A126917 -id:A126917 - cp's OEIS frontend

A243069 a(n) = smallest natural number that has not yet occurred among the first n terms of A126917.

2, 3, 4, 4, 6, 6, 7, 7, 7, 7, 9, 9, 12, 12, 12, 12, 13, 13, 15, 15, 15, 15, 16, 16, 16, 16, 16, 16, 19, 19, 21, 21, 21, 21, 21, 21, 22, 22, 22, 22, 27, 27, 28, 28, 28, 28, 30, 30, 30, 30, 30, 30, 31, 31, 31, 31, 31, 31, 36, 36, 37, 37, 37, 37, 37, 37, 39, 39, 39, 39, 42, 42, 45

Offset: 1

Antti Karttunen, Jun 20 2014

nonn

Facilitates the computing of A126917.

a(n) grows only when n is a prime. A243498 gives the values at those points.

			The first eight terms of A126917 are: 1, 2, 3, 5, 4, 8, 6, 11. In range [1,1] the first that has not yet occurred is 2, in range [1,2] it is 3, in range [1,3] it is 4, in range [1,4] it is still 4, in range [1,5] it is 6, in range [1,6] it is still 6, in range [1,7] it is 7, in range [1,8] it is still 7, thus the first eight terms of this sequence are 2, 3, 4, 4, 6, 6, 7, 7.

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A126917, A126918, A243488, A243498.

For all n, a(n) <= A126917(n+1).

For n >= 2, if A126917(n) > a(n-1) [when n is composite], a(n) = a(n-1).

A126918 Inverse permutation to sequence A126917.

Original entry on oeis.org

1, 2, 3, 5, 4, 7, 11, 6, 13, 9, 8, 17, 19, 10, 23, 29, 12, 15, 31, 14, 37, 41, 16, 25, 21, 18, 43, 47, 20, 53, 59, 22, 27, 35, 24, 61, 67, 26, 71, 33, 28, 73, 49, 30, 79, 83, 32, 39, 89, 34, 97, 101, 36, 55, 45, 38, 103, 107, 40, 109, 113, 42, 51, 65, 44, 127, 77, 46, 131, 57

Offset: 1

Leroy Quet, Dec 31 2006

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences that are permutations of the natural numbers

Cf. A126917, A243069.

Extended by Ray Chandler, Jan 04 2007

A243488 a(n) = A126917(n+1) - A243069(n).

Original entry on oeis.org

0, 0, 1, 0, 2, 0, 4, 3, 7, 0, 8, 0, 8, 6, 11, 0, 13, 0, 14, 10, 17, 0, 19, 8, 22, 17, 25, 0, 25, 0, 26, 19, 29, 13, 32, 0, 34, 26, 37, 0, 35, 0, 37, 27, 40, 0, 41, 13, 44, 33, 47, 0, 49, 23, 52, 39, 55, 0, 53, 0, 55, 41, 58, 27, 61, 0, 62, 46, 65, 0, 65, 0, 65, 48, 68, 22, 71, 0

Offset: 1

Antti Karttunen, Jun 20 2014

nonn

Zeros occur at positions given by A006093 (primes minus 1).

The sequence sheds some light on behavior of A126917, please see the plot.

Antti Karttunen, Table of n, a(n) for n = 1..10006

Cf. A006093, A126917, A243069, A243498.

(define (A243488 n) (- (A126917 (+ 1 n)) (A243069 n)))

a(n) = A126917(n+1) - A243069(n).

A246165 Permutation of natural numbers: a(1) = 1, a(n) = A064989(n)-th integer among those positive integers not occurring earlier in the sequence. [A064989(n) shifts the prime factorization of n one step right].

Original entry on oeis.org

1, 2, 4, 3, 7, 6, 11, 5, 12, 10, 17, 9, 23, 16, 19, 8, 29, 18, 35, 15, 28, 25, 41, 14, 31, 34, 30, 24, 51, 27, 59, 13, 44, 43, 47, 26, 67, 52, 58, 22, 77, 42, 83, 38, 49, 61, 89, 21, 70, 46, 73, 53, 99, 45, 69, 37, 88, 75, 111, 40, 119, 85, 72, 20, 94, 64, 127, 63, 103, 68, 137, 39, 143, 97, 79, 78, 106, 87, 151, 36

Offset: 1

Antti Karttunen, Aug 17 2014

nonn

Terms at a(2^n) are: 1, 2, 3, 5, 8, 13, 20, 32, 48, 71, 105, 156, 236, 354, 542, 815, 1228, ...

Fixed points begin as: 1, 2, 6, 10, 18, 42, 92, 26372, ...

			By definition, a(1) = 1.
After that, for n = 2, when its prime factorization is shifted once right, results A064989(2) = 1, so we select the 1st of still unused positive natural numbers, which is 2, thus a(2) = 2.
For n = 3 = p_2 (3 is the second prime), when its prime factorization is shifted once right, results A064989(3) = 2 = p_1, so we select 2nd of still unused numbers, which is 4, thus a(3) = 4.
For n = 4, like for all powers of two, the result of right shifting is 1, so we select the smallest still unused number, which is 3, thus a(4) = 3.
For n = 5 = p_3, A064989(5) = 3 = p_2, so we select the 3rd smallest still unused number from [5, 6, 7, 8, ...] which is 7, thus a(5) = 7.

