A280502 -id:A280502 - 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).

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

Original entry on oeis.org

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

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.

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.

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).

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

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

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

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

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