A280506 -id:A280506 - cp's OEIS frontend

A280500 Square array for division in ring GF(2)[X]: A(r,c) = r/c, or 0 if c is not a divisor of r, where the binary expansion of each number defines the corresponding (0,1)-polynomial.

1, 0, 2, 0, 1, 3, 0, 0, 0, 4, 0, 0, 1, 2, 5, 0, 0, 0, 0, 0, 6, 0, 0, 0, 1, 3, 3, 7, 0, 0, 0, 0, 0, 2, 0, 8, 0, 0, 0, 0, 1, 0, 0, 4, 9, 0, 0, 0, 0, 0, 0, 0, 0, 0, 10, 0, 0, 0, 0, 0, 1, 0, 2, 7, 5, 11, 0, 0, 0, 0, 0, 0, 0, 0, 0, 6, 0, 12, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 6, 13, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 4, 0, 14, 0, 0, 0, 0, 0, 0, 0, 1, 3, 3, 0, 3, 0, 7, 15

Offset: 1

Antti Karttunen, Jan 09 2017

The array A(row,col) is read by descending antidiagonals: A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), etc.

			The top left 17 X 17 corner of the array:
col: 1  2   3  4  5  6  7  8  9 10 11 12 13 14 15 16 17
     --------------------------------------------------
     1, 0,  0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
     2, 1,  0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
     3, 0,  1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
     4, 2,  0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
     5, 0,  3, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
     6, 3,  2, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
     7, 0,  0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
     8, 4,  0, 2, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0
     9, 0,  7, 0, 0, 0, 3, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0
    10, 5,  6, 0, 2, 3, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0
    11, 0,  0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0
    12, 6,  4, 3, 0, 2, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0
    13, 0,  0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0
    14, 7,  0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0
    15, 0,  5, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0
    16, 8,  0, 4, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 1, 0
    17, 0, 15, 0, 5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 1
    ---------------------------------------------------
7 ("111" in binary) encodes polynomial X^2 + X + 1, which is irreducible over GF(2) (7 is in A014580), thus it is divisible only by itself and 1, and for any other values of c than 1 and 7, A(7,c) = 0.
9 ("1001" in binary) encodes polynomial X^3 + 1, which is factored over GF(2) as (X+1)(X^2 + X + 1), and thus A(9,3) = 7 and A(9,7) = 3 because the polynomial X + 1 is encoded by 3 ("11" in binary).

Cf. A001317, A014580, A048720, A091255, A268389, A268669.
Cf. A280499 for the lower triangular region (A280494 for its transpose).
Cf. also A106449, A280502, A280504, A280506.

up_to = 10440;
A280500sq(a,b) = { my(Pa=Pol(binary(a))*Mod(1, 2), Pb=Pol(binary(b))*Mod(1, 2)); if(0!=lift(Pa % Pb), 0, fromdigits(Vec(lift(Pa/Pb)),2)); };
A280500list(up_to) = { my(v = vector(up_to), i=0); for(a=1,oo, for(col=1,a, i++; if(i > up_to, return(v)); v[i] = A280500sq(col,(a-(col-1))))); (v); };
v280500 = A280500list(up_to);
A280500(n) = v280500[n]; \\ Antti Karttunen, Jan 05 2025

(define (A280500 n) (A280500bi (A002260 n) (A004736 n)))
;; A very naive implementation:
(define (A280500bi row col) (let loop ((d row)) (cond ((zero? d) d) ((= (A048720bi d col) row) d) (else (loop (- d 1)))))) ;; A048720bi implements the carryless binary multiplication A048720.

A(row,col) = the unique d such that A048720(d,col) = row, provided that such d exists, otherwise zero.
Other identities. For all n >= 1:
A(n, A001317(A268389(n))) = A268669(n).

A280505 The palindromic kernel of n in base 2 (with carryless GF(2)[X] factorization): a(n) = A091255(n,A057889(n)).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 1, 12, 1, 14, 15, 16, 17, 18, 1, 20, 21, 2, 3, 24, 1, 2, 27, 28, 3, 30, 31, 32, 33, 34, 7, 36, 1, 2, 5, 40, 1, 42, 3, 4, 45, 6, 1, 48, 7, 2, 51, 4, 3, 54, 1, 56, 5, 6, 1, 60, 1, 62, 63, 64, 65, 66, 1, 68, 1, 14, 3, 72, 73, 2, 15, 4, 3, 10, 7, 80, 1, 2, 9, 84, 85, 6, 1, 8, 3, 90, 1, 12, 93, 2, 5, 96, 1, 14, 99, 4, 9, 102, 1, 8, 15, 6

Offset: 1

Antti Karttunen, Jan 09 2017

a(n) = the maximal GF(2)[X]-divisor of n which in base 2 is either a palindrome or becomes a palindrome if trailing 0's are omitted.
More precisely: a(n) = the unique term m of A057890 for which A280500(n,m) > 0 and A091222(m) >= A091222(k) for all such terms k of A057890 for which A280500(n,k) > 0.
All terms are in A057890 and each term of A057890 occurs an infinite number of times.

