A055396 -id:A055396 - cp's OEIS frontend

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

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

Antti Karttunen, Sep 06 2014

nonn

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.

			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.

Antti Karttunen, Table of n, a(n) for n = 1..2049
Index entries for sequences that are permutations of the natural numbers

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.

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].

A250244 Permutation of natural numbers: a(n) = A249741(A055396(n+1), a(A246277(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, 38, 33, 34, 35, 36, 37, 62, 51, 40, 41, 42, 43, 32, 45, 46, 47, 48, 49, 74, 39, 52, 53, 64, 55, 98, 57, 58, 59, 60, 61, 56, 75, 94, 65, 66, 67, 110, 69, 70, 71, 72, 73, 50, 123, 76, 101, 78, 79, 44, 81, 82, 83, 154, 85, 134, 63, 88, 89

Offset: 1

Antti Karttunen, Nov 16 2014

nonn

This is a "more recursed" variant of A249815. Preserves the parity of n.

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

Inverse: A250243.
Cf. A006093, A055396, A083221, A246277, A249741.
Similar or related permutations: A246683, A249814, A250245.
Differs from A249816 and A250243 for the first time at n=32, where a(32) = 38, while A249816(32) = A250243(32) = 44.
Differs from the "shallow variant" A249815 for the first time at n=39, where a(39) = 51, while A249815(39) = 39

a(n) = A249741(A055396(n+1), a(A246277(n+1))).
As a composition of other permutations:
a(n) = A249814(A246683(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].

A249813 Permutation of natural numbers: a(1) = 1, a(n) = A000079(A055396(n+1)-1) * ((2 * a(A078898(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, 14, 29, 256, 63, 12, 25, 18, 19, 512, 21, 1024, 127, 30, 33, 20, 255, 2048, 61, 26, 27, 4096, 57, 8192, 511, 22, 125, 16384, 23, 24, 49, 34, 35, 32768, 37, 28, 1023, 62, 41, 65536, 2047, 131072, 253, 58, 59, 36, 65, 262144, 39, 126, 509, 524288, 4095, 1048576, 121, 50, 51, 40, 53

Offset: 1

Antti Karttunen, Nov 06 2014

nonn

This sequence is a "recursed variant" of A249812.

See also the comments at the inverse permutation A249814.

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

Inverse: A249814.
Similar or related permutations: A246683, A249812, A250243.
Cf. A000079, A006093, A055396, A078898.
Differs from A246683 for the first time at n=20, where a(20) = 14, while A246683(20) = 18.

a(1) = 1, a(n) = A000079(A055396(n+1)-1) * ((2 * a(A078898(n+1))) - 1).
As a composition of other permutations:
a(n) = A246683(A250243(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)) = A000079(n-1).

A249815 Permutation of natural numbers: a(n) = A249741(A055396(n+1), A246277(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, 38, 33, 34, 35, 36, 37, 62, 39, 40, 41, 42, 43, 32, 45, 46, 47, 48, 49, 74, 51, 52, 53, 64, 55, 98, 57, 58, 59, 60, 61, 56, 63, 94, 65, 66, 67, 110, 69, 70, 71, 72, 73, 50, 75, 76, 77, 78, 79, 44, 81, 82, 83

Offset: 1

Antti Karttunen, Nov 06 2014

nonn

a(n) tells which number in square array A249741 (the sieve of Eratosthenes minus 1) is at the same position where n is in array A246275. As the topmost row in both arrays is A005408 (odd numbers), they are fixed, i.e. a(2n+1) = 2n+1 for all n. Also, as the leftmost column in both arrays is primes minus one (A006093), they are also among the fixed points.

Equally: a(n) tells which number in array A114881 is at the same position where n is in the array A246273, as they are the transposes of above two arrays.

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

Inverse: A249816
Similar or related permutations: A250244 ("deep variant"), A246675, A249811, A249817, A246273, A246275, A114881, A249741.
Cf. A005408, A006093, A055396, A246277.
Differs from A249816 and A250243 for the first time at n=32, where a(32) = 38, while A249816(32) = A250243(32) = 44.
Differs from A250244 for the first time at n=39, where a(39) = 39, while A250244(39) = 51.

(define (A249815 n) (+ -1 (A083221bi (A246277 (+ 1 n)) (A055396 (+ 1 n))))) ;; Code for A083221bi given in A083221

a(n) = A249741(A055396(n+1), A246277(n+1)).
As a composition of other permutations:
a(n) = A249811(A246675(n)).
a(n) = A249817(n+1) - 1.
Other identities. For all n >= 1:
a(A005408(n-1)) = A005408(n-1) and a(A006093(n)) = A006093(n). [Fixes odd numbers and precedents of primes. Cf. comments above].

