A280505 -id:A280505 - cp's OEIS frontend

A280506 Nonpalindromic part of n in base 2 (with carryless GF(2)[X] factorization): a(n) = A280500(n,A280505(n)).

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 11, 1, 13, 1, 1, 1, 1, 1, 19, 1, 1, 11, 13, 1, 25, 13, 1, 1, 11, 1, 1, 1, 1, 1, 13, 1, 37, 19, 11, 1, 41, 1, 25, 11, 1, 13, 47, 1, 11, 25, 1, 13, 19, 1, 55, 1, 13, 11, 59, 1, 61, 1, 1, 1, 1, 1, 67, 1, 69, 13, 61, 1, 1, 37, 13, 19, 59, 11, 25, 1, 81, 41, 11, 1, 1, 25, 87, 11, 55, 1, 91, 13, 1, 47, 19, 1, 97, 11, 1, 25, 13, 1, 103

Offset: 1

Antti Karttunen, Jan 09 2017

a(n) = number obtained when the maximal base-2 palindromic divisor of n, A280505(n), is divided out of n with carryless GF(2)[X] factorization (see examples of A280500 for the explanation).

Apart from 1, all terms are present in A164861 (form their proper subset).

Cf. A000265, A048720, A057889, A057891, A164861, A280500, A280502, A280504, A280508, A280509.

Cf. A057890 (positions of ones).

(define (A280506 n) (A280500bi n (A280505 n))) ;; Code for A280500bi given in A280500.

a(n) = A280500(n,A280505(n)).
Other identities. For all n >= 1:
a(2n) = a(A000265(n)) = a(n).
A048720(a(n), A280505(n)) = n.

Erroneous claim removed from comments by Antti Karttunen, May 13 2018

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

A091255 Square array computed from gcd(P(x),P(y)) where P(x) and P(y) are polynomials with coefficients in {0,1} given by the binary expansions of x and y, and the polynomial calculation is done over GF(2), with the result converted back to a binary number, and then expressed in decimal. Array is symmetric, and is read by falling antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 3, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 3, 2, 1, 1, 1, 3, 1, 1, 3, 1, 1, 1, 2, 1, 2, 5, 2, 1, 2, 1, 1, 1, 1, 1, 3, 3, 1, 1, 1, 1, 1, 2, 3, 4, 1, 6, 1, 4, 3, 2, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 2, 1, 2, 3, 2, 7, 2, 3, 2, 1, 2, 1, 1, 1, 3, 1, 5, 3, 1, 1, 3, 5, 1, 3, 1, 1

Offset: 1

Antti Karttunen, Jan 03 2004

Array is read by antidiagonals, with (x,y) = (1,1), (1,2), (2,1), (1,3), (2,2), (3,1), ...
Analogous to A003989.
"Coded in binary" means that a polynomial a(n)*X^n+...+a(0)*X^0 over GF(2) is represented by the binary number a(n)*2^n+...+a(0)*2^0 in Z (where a(k)=0 or 1).

			The top left 17 X 17 corner of the array:
      1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17
    +---------------------------------------------------------------
   1: 1, 1, 1, 1, 1, 1, 1, 1, 1,  1,  1,  1,  1,  1,  1,  1,  1, ...
   2: 1, 2, 1, 2, 1, 2, 1, 2, 1,  2,  1,  2,  1,  2,  1,  2,  1, ...
   3: 1, 1, 3, 1, 3, 3, 1, 1, 3,  3,  1,  3,  1,  1,  3,  1,  3, ...
   4: 1, 2, 1, 4, 1, 2, 1, 4, 1,  2,  1,  4,  1,  2,  1,  4,  1, ...
   5: 1, 1, 3, 1, 5, 3, 1, 1, 3,  5,  1,  3,  1,  1,  5,  1,  5, ...
   6: 1, 2, 3, 2, 3, 6, 1, 2, 3,  6,  1,  6,  1,  2,  3,  2,  3, ...
   7: 1, 1, 1, 1, 1, 1, 7, 1, 7,  1,  1,  1,  1,  7,  1,  1,  1, ...
   8: 1, 2, 1, 4, 1, 2, 1, 8, 1,  2,  1,  4,  1,  2,  1,  8,  1, ...
   9: 1, 1, 3, 1, 3, 3, 7, 1, 9,  3,  1,  3,  1,  7,  3,  1,  3, ...
  10: 1, 2, 3, 2, 5, 6, 1, 2, 3, 10,  1,  6,  1,  2,  5,  2,  5, ...
  11: 1, 1, 1, 1, 1, 1, 1, 1, 1,  1, 11,  1,  1,  1,  1,  1,  1, ...
  12: 1, 2, 3, 4, 3, 6, 1, 4, 3,  6,  1, 12,  1,  2,  3,  4,  3, ...
  13: 1, 1, 1, 1, 1, 1, 1, 1, 1,  1,  1,  1, 13,  1,  1,  1,  1, ...
  14: 1, 2, 1, 2, 1, 2, 7, 2, 7,  2,  1,  2,  1, 14,  1,  2,  1, ...
  15: 1, 1, 3, 1, 5, 3, 1, 1, 3,  5,  1,  3,  1,  1, 15,  1, 15, ...
  16: 1, 2, 1, 4, 1, 2, 1, 8, 1,  2,  1,  4,  1,  2,  1, 16,  1, ...
  17: 1, 1, 3, 1, 5, 3, 1, 1, 3,  5,  1,  3,  1,  1,  15, 1, 17, ...
  ...
3, which is "11" in binary, encodes polynomial X + 1, while 7 ("111" in binary) encodes polynomial X^2 + X + 1, whereas 9 ("1001" in binary), encodes polynomial X^3 + 1. Now (X + 1)(X^2 + X + 1) = (X^3 + 1) when the polynomials are multiplied over GF(2), or equally, when multiplication of integers 3 and 7 is done as a carryless base-2 product (A048720(3,7) = 9). Thus it follows that A(3,9) = A(9,3) = 3 and A(7,9) = A(9,7) = 7.
Furthermore, 5 ("101" in binary) encodes polynomial X^2 + 1 which is equal to (X + 1)(X + 1) in GF(2)[X], thus A(5,9) = A(9,5) = 3, as the irreducible polynomial (X + 1) is the only common factor for polynomials X^2 + 1 and X^3 + 1.

