A331166 -id:A331166 - cp's OEIS frontend

A331746 Lexicographically earliest infinite sequence such that a(i) = a(j) => A009194(i) = A009194(j) and A331166(i) = A331166(j) for all i, j.

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 11, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 18, 21, 24, 25, 22, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 33, 37, 38, 39, 40, 41, 42, 43, 31, 44, 45, 46, 38, 47, 48, 49, 35, 41, 48, 50, 42, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 63, 71, 72, 73, 74, 75, 76, 66, 77, 78, 79, 80, 81, 82, 83, 56

Offset: 1

Antti Karttunen, Feb 04 2020

nonn

Restricted growth sequence transform of the ordered pair [A009194(n), A331166(n)].
For all i, j:
   a(i) = a(j) => A331747(i) = A331747(j).

Cf. A009194, A331166, A331300.

Cf. also A331744, A331747.

\\ Needs also code from A331166.
up_to = 65537;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
A009194(n) = gcd(n, sigma(n));
Aux331746(n) = [A009194(n),A331166(n)];
v331746 = rgs_transform(vector(up_to, n, Aux331746(n)));
A331746(n) = v331746[n];

A057889 Bijective bit-reverse of n: keep the trailing zeros in the binary expansion of n fixed, but reverse all the digits up to that point.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 13, 12, 11, 14, 15, 16, 17, 18, 25, 20, 21, 26, 29, 24, 19, 22, 27, 28, 23, 30, 31, 32, 33, 34, 49, 36, 41, 50, 57, 40, 37, 42, 53, 52, 45, 58, 61, 48, 35, 38, 51, 44, 43, 54, 59, 56, 39, 46, 55, 60, 47, 62, 63, 64, 65, 66, 97, 68, 81, 98, 113

Offset: 0

Marc LeBrun, Sep 25 2000

The original name was "Bit-reverse of n, including as many leading as trailing zeros." - Antti Karttunen, Dec 25 2024
A permutation of integers consisting only of fixed points and pairs. a(n)=n when n is a binary palindrome (including as many leading as trailing zeros), otherwise a(n)=A003010(n) (i.e. n has no axis of symmetry). A057890 gives the palindromes (fixed points, akin to A006995) while A057891 gives the "antidromes" (pairs). See also A280505.
This is multiplicative in domain GF(2)[X], i.e. with carryless binary arithmetic. A193231 is another such permutation of natural numbers. - Antti Karttunen, Dec 25 2024

			a(6)=6 because 0110 is a palindrome, but a(11)=13 because 1011 reverses into 1101.

N. J. A. Sloane, Table of n, a(n) for n = 0..16384, May 30 2016 [First 8192 terms from _Ivan Neretin_, Jul 09 2015]
Index entries for sequences related to binary expansion of n
Index entries for sequences related to polynomials in ring GF(2)[X]
Index entries for sequences that are permutations of the natural numbers

Cf. A030101, A000265, A006519, A006995, A057890, A057891, A280505, A280508, A331166 [= min(n,a(n))], A366378 [k for which a(k) = k (mod 3)], A369044 [= A014963(a(n))].
Similar permutations for other bases: A263273 (base-3), A264994 (base-4), A264995 (base-5), A264979  (base-9).
Other related (binary) permutations: A056539, A193231.
Compositions of this permutation with other binary (or other base-related) permutations: A264965, A264966, A265329, A265369, A379471, A379472.
Compositions with permutations involving prime factorization: A245450, A245453, A266402, A266404, A293448, A366275, A366276.
Other derived permutations: A246200 [= a(3*n)/3], A266351, A302027, A302028, A345201, A356331, A356332, A356759, A366389.
See also A235027 (which is not a permutation).

Table[FromDigits[Reverse[IntegerDigits[n, 2]], 2]*2^IntegerExponent[n, 2], {n, 71}] (* Ivan Neretin, Jul 09 2015 *)

A030101(n) = if(n<1,0,subst(Polrev(binary(n)),x,2));
A057889(n) = if(!n,n,A030101(n/(2^valuation(n,2))) * (2^valuation(n, 2))); \\ Antti Karttunen, Dec 25 2024

def a(n):
    x = bin(n)[2:]
    y = x[::-1]
    return int(str(int(y))+(len(x) - len(str(int(y))))*'0', 2)
print([a(n) for n in range(101)]) # Indranil Ghosh, Jun 11 2017

def A057889(n): return int(bin(n>>(m:=(~n&n-1).bit_length()))[-1:1:-1],2)<Chai Wah Wu, Dec 25 2024

a(n) = A030101(A000265(n)) * A006519(n), with a(0)=0.