A246677 Permutation of natural numbers: a(1) = 1, a(2n) = A000079(A055396(2n+1)-1) * ((2A246277(2n+1))-1), a(2n+1) = 1 + 2a(n).

Original entry on oeis.org

1, 2, 3, 4, 5, 8, 7, 6, 9, 16, 11, 32, 17, 10, 15, 64, 13, 128, 19, 18, 33, 256, 23, 12, 65, 14, 35, 512, 21, 1024, 31, 26, 129, 20, 27, 2048, 257, 42, 39, 4096, 37, 8192, 67, 22, 513, 16384, 47, 24, 25, 50, 131, 32768, 29, 36, 71, 66, 1025, 65536, 43, 131072, 2049, 38, 63, 52, 53, 262144, 259, 74, 41

Offset: 1

Antti Karttunen, Sep 01 2014

nonn

See the comments in A246675. This is otherwise similar permutation, except for odd numbers, which are here recursively permuted by the emerging permutation itself. The even bisection halved gives A246679, the odd bisection from a(3) onward with one subtracted and then halved gives this sequence back.

Antti Karttunen, Table of n, a(n) for n = 1..2048
Index entries for sequences that are permutations of the natural numbers

Inverse: A246678. Variants: A246675, A246683.
Even bisection halved: A246679.
Cf. A000079, A055396, A246277.
a(n) differs from A156552(n+1) for the first time at n=32, where a(32) = 26, while A156552(33) = 34.

a(1) = 1, a(2n) = A000079(A055396(2n+1)-1) * ((2*A246277(2n+1))-1), a(2n+1) = 1 + 2*a(n).

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

Antti Karttunen, Nov 06 2014

nonn

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.

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

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.

(define (A249812 n) (* (A000079 (- (A055396 (+ 1 n)) 1)) (+ -1 (* 2 (A078898 (+ 1 n))))))

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

Antti Karttunen, Nov 16 2014

nonn

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

Antti Karttunen, Table of n, a(n) for n = 1..5749
Index entries for sequences that are permutations of the natural numbers

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.

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].

A250247 Permutation of natural numbers: a(1) = 1, a(n) = A083221(a(A055396(n)),A246277(n)).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 27, 22, 23, 24, 25, 26, 21, 28, 29, 30, 31, 32, 39, 34, 35, 36, 37, 38, 63, 40, 41, 42, 43, 44, 33, 46, 47, 48, 49, 50, 75, 52, 53, 54, 65, 56, 99, 58, 59, 60, 61, 62, 57, 64, 95, 66, 67, 68, 111, 70, 71, 72, 103, 74, 51, 76, 77, 78, 79, 80, 45, 82

Offset: 1

Antti Karttunen, Nov 17 2014

nonn

The first 7-cycle occurs at: (33 39 63 57 99 81 45), which is mirrored at the cycle (137 167 307 269 523 419 197), consisting of primes (p_33, p_39, p_63, ...).

			As a(21) = 27, and A000040(21) = 73 and A000040(27) = 103, a(73) = 103.

Antti Karttunen, Table of n, a(n) for n = 1..4240
Index entries for sequences that are permutations of the natural numbers

Inverse: A250248.
Cf. A000040, A005843, A049076, A049084, A055396, A078442, A083221, A246277.
Differs from its inverse A250248 for the first time at n = 33, where a(33) = 39, while A250248(33) = 45.
Differs from the "vanilla version" A249817 for the first time at n=73, where a(73) = 103, while A249817(73) = 73.
Differs from "doubly recursed" version A250249 for the first time at n=42, where a(42) = 42, while A250249(42) = 54, thus the first prime where they get different values is p_42 = 181, where a(181) = 181, while A250249(181) = 251 = p_54.

a(1) = 1, a(n) = A083221(a(A055396(n)),A246277(n)).
Other identities. For all n >= 1:
a(A005843(n)) = A005843(n). [Fixes even numbers].
a(p_n) = p_{a(n)}, or equally, a(n) = A049084(a(A000040(n))). [Restriction to primes induces the same sequence].
A078442(a(n)) = A078442(n), A049076(a(n)) = A049076(n). [Preserves the "order of primeness of n"].

A249816 Permutation of natural numbers: a(n) = A246275(A055396(n+1), 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, 39, 40, 41, 42, 43, 80, 45, 46, 47, 48, 49, 74, 51, 52, 53, 124, 55, 62, 57, 58, 59, 60, 61, 38, 63, 54, 65, 66, 67, 134, 69, 70, 71, 72, 73, 50, 75, 76, 77, 78, 79, 98, 81, 82, 83

Offset: 1

Antti Karttunen, Nov 06 2014

nonn

a(n) tells which number in square array A246275 is at the same position where n is in 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. Also, as the leftmost column in both arrays is primes minus one (A006093), they are also among the fixed points.

