A193231 -id:A193231 - cp's OEIS frontend

A268386 a(n) = A193231(A268387(n)).

0, 1, 1, 3, 1, 0, 1, 2, 3, 0, 1, 2, 1, 0, 0, 5, 1, 2, 1, 2, 0, 0, 1, 3, 3, 0, 2, 2, 1, 1, 1, 4, 0, 0, 0, 0, 1, 0, 0, 3, 1, 1, 1, 2, 2, 0, 1, 4, 3, 2, 0, 2, 1, 3, 0, 3, 0, 0, 1, 3, 1, 0, 2, 6, 0, 1, 1, 2, 0, 1, 1, 1, 1, 0, 2, 2, 0, 1, 1, 4, 5, 0, 1, 3, 0, 0, 0, 3, 1, 3, 0, 2, 0, 0, 0, 5, 1, 2, 2, 0, 1, 1, 1, 3, 1, 0, 1, 1, 1, 1, 0, 4, 1, 1, 0, 2, 2, 0, 0, 2

Offset: 1

Antti Karttunen, Feb 10 2016

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16384

A003987, A048720, A059897, A193231, A268385, A268387 are used in definitions of this sequence.

Cf. A000028 (indices of odd numbers), A000379 (indices of even numbers), A268390 (indices of zeros).

f[n_] := Which[0 <= # <= 1, #, EvenQ@ #, BitXor[2 #, #] &[f[#/2]], True, BitXor[#, 2 # + 1] &[f[(# - 1)/2]]] &@ Abs@ n; {0}~Join~Table[f[BitXor @@ Map[Last, FactorInteger@ n]], {n, 2, 120}] (* Michael De Vlieger, Feb 12 2016, after Robert G. Wilson v at A048724 and A065621 *)

a268387(n) = {my(f = factor(n), b = 0); for (k=1, #f~, b = bitxor(b, f[k, 2]); ); b; }
a193231(n) = {my(x='x); subst(lift(Mod(1, 2)*subst(Pol(binary(n), x), x, 1+x)), x, 2)};
a(n) = a193231(a268387(n)); \\ Michel Marcus, May 09 2020

(define (A268386 n) (A193231 (A268387 n)))

The following two formulas are equivalent because A193231 distributes over bitwise XOR (A003987):
a(n) = A193231(A268387(n)) and
a(n) = A268387(A268385(n)).
a(2^k) = A193231(k). - Peter Munn, May 07 2020
From Peter Munn, Jun 02 2020: (Start)
Alternative definition, for n, k >= 1, where XOR denotes A003987:
a(prime(n)) = 1, where prime(n) = A000040(n);
a(n^2) = a(n) XOR (2 * a(n)) = A048720(a(n), 3);
a(A059897(n, k)) = a(n) XOR a(k).
(End)

A286601 a(n) = A278222(A193231(n)).

Original entry on oeis.org

1, 2, 4, 2, 6, 2, 4, 8, 16, 8, 4, 12, 6, 12, 6, 2, 6, 2, 6, 12, 6, 30, 24, 12, 16, 32, 24, 8, 36, 12, 4, 12, 36, 12, 4, 12, 36, 72, 60, 12, 16, 48, 64, 32, 24, 8, 24, 72, 6, 12, 6, 2, 24, 12, 6, 30, 60, 12, 24, 48, 6, 30, 60, 30, 210, 30, 60, 120, 6, 30, 60, 30, 60, 180, 60, 12, 96, 48, 24, 120, 6, 30, 24, 12, 6, 2, 6, 12, 60, 30, 6, 30

Offset: 0

Antti Karttunen, Jun 04 2017

Antti Karttunen, Table of n, a(n) for n = 0..8191
Indranil Ghosh, Python program to generate the sequence
Index entries for sequences related to binary expansion of n

Cf. A193231, A234022, A278222, A278231, A278233, A286602 (rgs-version of this sequence).

(define (A286601 n) (A278222 (A193231 n)))

a(n) = A278222(A193231(n)).

A302793 Permutation of nonnegative integers: a(0) = 0; for n >= 1, a(n) = A193231(1+A193231(n-1)), where A193231(n) is blue code of n.

Original entry on oeis.org

0, 1, 3, 5, 2, 6, 4, 7, 15, 17, 8, 11, 9, 13, 10, 12, 14, 18, 16, 19, 20, 21, 23, 30, 22, 25, 51, 24, 26, 27, 28, 31, 29, 39, 32, 35, 33, 37, 45, 36, 38, 41, 43, 85, 42, 46, 44, 47, 40, 49, 54, 48, 50, 60, 52, 55, 53, 58, 56, 59, 34, 61, 63, 57, 62, 66, 64, 67, 75, 69, 71, 65, 70, 73, 78, 72, 74, 102, 76, 79, 77, 81

Offset: 0

Antti Karttunen, Apr 26 2018

Antti Karttunen, Table of n, a(n) for n = 0..16383
David Newman, et al., New sequences from old, Discussion on SeqFan-mailing list, January 2018.
Index entries for sequences related to binary expansion of n
Index entries for sequences that are permutations of the natural numbers

Cf. A302794 (inverse).
Cf. A193231, A302795.
Cf. also A268717, A302027.