Cf. A003987, A048720, A091256, A091257, A106449, A280500, A280501, A280503, A280505, A286153, A325634, A325635, A325825.

Cf. also A327856 (the upper left triangular section of this array), A327857.

A091255sq(a,b) = fromdigits(Vec(lift(gcd(Pol(binary(a))*Mod(1, 2),Pol(binary(b))*Mod(1, 2)))),2); \\ Antti Karttunen, Aug 12 2019

A(x,y) = A(y,x) = A(x, A003987(x,y)) = A(A003987(x,y), y), where A003987 gives the bitwise-XOR of its two arguments. - Antti Karttunen, Sep 28 2019

Data section extended up to a(105), examples added by Antti Karttunen, Sep 28 2019

A280508 a(n) = n XOR A057889(n).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 6, 0, 6, 0, 0, 0, 0, 0, 10, 0, 0, 12, 10, 0, 10, 12, 0, 0, 10, 0, 0, 0, 0, 0, 18, 0, 12, 20, 30, 0, 12, 0, 30, 24, 0, 20, 18, 0, 18, 20, 0, 24, 30, 0, 12, 0, 30, 20, 12, 0, 18, 0, 0, 0, 0, 0, 34, 0, 20, 36, 54, 0, 0, 24, 34, 40, 20, 60, 54, 0, 20, 24, 54, 0, 0, 60, 34, 48, 20, 0, 54, 40, 0, 36, 34, 0, 34, 36, 0, 40, 54, 0, 20, 48

Offset: 0

Antti Karttunen, Jan 09 2017

Cf. A003987, A057889, A280505, A280506, A280507, A280509.

Cf. A057890 (positions of zeros).

(define (A280508 n) (A003987bi n (A057889 n))) ;; Where A003987bi implements A003987 (bitwise-XOR).

a(n) = A003987(n,A057889(n)) = n XOR A057889(n).
Other identities. For all n >= 0:
a(A057889(n)) = a(n).

A280501 "Blue kernel" of n: a(n) = A091255(n, A193231(n)).

Original entry on oeis.org

1, 1, 1, 1, 1, 6, 7, 1, 7, 6, 1, 6, 1, 7, 1, 1, 1, 18, 19, 20, 21, 1, 1, 6, 1, 1, 7, 7, 1, 6, 1, 1, 1, 6, 7, 18, 1, 19, 1, 20, 1, 21, 1, 1, 7, 6, 1, 6, 7, 1, 1, 1, 19, 18, 1, 7, 1, 6, 1, 20, 1, 1, 21, 1, 21, 6, 1, 20, 1, 7, 1, 18, 1, 1, 1, 19, 1, 6, 7, 20, 1, 1, 7, 21, 1, 6, 1, 1, 1, 18, 1, 6, 7, 1, 19, 6, 1, 7, 1, 1, 7, 6, 1, 1, 1, 106, 107, 108, 109, 1, 1, 7

Offset: 1

Antti Karttunen, Jan 09 2017

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537
Index entries for sequences related to binary expansion of n
Index entries for sequences operating on polynomials in ring GF(2)[X]

Cf. A048720, A091255, A193231, A280502, A280505, A331167.

