 9, 16, 15, 32, 13, 10, 11, 64, 17, 12, 31, 14, 29, 128, 63, 256, 25, 18, 19, 512, 21, 24, 127, 30, 33, 20, 23, 1024, 61, 26, 27, 2048, 57, 4096, 255, 22, 125, 8192, 511, 28, 49, 34, 35, 16384, 37, 48, 1023, 62, 41, 40, 47, 32768, 253, 58, 59, 36, 65, 65536, 39, 126, 45, 131072, 2047, 96, 121, 50, 51, 262144, 53, 60
Offset: 1

Views

Author

Antti Karttunen, Mar 01 2016

Keywords

Links

Crossrefs

Inverse: A269384.
Cf. A000035, A000079, A260738, A260739.
Cf. also A249813, A269373.
Differs from both A246683 and A249813 for the first time at n=18, which here a(18)=12, instead of 128.

Formula

a(1) = 1, a(n) = A000079(A260738(n+1)-1) * ((2 * a(A260739(n+1))) - 1).
Other identities. For all n >= 0:
A000035(a(n)) = A000035(n). [This permutation preserves the parity of n.]
Showing 1-7 of 7 results.

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