A266403 -id:A266403 - cp's OEIS frontend

A265369 Self-inverse permutation of nonnegative integers: a(n) = A057889(A263273(A057889(n))).

0, 1, 2, 3, 4, 7, 6, 5, 8, 9, 10, 11, 12, 25, 26, 21, 16, 19, 18, 17, 20, 15, 22, 59, 24, 13, 14, 27, 28, 29, 30, 41, 64, 39, 58, 53, 36, 97, 98, 33, 40, 31, 66, 121, 44, 63, 50, 61, 48, 73, 46, 105, 100, 35, 54, 65, 56, 57, 34, 23, 60, 47, 82, 45, 32, 55, 42, 137, 68, 69, 142, 131, 72, 49, 74, 219, 76, 155, 234, 79, 80, 81, 62, 173

Offset: 0

Antti Karttunen, Jan 02 2016

Antti Karttunen, Table of n, a(n) for n = 0..19684
Indranil Ghosh, Python program to generate the sequence
Index entries for sequences that are permutations of the natural numbers

Cf. A000035, A057889, A263273, A264965, A264966.

Cf. also A265329, A265904, A266190, A266401, A266403, A266407, A266408.

f[n_] := Block[{g, h}, g[x_] := x/3^IntegerExponent[x, 3]; h[x_] := x/g@ x; If[n == 0, 0, FromDigits[Reverse@ IntegerDigits[#, 3], 3] &@ g[n] h[n]]]; g[n_] := FromDigits[Reverse@ IntegerDigits[n, 2], 2] 2^IntegerExponent[n, 2]; Prepend[#, 0] &@ Table[g@ f@ g@ n, {n, 83}] (* Michael De Vlieger, Jan 04 2016, after Jean-François Alcover at A263273 and Ivan Neretin at A057889 *)

(define (A265369 n) (A057889 (A263273 (A057889 n))))

a(n) = A057889(A263273(A057889(n))).
As a composition of related permutations:
a(n) = A264966(A057889(n)).
a(n) = A057889(A264965(n)).
Other identities. For all n >= 0:
A000035(a(n)) = A000035(n). [This permutation preserves the parity of n.]

A266645 Permutation of natural numbers: a(n) = A064989(A250469(n)).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 10, 9, 8, 11, 14, 13, 22, 15, 12, 17, 26, 19, 34, 21, 20, 23, 38, 25, 18, 33, 16, 29, 46, 31, 58, 39, 28, 35, 30, 37, 62, 51, 44, 41, 74, 43, 82, 57, 24, 47, 86, 49, 50, 27, 52, 53, 94, 55, 42, 69, 68, 59, 106, 61, 118, 87, 40, 65, 66, 67, 122, 45, 76, 71, 134, 73, 142, 93, 36, 77, 70, 79, 146, 111, 32, 83, 158, 85, 78, 123

Offset: 1

Antti Karttunen, Jan 02 2016

nonn

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

Inverse: A266646.
Cf. A000035, A003961, A020639, A055396, A064989, A250469.
Related permutations: A266403, A266416, A249817, A249818.

f[n_] := Times @@ Power[Which[# == 1, 1, # == 2, 1, True, NextPrime[#, -1]] & /@ First@ #, Last@ #] &@ Transpose@ FactorInteger@ n; g[n_] := If[n == 1, 0, PrimePi@ FactorInteger[n][[1, 1]]]; Function[s, MapIndexed[ Function[{m, n}, f[Lookup[s, g[n] + 1][[m]] - Boole[n == 1]]][#1, First@ #2] &, #] &@ Map[Position[Lookup[s, g@ #], #][[1, 1]] &, Range@ 120]]@ PositionIndex@ Array[g, 10^4] (* Michael De Vlieger, Mar 09 2017, Version 10 *)

a(n) = A064989(A250469(n)).
a(n) = A064989(A249817(A003961(A249818(n)))).
As a composition of related permutations:
a(n) = A266416(A266403(n)).
Other identities. For all n >= 0:
A000035(a(n)) = A000035(n). [This permutation preserves the parity of n.]
A020639(a(n)) = A020639(n). [More generally, it preserves the smallest prime dividing n.]
A055396(a(n)) = A055396(n).