Clarified the name with May 30 2016 comment from N. J. A. Sloane, and moved the old name to the comments - Antti Karttunen, Dec 25 2024

A331300 Lexicographically earliest infinite sequence such that a(i) = a(j) => f(i) = f(j), where f(n) = min(n, A057889(n)), and A057889 is a bijective base-2 reverse.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 12, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 19, 22, 25, 26, 23, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 34, 38, 39, 40, 41, 42, 43, 44, 32, 35, 45, 40, 39, 46, 47, 48, 36, 42, 47, 49, 43, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 57, 62, 69, 70, 71, 72, 73, 74, 65, 75, 76, 77, 78, 79, 80, 81, 55, 58, 82, 64, 69, 83, 84, 74, 63

Offset: 0

Antti Karttunen, Jan 18 2020

Restricted growth sequence transform of A331166. See comments in that sequence.

Antti Karttunen, Table of n, a(n) for n = 0..100000
Index entries for sequences related to binary expansion of n

Cf. A057889, A331166.

Cf. also A324400, A331303, A305801, A305801, A305900, A295300 for other "top level" filtering sequences.

up_to = 100000;
rgs_transform(invec) = { my(om = Map(), outvec = vector(length(invec)), u=1); for(i=1, length(invec), if(mapisdefined(om,invec[i]), my(pp = mapget(om, invec[i])); outvec[i] = outvec[pp] , mapput(om,invec[i],i); outvec[i] = u; u++ )); outvec; };
A030101(n) = if(n<1,0,subst(Polrev(binary(n)),x,2));
A057889(n) = if(!n,n,A030101(n/(2^valuation(n,2))) * (2^valuation(n, 2)));
A331166(n) = min(n, A057889(n));
v331300 = rgs_transform(vector(1+up_to,n,A331166(n-1)));
A331300(n) = v331300[1+n];
for(n=0,up_to,write("b331300.txt", n, " ", A331300(n)));

A331167 a(n) = min(n, A193231(n)), where A193231(n) is blue code of n.

Original entry on oeis.org

0, 1, 2, 2, 4, 4, 6, 7, 8, 9, 10, 11, 10, 11, 9, 8, 16, 16, 18, 19, 20, 21, 22, 22, 24, 25, 26, 27, 27, 26, 24, 25, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 34, 35, 33, 32, 39, 38, 36, 37, 45, 44, 46, 47, 40, 41, 43, 42, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 68, 69, 71, 70, 65, 64, 66, 67, 75, 74, 72, 73, 78, 79, 77, 76

Offset: 0

Antti Karttunen, Jan 12 2020

nonn

For all i, j > 0: a(i) = a(j) => A280501(i) = A280501(j).

Antti Karttunen, Table of n, a(n) for n = 0..16383
Antti Karttunen, Data supplement: n, a(n) computed for n = 0..65537
Index entries for sequences related to binary expansion of n
Index entries for sequences related to polynomials in ring GF(2)[X]

Cf. A193231, A280501.

Cf. also A331166.

A331167(n) = { my(x='x); min(n,subst(lift(Mod(1, 2)*subst(Pol(binary(n), x), x, 1+x)), x, 2)); };

a(n) = min(n, A193231(n)).

A331173 a(n) = min(n, A263273(n)), where A263273 is bijective base-3 reverse.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 5, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 11, 20, 15, 14, 23, 24, 17, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 31, 38, 39, 40, 41, 42, 43, 44, 45, 34, 47, 48, 43, 50, 51, 52, 53, 54, 29, 56, 33, 38, 59, 60, 47, 62, 45, 32, 59, 42, 41, 68, 69, 50, 71, 72, 35, 62, 51, 44, 71, 78, 53, 80, 81

Offset: 0

Antti Karttunen, Jan 12 2020

For all i, j:
  a(i) = a(j) => A290094(i) = A290094(j).
For all i, j > 0:
  a(i) = a(j) => A007949(i) = A007949(j).

Antti Karttunen, Table of n, a(n) for n = 0..59048

Cf. A007949, A030102, A263273, A290094.

Cf. also A330740, A331166.

A030102(n) = { my(r=[n%3]); while(0A030102.
A263273 = n -> if(!n,n,A030102(n/(3^valuation(n,3))) * (3^valuation(n, 3)));
A331173(n) = min(n, A263273(n));

cp's OEIS Frontend

A331746 Lexicographically earliest infinite sequence such that a(i) = a(j) => A009194(i) = A009194(j) and A331166(i) = A331166(j) for all i, j.

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A057889 Bijective bit-reverse of n: keep the trailing zeros in the binary expansion of n fixed, but reverse all the digits up to that point.

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A331300 Lexicographically earliest infinite sequence such that a(i) = a(j) => f(i) = f(j), where f(n) = min(n, A057889(n)), and A057889 is a bijective base-2 reverse.

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A331167 a(n) = min(n, A193231(n)), where A193231(n) is blue code of n.

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A331173 a(n) = min(n, A263273(n)), where A263273 is bijective base-3 reverse.

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