Cf. A048720, A057889, A057890, A091222, A091255, A280501, A280503, A280506.

(define (A280505 n) (A091255bi n (A057889 n))) ;; A091255bi implements the 2-argument GF(2)[X] GCD-function (A091255).

a(n) = A091255(n,A057889(n)).
Other identities. For all n >= 1:
a(A057889(n)) = a(n).
A048720(a(n), A280506(n)) = n.

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).

A280502 a(n) = A280500(n, A280501(n)).

Original entry on oeis.org

1, 2, 3, 4, 5, 1, 1, 8, 3, 3, 11, 2, 13, 2, 15, 16, 17, 1, 1, 1, 1, 22, 23, 4, 25, 26, 5, 4, 29, 5, 31, 32, 33, 15, 13, 2, 37, 2, 39, 2, 41, 2, 43, 44, 15, 13, 47, 8, 11, 50, 51, 52, 3, 3, 55, 8, 57, 11, 59, 3, 61, 62, 3, 64, 5, 31, 67, 5, 69, 26, 71, 4, 73, 74, 75, 4, 77, 29, 25, 4, 81, 82, 29, 4, 85, 25, 87, 88, 89, 5, 91, 26, 31, 94, 5, 16, 97, 22, 99, 100, 23, 17

Offset: 1

Antti Karttunen, Jan 09 2017

Cf. A193231, A280500, A280501, A280506, A280507.

Cf. A118666 (positions of ones).

(define (A280502 n) (A280500bi n (A280501 n))) ;; Code for A280500bi given in A280500.

a(n) = A280500(n, A280501(n)).
Other identities. For all n >= 1:
A048720(a(n), A280501(n)) = n.

A280504 a(n) = A280500(n,A280503(n)).

Original entry on oeis.org

1, 1, 1, 2, 1, 3, 1, 4, 1, 1, 11, 1, 13, 7, 1, 8, 1, 9, 19, 10, 1, 11, 13, 6, 25, 13, 1, 7, 11, 15, 1, 16, 1, 15, 13, 2, 37, 1, 11, 12, 41, 1, 25, 22, 1, 13, 47, 4, 11, 25, 1, 1, 19, 3, 55, 1, 13, 11, 59, 5, 61, 31, 1, 32, 1, 33, 67, 34, 69, 35, 61, 36, 1, 37, 13, 38, 59, 39, 25, 40, 81, 41, 11, 42, 1, 43, 87, 44, 55, 45, 91, 46, 1, 47, 19, 24, 97, 49, 1, 25, 13

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, A280500, A280503, A280506.

Cf. A044918 (very likely gives the positions of all ones).

(define (A280504 n) (A280500bi n (A280503 n))) ;; Code for A280500bi given in A280500.

a(n) = A280500(n,A280503(n)).
Other identities. For all n >= 1:
A048720(a(n), A280503(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

A303534 Amount by which n exceeds the largest binary palindrome less than or equal to n.

Original entry on oeis.org

0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 2, 3, 4, 5, 0, 1, 0, 1, 2, 3, 0, 1, 2, 3, 4, 5, 0, 1, 2, 3, 0, 1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 0, 1, 2, 3, 4, 5, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 0, 1, 0, 1, 2, 3, 4, 5, 6, 7, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 0, 1, 2, 3, 4, 5, 6, 7, 0, 1, 2, 3

Offset: 0

Allan C. Wechsler, Apr 25 2018

			The largest binary palindrome that doesn't exceed 30 is 27 (11011 r2). 30 - 27 = 3, so a(30) = 3.

Antti Karttunen, Table of n, a(n) for n = 0..65537
Index entries for sequences related to binary expansion of n

A006995 lists the binary palindromes.
A206913 gives the largest binary palindrome that does not exceed n.
Cf. also A261424 (analog in base 10), A280506, A303536.

isA006995(n) = Vecrev(n=binary(n))==n;
a(n) = {my(k=0); while(!isA006995(n-k), k++); k;} \\ Altug Alkan, Apr 25 2018

a(n) = n - A206913(n).

More terms from Altug Alkan, Apr 25 2018

cp's OEIS Frontend

A280500 Square array for division in ring GF(2)[X]: A(r,c) = r/c, or 0 if c is not a divisor of r, where the binary expansion of each number defines the corresponding (0,1)-polynomial.

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A280505 The palindromic kernel of n in base 2 (with carryless GF(2)[X] factorization): a(n) = A091255(n,A057889(n)).

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

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A280502 a(n) = A280500(n, A280501(n)).

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A280504 a(n) = A280500(n,A280503(n)).

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

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A303534 Amount by which n exceeds the largest binary palindrome less than or equal to n.

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