A266401 Self-inverse permutation of natural numbers: a(n) = A064989(A263273(A003961(n))).

Original entry on oeis.org

1, 2, 5, 4, 3, 10, 17, 8, 13, 6, 11, 20, 9, 34, 71, 16, 7, 26, 19, 12, 23, 22, 21, 40, 41, 18, 227, 68, 31, 142, 29, 32, 53, 14, 67, 52, 61, 38, 107, 24, 25, 46, 59, 44, 65, 42, 73, 80, 49, 82, 197, 36, 33, 454, 55, 136, 137, 62, 43, 284, 37, 58, 571, 64, 45, 106, 35, 28, 89, 134, 15, 104, 47

Offset: 1

Antti Karttunen, Jan 02 2016

Shift primes in the prime factorization of n one step towards larger primes (A003961), then apply the bijective base-3 reverse (A263273) to the resulting odd number, which yields another (or same) odd number, then shift primes in the prime factorization of that second odd number one step back towards smaller primes (A064989).

Antti Karttunen, Table of n, a(n) for n = 1..6560
Index entries for sequences that are permutations of the natural numbers

Cf. A000035, A003961, A064989, A263273.
Cf. A265369, A265904, A266190, A266403 (other conjugates or similar sequences derived from A263273).
Cf. also A048673, A064216, A264996, A266402, A266407, A266408.

f[n_] := Block[{g, h}, g[x_] := x/3^IntegerExponent[x, 3]; h[x_] := x/g@ x; If[n == 0, 0, FromDigits[Reverse@ IntegerDigits[#, 3], 3] &@ g[n] h[n]]]; g[p_?PrimeQ] := g[p] = Prime[PrimePi@ p + 1]; g[1] = 1; g[n_] := g[n] = Times @@ (g[First@ #]^Last@ # &) /@ FactorInteger@ n; h[n_] := Times @@ Power[Which[# == 1, 1, # == 2, 1, True, NextPrime[#, -1]] & /@ First@ #, Last@ #] &@ Transpose@ FactorInteger@ n; Table[h@ f@ g@ n, {n, 82}] (* Michael De Vlieger, Jan 04 2016, after Jean-François Alcover at A003961 and A263273 *)

A030102(n) = { my(r=[n%3]); while(0M. F. Hasler's Nov 04 2011 code in A030102.
A263273 = n -> if(!n,n,A030102(n/(3^valuation(n,3))) * (3^valuation(n, 3))); \\ Taking of the quotient probably unnecessary.
A003961(n) = my(f = factor(n)); for (i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); \\ Using code of Michel Marcus
A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
A266401 = n -> A064989(A263273(A003961(n)));
for(n=1, 6560, write("b266401.txt", n, " ", A266401(n)));

(define (A266401 n) (A064989 (A263273 (A003961 n))))

a(n) = A064989(A263273(A003961(n))).
As a composition of related permutations:
a(n) = A064216(A264996(A048673(n))).
Other identities. For all n >= 0:
A000035(a(n)) = A000035(n). [This permutation preserves the parity of n.]

A266646 Permutation of natural numbers: a(n) = A250470(A003961(n)).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 10, 9, 8, 11, 16, 13, 12, 15, 28, 17, 26, 19, 22, 21, 14, 23, 46, 25, 18, 51, 34, 29, 36, 31, 82, 27, 20, 35, 76, 37, 24, 33, 64, 41, 56, 43, 40, 69, 30, 47, 136, 49, 50, 39, 52, 53, 126, 55, 100, 45, 32, 59, 106, 61, 38, 111, 244, 65, 66, 67, 58, 57, 78, 71, 226, 73, 42, 99, 70, 77, 86, 79, 190, 249, 44, 83, 166, 85

Offset: 1

Antti Karttunen, Jan 02 2016

nonn

Antti Karttunen, Table of n, a(n) for n = 1..767
Index entries for sequences that are permutations of the natural numbers

Inverse: A266645.
Cf. A000035, A020639, A055396, A003961, A250470.
Related permutations: A249817, A249818, A266403, A266415.