A193231(n) = { my(x='x); subst(lift(Mod(1, 2)*subst(Pol(binary(n), x), x, 1+x)), x, 2) }; \\ From A193231
A302793(n) = if(!n,n,A193231(1+A193231(n-1)));

a(0) = 0; for n >= 1, a(n) = A193231(1+A193231(n-1)).

A234613 Self-inverse permutation of nonnegative integers, "gray-blue" code: a(n) = A193231(A003188(n)).

Original entry on oeis.org

0, 1, 2, 3, 6, 7, 4, 5, 10, 11, 8, 9, 12, 13, 14, 15, 30, 31, 28, 29, 24, 25, 26, 27, 20, 21, 22, 23, 18, 19, 16, 17, 34, 35, 32, 33, 36, 37, 38, 39, 40, 41, 42, 43, 46, 47, 44, 45, 60, 61, 62, 63, 58, 59, 56, 57, 54, 55, 52, 53, 48, 49, 50, 51, 102, 103, 100

Offset: 0

Antti Karttunen, Dec 28 2013

nonn

Antti Karttunen, Table of n, a(n) for n = 0..8191
Joerg Arndt, Matters Computational (The Fxtbook), section 1.19 "Invertible transforms on words", pp. 49--55 [Cf. especially the identity 1.19-9d: gB = Bg^{-1} given on page 53].
Index entries for sequences that are permutations of the natural numbers

Cf. A003188, A006068, A193231, A234612, A234024-A234027.

(define (A234613 n) (A193231 (A003188 n)))
(define (A234613v2 n) (A006068 (A193231 n))) ;; Alternative two.

a(n) = A193231(A003188(n)).
a(n) = A006068(A193231(n)).
a(n) = A193231(A234612(A193231(n))).

A278231 Least number with the same prime signature as the n-th number in Blue-code: a(n) = A046523(A193231(n)).

Original entry on oeis.org

1, 2, 2, 2, 4, 6, 2, 6, 6, 12, 2, 6, 2, 4, 8, 2, 16, 12, 2, 12, 6, 2, 6, 30, 2, 2, 12, 8, 6, 24, 4, 6, 12, 48, 4, 24, 6, 2, 12, 60, 2, 12, 6, 6, 24, 6, 2, 6, 6, 6, 32, 6, 6, 36, 2, 12, 12, 6, 2, 24, 2, 2, 30, 6, 60, 6, 6, 48, 16, 2, 6, 60, 6, 2, 24, 6, 6, 12, 6, 12, 6, 2, 30, 6, 64, 30, 2, 12, 6, 72, 2, 30, 2

Offset: 1

Antti Karttunen, Nov 16 2016

nonn

Antti Karttunen, Table of n, a(n) for n = 1..65537
Indranil Ghosh, Python program to generate the sequence
Index entries for sequences computed from exponents in factorization of n

Cf. A046523, A193231, A278233.

(define (A278231 n) (A046523 (A193231 n)))

a(n) = A046523(A193231(n)).

A280507 a(n) = n XOR A193231(n).

Original entry on oeis.org

0, 0, 1, 1, 1, 1, 0, 0, 7, 7, 6, 6, 6, 6, 7, 7, 1, 1, 0, 0, 0, 0, 1, 1, 6, 6, 7, 7, 7, 7, 6, 6, 19, 19, 18, 18, 18, 18, 19, 19, 20, 20, 21, 21, 21, 21, 20, 20, 18, 18, 19, 19, 19, 19, 18, 18, 21, 21, 20, 20, 20, 20, 21, 21, 21, 21, 20, 20, 20, 20, 21, 21, 18, 18, 19, 19, 19, 19, 18, 18, 20, 20, 21, 21, 21, 21, 20, 20, 19, 19, 18, 18, 18, 18, 19, 19, 6, 6

Offset: 0

Antti Karttunen, Jan 09 2017

Cf. A003987, A193231, A280501, A280502, A280508.

Cf. A118666 (positions of zeros).

A193231(n) = { my(x='x); subst(lift(Mod(1, 2)*subst(Pol(binary(n), x), x, 1+x)), x, 2) }; \\ From A193231
A280507(n) = bitxor(n,A193231(n)); \\ Antti Karttunen, Feb 15 2021

(define (A280507 n) (A003987bi n (A193231 n))) ;; Where A003987bi implements A003987 (bitwise-XOR).

a(n) = A003987(n,A193231(n)) = n XOR A193231(n).
Other identities. For all n >= 0:
a(A193231(n)) = a(n).