Equally: a(n) tells which number in array A246273 is at the same position where n is in the array A114881, as they are the transposes of above two arrays.

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

Inverse: A249815.
Cf. A005408, A006093, A055396, A078898, A246278.
Similar or related permutations: A250243 ("deep variant"), A246676, A249812, A249818, A246273, A246275, A114881, A249741.
Differs from A249815 and A250244 for the first time at n=32, where a(32) = 44, while A249815(32) = A250244(32) = 38.
Differs from A250244 for the first time at n=39, where a(39) = 39, while A250243(39) = 51.

(define (A249816 n) (+ -1 (A246278bi (A055396 (+ 1 n)) (A078898 (+ 1 n)))))

a(n) = A246275(A055396(n+1), A078898(n+1)).
As a composition of other permutations:
a(n) = A246676(A249812(n)).
a(n) = A249818(n+1) - 1.
Other identities. For all n >= 1:
a(A005408(n-1)) = A005408(n-1) and a(A006093(n)) = A006093(n). [Fixes odd numbers and precedents of primes. Cf. comments above].

A252461 Shift one instance of the smallest prime one step towards smaller primes: a(1) = 1, a(2n) = n, and for odd numbers > 1: a(n) = (n / prime(s)) * prime(s-1), where s = A055396(n), index of the smallest prime dividing n.

Original entry on oeis.org

1, 1, 2, 2, 3, 3, 5, 4, 6, 5, 7, 6, 11, 7, 10, 8, 13, 9, 17, 10, 14, 11, 19, 12, 15, 13, 18, 14, 23, 15, 29, 16, 22, 17, 21, 18, 31, 19, 26, 20, 37, 21, 41, 22, 30, 23, 43, 24, 35, 25, 34, 26, 47, 27, 33, 28, 38, 29, 53, 30, 59, 31, 42, 32, 39, 33, 61, 34, 46, 35, 67, 36, 71, 37, 50, 38, 55, 39, 73, 40, 54, 41, 79, 42

Offset: 1

Antti Karttunen, Dec 20 2014

nonn

Iterating from any n as a(n), a(a(n)), a(a(a(n))), etc. reaches 1 after A056239(n) iterations.

Even bisection gives the natural numbers A000027, the odd bisection from the third term onward is A129128: 2, 3, 5, 6, 7, 11, 10, ...

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

Variants: A252462, A252463.
Bisections: A000027, A129128.
Cf. A001222, A008578, A032742, A055396, A056239, A064989.

a252461[n_Integer] := Block[{a008578, a032742, a055396, a},
  a008578[x_] := If[x == 1, 1, Prime[x - 1]];
  a032742[x_] := If[x == 1, 1, Divisors[x][[-2]]];
  a055396[x_] := PrimePi[FactorInteger[x][[1]][[1]]];
  a[1] = 1;
  a[x_] := a008578[a055396[x]]*a032742[x];
Array[a, n]]; a252461[84] (* Michael De Vlieger, Dec 21 2014 *)

(define (A252461 n) (if (= 1 n) n (* (A008578 (A055396 n)) (A032742 n))))

a(1) = 1; for n>1: a(n) = A008578(A055396(n)) * A032742(n). [Compare to the similar formula of A064989.]
Other identities. For all n >= 1:
a(2n) = n.
If n is odd, A001222(a(n)) = A001222(n).
If n is even, A001222(a(n)) = A001222(n) - 1.

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A250244 Permutation of natural numbers: a(n) = A249741(A055396(n+1), a(A246277(n+1))).

Views

Author

Keywords

Comments

Links

Crossrefs

Formula

A249813 Permutation of natural numbers: a(1) = 1, a(n) = A000079(A055396(n+1)-1) * ((2 * a(A078898(n+1))) - 1).

Views

Author

Keywords

Comments

Links

Crossrefs

Formula

A249815 Permutation of natural numbers: a(n) = A249741(A055396(n+1), A246277(n+1)).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Scheme

Formula

A246677 Permutation of natural numbers: a(1) = 1, a(2n) = A000079(A055396(2n+1)-1) * ((2*A246277(2n+1))-1), a(2n+1) = 1 + 2*a(n).

Views

Author

Keywords

Comments

Links

Crossrefs

Formula

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Scheme

Formula

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

Views

Author

Keywords

Comments

Links

Crossrefs

Formula

A250247 Permutation of natural numbers: a(1) = 1, a(n) = A083221(a(A055396(n)),A246277(n)).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A249816 Permutation of natural numbers: a(n) = A246275(A055396(n+1), A078898(n+1)).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Scheme

A246677 Permutation of natural numbers: a(1) = 1, a(2n) = A000079(A055396(2n+1)-1) * ((2A246277(2n+1))-1), a(2n+1) = 1 + 2a(n).