(define (A266646 n) (A250470 (A003961 n)))

a(n) = A250470(A003961(n)).
As a composition of related permutations:
a(n) = A266403(A266415(n)).
Other identities. For all n >= 0:
A000035(a(n)) = A000035(n). [This permutation preserves the parity of n.]
A020639(a(n)) = A020639(n). [More generally, it preserves the smallest prime dividing n.]
A055396(a(n)) = A055396(n).

A265904 Self-inverse permutation of nonnegative integers: a(n) = A263272(A263273(A263272(n))).

Original entry on oeis.org

0, 1, 2, 3, 4, 11, 6, 29, 8, 9, 10, 5, 12, 13, 38, 33, 92, 17, 18, 83, 20, 87, 110, 35, 24, 89, 26, 27, 28, 7, 30, 37, 32, 15, 86, 23, 36, 31, 14, 39, 40, 119, 114, 281, 44, 99, 254, 65, 276, 335, 98, 51, 260, 71, 54, 245, 56, 249, 326, 101, 60, 263, 74, 261, 272, 47, 330, 353, 116, 105, 278, 53, 72, 251, 62, 267, 332, 107, 78, 269, 80, 81, 82, 19

Offset: 0

Antti Karttunen, Jan 02 2016

A263273 conjugated with the permutation obtained from its even bisection.

Antti Karttunen, Table of n, a(n) for n = 0..6561
Index entries for sequences that are permutations of the natural numbers

Cf. A000035, A263272, A263273, A265351, A265352.
Cf. also A265902.
Cf. A265369, A266190, A266401, A266403 (other conjugates or similar derivations of A263273).

f[n_] := Block[{g, h}, g[x_] := x/3^IntegerExponent[x, 3]; h[x_] := x/g@ x; If[n == 0, 0, FromDigits[Reverse@ IntegerDigits[#, 3], 3] &@g[n] h[n]]]; t = Table[f[2 n]/2, {n, 0, 1000}]; Table[t[[f[t[[n + 1]]] + 1]], {n, 0, 83}] (* Michael De Vlieger, Jan 04 2016, after Jean-François Alcover at A263273 *)

from sympy import factorint
from sympy.ntheory.factor_ import digits
from operator import mul
def a030102(n): return 0 if n==0 else int(''.join(map(str, digits(n, 3)[1:][::-1])), 3)
def a038502(n):
    f=factorint(n)
    return 1 if n==1 else reduce(mul, [1 if i==3 else i**f[i] for i in f])
def a038500(n): return n/a038502(n)
def a263273(n): return 0 if n==0 else a030102(a038502(n))*a038500(n)
def a263272(n): return a263273(2*n)/2
def a(n): return a263272(a263273(a263272(n))) # Indranil Ghosh, May 25 2017

(define (A265904 n) (A263272 (A263273 (A263272 n))))

a(n) = A263272(A263273(A263272(n))).
As a composition of related permutations:
a(n) = A263272(A265352(n)).
a(n) = A265351(A263272(n)).
Other identities. For all n >= 0:
a(3*n) = 3*a(n).
A000035(a(n)) = A000035(n). [This permutation preserves the parity of n.]

A266190 Self-inverse permutation of nonnegative integers: a(n) = A264985(A263273(A264985(n))).

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 10, 9, 19, 12, 13, 14, 15, 16, 17, 24, 11, 69, 21, 25, 23, 18, 22, 26, 31, 28, 73, 30, 27, 46, 33, 55, 58, 37, 36, 64, 39, 40, 41, 42, 43, 44, 51, 32, 150, 48, 52, 50, 45, 49, 53, 78, 34, 213, 57, 35, 204, 60, 61, 231, 75, 38, 210, 66, 79, 68, 20, 70, 77, 72, 29, 207, 63, 76, 71, 54, 67, 80, 94, 85, 235

Offset: 0

Antti Karttunen, Jan 02 2016

A263273 conjugated with the permutation obtained from its odd bisection.