A286600 a(n) = A286622(A193231(n)).

Original entry on oeis.org

1, 2, 3, 2, 4, 2, 3, 5, 7, 5, 3, 6, 4, 6, 4, 2, 4, 2, 4, 6, 4, 8, 9, 6, 7, 11, 9, 5, 10, 6, 3, 6, 10, 6, 3, 6, 10, 14, 12, 6, 7, 13, 15, 11, 9, 5, 9, 14, 4, 6, 4, 2, 9, 6, 4, 8, 12, 6, 9, 13, 4, 8, 12, 8, 16, 8, 12, 17, 4, 8, 12, 8, 12, 18, 12, 6, 19, 13, 9, 17, 4, 8, 9, 6, 4, 2, 4, 6, 12, 8, 4, 8, 9, 13, 12, 6, 10, 14, 12, 6, 10, 6, 3, 6

Offset: 0

Antti Karttunen, Jun 04 2017

nonn

Antti Karttunen, Table of n, a(n) for n = 0..65535

Cf. A193231, A286602, A286622.

a(n) = A286622(A193231(n)).

A302794 Permutation of nonnegative integers: a(0) = 0; for n >= 1, a(n) = 1+A193231(A193231(n)-1), where A193231(n) is blue code of n.

Original entry on oeis.org

0, 1, 4, 2, 6, 3, 5, 7, 10, 12, 14, 11, 15, 13, 16, 8, 18, 9, 17, 19, 20, 21, 24, 22, 27, 25, 28, 29, 30, 32, 23, 31, 34, 36, 60, 35, 39, 37, 40, 33, 48, 41, 44, 42, 46, 38, 45, 47, 51, 49, 52, 26, 54, 56, 50, 55, 58, 63, 57, 59, 53, 61, 64, 62, 66, 71, 65, 67, 94, 69, 72, 70, 75, 73, 76, 68, 78, 80, 74, 79, 88, 81, 84, 82, 86, 43

Offset: 0

Antti Karttunen, Apr 26 2018

Antti Karttunen, Table of n, a(n) for n = 0..16383
David Newman, et al., New sequences from old, Discussion on SeqFan-mailing list, January 2018.
Index entries for sequences related to binary expansion of n
Index entries for sequences that are permutations of the natural numbers

Cf. A302793 (inverse).
Cf. A193231.
Cf. also A268718, A302028.

A193231(n) = { my(x='x); subst(lift(Mod(1, 2)*subst(Pol(binary(n), x), x, 1+x)), x, 2) }; \\ From A193231
A302794(n) = if(!n,n,1+A193231(A193231(n)-1));

a(0) = 0; for n >= 1, a(n) = 1+A193231(A193231(n)-1).

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)).

A379471 Composition of bijective bit-reverse and Blue code, in this order: a(n) = A193231(A057889(n)).

Original entry on oeis.org

0, 1, 3, 2, 5, 4, 6, 7, 15, 14, 12, 11, 10, 13, 9, 8, 17, 16, 18, 31, 20, 21, 29, 26, 30, 19, 23, 28, 27, 22, 24, 25, 51, 50, 48, 35, 54, 61, 33, 44, 60, 55, 63, 38, 39, 56, 46, 41, 34, 49, 53, 32, 57, 62, 36, 47, 45, 52, 58, 37, 40, 59, 43, 42, 85, 84, 86, 103, 80, 69, 101, 118, 90, 91, 71, 104, 99, 74, 116, 121

Offset: 0

Antti Karttunen, Dec 25 2024

Cf. A057889, A193231, A379472 (inverse permutation).

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)));
A193231(n) = { my(x='x); subst(lift(Mod(1, 2)*subst(Pol(binary(n), x), x, 1+x)), x, 2) };
A379471(n) = A193231(A057889(n));

cp's OEIS Frontend

A268386 a(n) = A193231(A268387(n)).

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A286601 a(n) = A278222(A193231(n)).

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A302793 Permutation of nonnegative integers: a(0) = 0; for n >= 1, a(n) = A193231(1+A193231(n-1)), where A193231(n) is blue code of n.

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A234613 Self-inverse permutation of nonnegative integers, "gray-blue" code: a(n) = A193231(A003188(n)).

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A278231 Least number with the same prime signature as the n-th number in Blue-code: a(n) = A046523(A193231(n)).

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A280507 a(n) = n XOR A193231(n).

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A286600 a(n) = A286622(A193231(n)).

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A302794 Permutation of nonnegative integers: a(0) = 0; for n >= 1, a(n) = 1+A193231(A193231(n)-1), where A193231(n) is blue code of n.

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

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