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 1-5 of 5 results.

A243065 Permutation of natural numbers, the odd bisection of A241909 halved; equally, a composition of A064216 and A241909: a(n) = A241909(A064216(n)).

Original entry on oeis.org

1, 2, 4, 8, 3, 16, 32, 9, 64, 128, 27, 256, 6, 5, 512, 1024, 81, 18, 2048, 243, 4096, 8192, 25, 16384, 12, 729, 32768, 54, 2187, 65536, 131072, 125, 162, 262144, 6561, 524288, 1048576, 15, 36, 2097152, 7, 4194304, 486, 19683, 8388608, 108, 59049, 1458, 16777216, 625, 33554432, 67108864, 75
Offset: 1

Views

Author

Antti Karttunen, Jun 01 2014

Keywords

Comments

Are there any other fixed points than 1, 2, 18 and 72?

Links

Crossrefs

Inverse: A243066.
Cf. A064216, A241909, A243505-A243506, A244152-A244154, A243061-A243062, A006254.

Programs

Formula

a(1) = 1, and for n>=2, a(n) = A241909(2n-1)/2. Equally, a(n) = ceiling(A241909(2n-1)/2) for all n.
As a composition of related permutations:
a(n) = A241909(A064216(n)).
a(n) = A241909(A243061(A241909(n))).
For all n, a(A006254(n)) = 2^n.

A243066 Permutation of natural numbers, the even bisection of A241909 incremented by one and halved; equally, a composition of A241909 and A048673: a(n) = A048673(A241909(n)).

Original entry on oeis.org

1, 2, 5, 3, 14, 13, 41, 4, 8, 63, 122, 25, 365, 313, 38, 6, 1094, 18, 3281, 172, 188, 1563, 9842, 61, 23, 7813, 11, 1201, 29525, 123, 88574, 7, 938, 39063, 113, 39, 265721, 195313, 4688, 666, 797162, 858, 2391485, 8404, 74, 976563, 7174454, 85, 68, 88, 23438, 58825, 21523361, 28
Offset: 1

Views

Author

Antti Karttunen, Jun 01 2014

Keywords

Comments

For n > 1, 2n is found in A241909 from the position (2*a(n))-1. I.e., A241909((2*a(n))-1) = 2n for all n >= 2.
Or in other words, a(n) gives the position in the odd bisection of A241909 where 2n is located at.
Are there any other fixed points than 1, 2, 18 and 72?

Links

Crossrefs

Inverse: A243065.
Cf. A048673, A241909, A243505-A243506, A244152-A244154, A243061-A243062.

Formula

a(1) = 1, a(n) = (A241909(2*n)+1)/2.
As a composition of related permutations:
a(n) = A048673(A241909(n)).
a(n) = A241909(A243062(A241909(n))).
For all n>=1, a(2^n) = A006254(n).

A244319 Self-inverse permutation of natural numbers: a(1) = 1, a(2n) = A003961(1+a(A064989(2n-1))), a(2n+1) = 1+A003961(a(A064989(2n+1)-1)).

Original entry on oeis.org

1, 3, 2, 9, 6, 5, 26, 11, 4, 21, 8, 125, 56, 25, 16, 15, 344, 115, 36, 1015, 10, 39, 204, 41, 14, 7, 52, 45, 86, 301, 176, 155, 298, 51, 50, 19, 518, 305, 22, 189, 24, 895, 1376, 49, 28, 825, 1268, 11875, 44, 35, 34, 27, 3186, 6625, 2388, 13, 454, 153, 126, 3191, 476, 131
Offset: 1

Views

Author

Antti Karttunen, Jul 18 2014; description corrected and PARI code added Jul 30 2014

Keywords

Comments

After 1, maps each even number to a unique odd number and vice versa, i.e., for all n > 1, A000035(a(n)) XOR A000035(n) = 1, where XOR is given in A003987.

Links

Crossrefs

Cf. A000035, A003987, A003961, A064989, A243501.
Related permutations: A048673, A064216, A245609-A245610.
Similar entanglement permutations: A245605-A245606, A235491, A236854, A243347, A244152.

Programs

Formula

a(1) = 1, a(2n) = A003961(1+a(A064989(2n-1))), a(2n+1) = A243501(a(A064989(2n+1)-1)).
As a composition of related permutations:
a(n) = A245609(A048673(n)) = A064216(A245610(n)).

A245603 Permutation of natural numbers: a(1) = 1; thereafter, if n is k-th number with an odd number of prime divisors (counted with multiplicity) [i.e., n = A026424(k)], a(n) = 2*a(k), otherwise, when n is k-th number > 1 with an even number of prime divisors [i.e., n = A028260(1+k)], a(n) = 1+(2*a(k)).

Original entry on oeis.org

1, 2, 4, 3, 8, 5, 6, 16, 9, 7, 10, 12, 32, 17, 11, 13, 18, 14, 20, 24, 33, 19, 64, 15, 21, 25, 34, 22, 26, 36, 28, 40, 65, 35, 23, 27, 48, 37, 29, 41, 66, 38, 128, 30, 42, 49, 50, 68, 67, 44, 39, 52, 72, 129, 31, 43, 51, 69, 56, 45, 80, 53, 130, 73, 57, 70, 46, 54, 81, 96, 74, 58, 82, 131, 132, 76, 71, 256, 60
Offset: 1

Views

Author

Antti Karttunen, Jul 27 2014

Keywords

Links

Crossrefs

Inverse: A245604.
Cf. A026424, A028260, A066829, A055037, A055038.
Similar permutations: A143692, A244152, A244321, A245613, A245605, A245607.

Formula

a(1) = 1, and for n > 1, if A066829(n) = 1, then a(n) = 2 * A245603(A055038(n)), otherwise a(n) = 1 + (2 * A245603(A055037(n)-1)).
As a composition of related permutations:
a(n) = A244321(A245613(n)).
For all n >= 1, A000035(a(n)) = 1 - A066829(n). [Permutation A143692 has the same property.]

A245604 Permutation of natural numbers: a(1)=1, a(2n) = A026424(a(n)), a(2n+1) = A028260(1+a(n)).

Original entry on oeis.org

1, 2, 4, 3, 6, 7, 10, 5, 9, 11, 15, 12, 16, 18, 24, 8, 14, 17, 22, 19, 25, 28, 35, 20, 26, 29, 36, 31, 39, 44, 55, 13, 21, 27, 34, 30, 38, 42, 51, 32, 40, 45, 56, 50, 60, 67, 81, 37, 46, 47, 57, 52, 62, 68, 82, 59, 65, 72, 86, 79, 93, 105, 121, 23, 33, 41, 49, 48, 58, 66, 77, 53, 64, 71, 85, 76, 90, 99, 111, 61
Offset: 1

Views

Author

Antti Karttunen, Jul 27 2014

Keywords

Links

Crossrefs

Inverse: A245603.
Similar permutations: A143691, A244152, A244322, A245614, A245606, A245608.

Formula

a(1)=1, a(2n) = A026424(a(n)), a(2n+1) = A028260(1+a(n)).
As a composition of related permutations:
a(n) = A245614(A244322(n)).
For all n >= 1, A066829(a(n)) = 1 - A000035(n). [Permutation A143691 has the same property].
Equally, A066829(a(n)*a(n+1)) = 1 for all n.
Showing 1-5 of 5 results.

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