Antti Karttunen, Table of n, a(n) for n = 0..6561
Index entries for sequences that are permutations of the natural numbers

Cf. A263273, A264985, A265353, A265354.
Cf. A265369, A265904, A266401, A266403 (other conjugates or similar derivations of A263273).
Cf. also A266189.

f[n_] := Block[{g, h}, g[x_] := x/3^IntegerExponent[x, 3]; h[x_] := x/g@ x; If[n == 0, 0, FromDigits[Reverse@ IntegerDigits[#, 3], 3] &@ g[n] h[n]]]; s = Select[f /@ Range@ 5000, OddQ]; t = Table[(s[[n + 1]] - 1)/2, {n, 0, 1000}]; Table[t[[f[t[[n + 1]]] + 1]], {n, 0, 83}] (* Michael De Vlieger, Jan 04 2016, after Jean-François Alcover at A263273 *)

(define (A266190 n) (A264985 (A263273 (A264985 n))))

a(n) = A264985(A263273(A264985(n))).
As a composition of related permutations:
a(n) = A265353(A264985(n)).
a(n) = A264985(A265354(n)).

A266415 Permutation of natural numbers: a(n) = A250470(A263273(A003961(n))).

Original entry on oeis.org

1, 2, 5, 4, 3, 8, 17, 10, 13, 6, 11, 22, 9, 20, 71, 28, 7, 18, 19, 16, 23, 14, 21, 64, 41, 26, 227, 58, 31, 74, 29, 82, 53, 12, 67, 52, 61, 24, 107, 46, 25, 30, 59, 40, 65, 56, 73, 190, 49, 44, 197, 76, 27, 230, 55, 172, 137, 38, 43, 220, 37, 32, 571, 244, 69, 60, 35, 34, 89, 72, 15, 154, 47, 68, 479, 70

Offset: 1

Antti Karttunen, Jan 02 2016

Antti Karttunen, Table of n, a(n) for n = 1..767
Index entries for sequences that are permutations of the natural numbers

Inverse: A266416.
Cf. A000035, A003961, A250470, A263273.
Related permutations: A048673, A250472, A264985, A264996, A266403, A266646.

(define (A266415 n) (A250470 (A263273 (A003961 n))))

a(n) = A250470(A263273(A003961(n))).
As a composition of related permutations:
a(n) = A266403(A266646(n)).
a(n) = A250472(A264996(A048673(n))) = A250472(1+A264985(-1+A048673(n))).
Other identities. For all n >= 0:
A000035(a(n)) = A000035(n). [This permutation preserves the parity of n.]

A266416 Permutation of natural numbers: a(n) = A064989(A263273(A250469(n))).

Original entry on oeis.org

1, 2, 5, 4, 3, 10, 17, 6, 13, 8, 11, 34, 9, 22, 71, 20, 7, 18, 19, 14, 23, 12, 21, 38, 41, 26, 53, 16, 31, 42, 29, 62, 107, 68, 67, 142, 61, 58, 197, 44, 25, 122, 59, 50, 137, 40, 73, 118, 49, 82, 227, 36, 33, 146, 55, 46, 89, 28, 43, 66, 37, 86, 91, 24, 45, 106, 35, 74, 65, 76, 15, 70, 47, 30, 119, 52

Offset: 1

Antti Karttunen, Jan 02 2016

Antti Karttunen, Table of n, a(n) for n = 1..6562
Index entries for sequences that are permutations of the natural numbers

Inverse: A266415.
Cf. A000035, A064989, A250469, A263273.
Related permutations: A064216, A250471, A264985, A264996, A266403, A266645.