A091255sq(a,b) = fromdigits(Vec(lift(gcd(Pol(binary(a))*Mod(1, 2),Pol(binary(b))*Mod(1, 2)))),2);
A193231(n) = { my(x='x); subst(lift(Mod(1, 2)*subst(Pol(binary(n), x), x, 1+x)), x, 2) }; \\ From A193231
A280501(n) = A091255sq(n, A193231(n)); \\ Antti Karttunen, Jan 12 2020

(define (A280501 n) (A091255bi n (A193231 n))) ;; A091255bi implements the 2-argument GF(2)[X] GCD-function (A091255).

a(n) = A091255(n, A193231(n)).
Other identities. For all n >= 1:
a(A193231(n)) = a(n).
A048720(a(n), A280502(n)) = n.

A331166 a(n) = min(n, A057889(n)), where A057889 is bijective base-2 reverse.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 11, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 19, 22, 27, 28, 23, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 37, 42, 43, 44, 45, 46, 47, 48, 35, 38, 51, 44, 43, 54, 55, 56, 39, 46, 55, 60, 47, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 69, 74, 83, 84, 85, 86, 87, 88, 77, 90, 91, 92, 93, 94, 95, 96, 67, 70

Offset: 0

Antti Karttunen, Jan 12 2020

There is a large number of sequences b, related to binary expansion of n (A007088), for which it holds that b(n) = b(a(n)) for all n >= 0 (or n >= 1). For example, we have:
For all i, j:
  a(i) = a(j) => A002487(i) = A002487(j),
  a(i) = a(j) => A005811(i) = A005811(j),
  a(i) = a(j) => A286622(i) = A286622(j) => A000120(i) = A000120(j).
For all i, j > 0:
  a(i) = a(j) => A007814(i) = A007814(j),
  a(i) = a(j) => A280505(i) = A280505(j),
  a(i) = a(j) => A305788(i) = A305788(j) => A091222(i) = A091222(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

Cf. A000120, A002487, A005811, A007088, A007814, A030101, A057889, A091222, A280505, A286622, A305788.

Cf. also A330740, A331167, A331173.

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));

a(n) = min(n, A057889(n)).

A280503 a(n) = A091255(n,A056539(n)).

Original entry on oeis.org

1, 2, 3, 2, 5, 2, 7, 2, 9, 10, 1, 12, 1, 2, 15, 2, 17, 2, 1, 2, 21, 2, 3, 4, 1, 2, 27, 4, 3, 2, 31, 2, 33, 6, 7, 18, 1, 38, 5, 6, 1, 42, 3, 2, 45, 6, 1, 12, 7, 2, 51, 52, 3, 18, 1, 56, 5, 6, 1, 12, 1, 2, 63, 2, 65, 2, 1, 2, 1, 2, 3, 2, 73, 2, 15, 2, 3, 2, 7, 2, 1, 2, 9, 2, 85, 2, 1, 2, 3, 2, 1, 2, 93, 2, 5, 4, 1, 2, 99, 4, 9, 2, 1, 4, 15, 2, 107, 4, 1, 2, 3, 8

Offset: 1

Antti Karttunen, Jan 09 2017

Antti Karttunen, Table of n, a(n) for n = 1..8192
Index entries for sequences related to binary expansion of n

Cf. A048720, A056539, A091255, A280502, A280505.

(define (A280503 n) (A091255bi n (A056539 n))) ;; A091255bi implements the 2-argument GF(2)[X] GCD-function (A091255).

a(n) = A091255(n,A056539(n)).
Other identities. For all n >= 1:
a(A056539(n)) = a(n).
A048720(a(n), A280504(n)) = n.

A178226 Characteristic function of A154809 (numbers that are not binary palindromes).

Original entry on oeis.org

0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 1, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1

Offset: 0

Jeremy Gardiner, May 23 2010

a(n)=1 if n is in A154809, a(n)=0 otherwise.

Identical to parity of A086757 (Smallest prime p such that n is a palindrome in base p representation)

Cf. A030101, A057891, A086757, A154809, A178225, A280505, A280506.

Table[If[IntegerDigits[n,2]==Reverse[IntegerDigits[n,2]],0,1],{n,0,120}] (* Harvey P. Dale, Aug 07 2023 *)

A178226(n) = (n!=subst(Polrev(binary(n)),x,2)); \\ Antti Karttunen, Dec 15 2017

a(n) = 1 - A178225(n). - Antti Karttunen, Dec 15 2017

cp's OEIS Frontend

A280506 Nonpalindromic part of n in base 2 (with carryless GF(2)[X] factorization): a(n) = A280500(n,A280505(n)).

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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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A280508 a(n) = n XOR A057889(n).

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A280501 "Blue kernel" of n: a(n) = A091255(n, A193231(n)).

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A331166 a(n) = min(n, A057889(n)), where A057889 is bijective base-2 reverse.

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A280503 a(n) = A091255(n,A056539(n)).

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A178226 Characteristic function of A154809 (numbers that are not binary palindromes).

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