Antti Karttunen, Table of n, a(n) for n = 1..8192
Index entries for sequences that are permutations of the natural numbers

Inverse: A246166.
Similar permutations: A119435, A126917.
Cf. A064989.

A266411 a(1) = 1, after which each a(n) = (A004074(n)+1)-th number selected from those not yet in the sequence.

Original entry on oeis.org

1, 2, 4, 3, 6, 8, 7, 5, 10, 12, 14, 13, 16, 15, 11, 9, 18, 20, 22, 24, 23, 26, 28, 27, 30, 29, 25, 32, 31, 21, 19, 17, 34, 36, 38, 40, 42, 41, 44, 46, 48, 47, 50, 52, 51, 54, 53, 49, 56, 58, 57, 60, 59, 55, 62, 61, 45, 43, 64, 63, 39, 37, 35, 33, 66, 68, 70, 72, 74, 76, 75, 78, 80, 82, 84, 83, 86, 88, 90, 89, 92, 94, 93, 96, 95, 91

Offset: 1

Antti Karttunen, Dec 29 2015

nonn

Inverse: A266412.
Cf. A004074.
Similar permutations in Quetian style: A119435, A126917, A246165, A266413.
Cf. also A265901, A265903.

f[n_] := Block[{a = {1}, g, b = Range[2, n]}, g[1] = g[2] = 1; g[x_] := g[x] = g[g[x - 1]] + g[x - g[x - 1]]; Do[{AppendTo[a, #[[1, 1]]], Set[b, Last@ #]} &@ If[# > Length@ b, Break[], TakeDrop[b, {#}]] &@ (2 g[#] - # + 1) &@ k, {k, 2, n}]; a]; f@ 97 (* Michael De Vlieger, Dec 29 2015, Version 10.2, based on Harvey P. Dale at A004074 *)

A243498 Records (and distinct values) of A243069: a(n) = A243069(A008578(n)).

Original entry on oeis.org

2, 3, 4, 6, 7, 9, 12, 13, 15, 16, 19, 21, 22, 27, 28, 30, 31, 36, 37, 39, 42, 45, 46, 49, 51, 52, 57, 58, 60, 61, 66, 69, 72, 73, 75, 76, 81, 82, 87, 88, 91, 96, 97, 99, 102, 105, 109, 111, 112, 118, 120, 126, 127, 129, 132, 133, 135, 136, 142, 147, 148, 150

Offset: 1

Antti Karttunen, Jun 21 2014

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A243069, A243488, A126917, A008578.

(define (A243498 n) (A243069 (A008578 n)))

a(n) = A243069(A008578(n)).

A266413 a(1) = 1, after which each a(n) = A002487(n)-th number selected from those not yet in the sequence.

Original entry on oeis.org

1, 2, 4, 3, 7, 6, 9, 5, 12, 11, 15, 10, 17, 14, 18, 8, 21, 20, 25, 19, 28, 24, 29, 16, 31, 27, 34, 23, 35, 30, 33, 13, 38, 37, 43, 36, 47, 42, 48, 32, 51, 46, 55, 41, 56, 49, 53, 26, 57, 52, 62, 45, 65, 59, 64, 40, 66, 60, 69, 50, 68, 58, 63, 22, 71, 70, 77, 67, 82, 76, 83, 61, 87, 81, 92, 75, 93, 84, 89, 54, 94, 88, 101, 80

Offset: 1

Antti Karttunen, Dec 29 2015

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16385
Index entries for sequences that are permutations of the natural numbers

Inverse: A266414.
Cf. A002487.
Similar permutations in Quetian style: A119435, A126917, A246165, A266411.
Cf. also A266405.

f[n_] := Block[{a = {1}, g, b = Range[2, n]}, g[1] = 1; g[x_] := g[x] = If[EvenQ@ x, g[x/2], g[(x - 1)/2] + g[(x + 1)/2]]; Do[{AppendTo[a, #[[1, 1]]], Set[b, Last@ #]} &@ If[# > Length@ b, Break[], TakeDrop[b, {#}]] &@ g@ k, {k, 2, n}]; a]; f@ 103 (* Michael De Vlieger, Dec 29 2015, Version 10.2, after N. J. A. Sloane at A002487 *)

cp's OEIS Frontend

A243069 a(n) = smallest natural number that has not yet occurred among the first n terms of A126917.

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A126918 Inverse permutation to sequence A126917.

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A243488 a(n) = A126917(n+1) - A243069(n).

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A246165 Permutation of natural numbers: a(1) = 1, a(n) = A064989(n)-th integer among those positive integers not occurring earlier in the sequence. [A064989(n) shifts the prime factorization of n one step right].

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A266411 a(1) = 1, after which each a(n) = (A004074(n)+1)-th number selected from those not yet in the sequence.

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A243498 Records (and distinct values) of A243069: a(n) = A243069(A008578(n)).

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A266413 a(1) = 1, after which each a(n) = A002487(n)-th number selected from those not yet in the sequence.

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