(define (A266416 n) (A064989 (A263273 (A250469 n))))

a(n) = A064989(A263273(A250469(n))).
As a composition of related permutations:
a(n) = A266645(A266403(n)).
a(n) = A064216(A264996(A250471(n))) = A064216(1+A264985(-1+A250471(n))).
Other identities. For all n >= 0:
A000035(a(n)) = A000035(n). [This permutation preserves the parity of n.]

A264996 Self-inverse permutation of natural numbers: a(n) = (1/2) * (1+A263273(2n -1)) = 1 + A264985(n-1).

Original entry on oeis.org

1, 2, 4, 3, 5, 10, 7, 11, 13, 6, 8, 12, 9, 14, 28, 19, 29, 37, 16, 20, 34, 25, 32, 31, 22, 38, 40, 15, 17, 33, 24, 23, 30, 21, 35, 39, 18, 26, 36, 27, 41, 82, 55, 83, 109, 46, 56, 100, 73, 86, 91, 64, 110, 118, 43, 47, 97, 70, 59, 88, 61, 101, 115, 52, 74, 106, 79, 95, 85, 58, 92, 112, 49, 65, 103, 76

Offset: 1

Antti Karttunen, Jan 04 2016

Antti Karttunen, Table of n, a(n) for n = 1..6561
Index entries for sequences that are permutations of the natural numbers

Cf. A263273, A264985.

Cf. also A266401, A266403, A266415, A266416.

f[n_] := Block[{g, h}, g[x_] := x/3^IntegerExponent[x, 3]; h[x_] := x/g@ x; If[n == 0, 0, FromDigits[Reverse@ IntegerDigits[#, 3], 3] &@ g[n] h[n]]]; Array[(1/2) (1 + f[2 # - 1]) &, {76}] (* Michael De Vlieger, Jan 04 2016, after Jean-François Alcover at A263273 *)

a(n) = (1/2) * (1+A263273((2*n)-1)).

a(n) = 1 + A264985(n-1).

A266404 Self-inverse permutation of natural numbers: a(n) = A250470(A030101(A250469(n))).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 11, 8, 17, 10, 7, 12, 13, 20, 25, 16, 9, 18, 23, 14, 53, 22, 19, 28, 15, 36, 27, 24, 29, 40, 37, 32, 33, 34, 83, 26, 31, 42, 51, 30, 47, 38, 59, 44, 101, 76, 41, 60, 73, 68, 39, 52, 21, 84, 107, 56, 131, 72, 43, 48, 89, 80, 125, 64, 65, 66, 109, 50, 99, 82, 71, 58, 49, 74, 151, 46, 239, 78, 97, 62, 173, 70, 35, 54

Offset: 1

Antti Karttunen, Jan 02 2016

Antti Karttunen, Table of n, a(n) for n = 1..4096
Index entries for sequences that are permutations of the natural numbers

Cf. A000035, A030101, A057889, A250469, A250470.
Cf. A265329, A266402 (other conjugates or similar derivations of A057889).
Cf. also A266403.

(define (A266404 n) (A250470 (A030101 (A250469 n))))

a(n) = A250470(A030101(A250469(n))) = A250470(A057889(A250469(n))).
Other identities. For all n >= 0:
A000035(a(n)) = A000035(n). [This permutation preserves the parity of n.]

cp's OEIS Frontend

A265369 Self-inverse permutation of nonnegative integers: a(n) = A057889(A263273(A057889(n))).

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A266645 Permutation of natural numbers: a(n) = A064989(A250469(n)).

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A266401 Self-inverse permutation of natural numbers: a(n) = A064989(A263273(A003961(n))).

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A266646 Permutation of natural numbers: a(n) = A250470(A003961(n)).

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A265904 Self-inverse permutation of nonnegative integers: a(n) = A263272(A263273(A263272(n))).

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A266190 Self-inverse permutation of nonnegative integers: a(n) = A264985(A263273(A264985(n))).

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A266415 Permutation of natural numbers: a(n) = A250470(A263273(A003961(n))).

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A266416 Permutation of natural numbers: a(n) = A064989(A263273(A250469(n))).

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