cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

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.

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Jan 09 2017

Keywords

Comments

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.

Examples

			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).

Links

Crossrefs

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

Programs

  • PARI
    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
  • Scheme
    (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.

Formula

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).

A280493 Sum of GF(2)[X] divisors of n: the sum is ordinary sum of integers, the summands being all the natural numbers whose binary expansions encode such (0,1)-polynomials that divide the (0,1)-polynomial encoded by n when the polynomial factorization is done over the field GF(2).

Original entry on oeis.org

1, 3, 4, 7, 9, 12, 8, 15, 20, 27, 12, 28, 14, 24, 24, 31, 41, 60, 20, 63, 29, 36, 40, 60, 26, 42, 52, 56, 44, 72, 32, 63, 68, 123, 56, 140, 38, 60, 88, 135, 42, 87, 72, 84, 112, 120, 48, 124, 68, 78, 92, 98, 76, 156, 56, 120, 102, 132, 60, 168, 62, 96, 104, 127, 201, 204, 68, 287, 81, 168, 136, 300, 74, 114, 192, 140, 140, 264, 112, 279, 95, 126, 192, 203
Offset: 1

Views

Author

Antti Karttunen, Jan 09 2017

Keywords

Comments

This is roughly a GF(2)[X]-analog of A000203. A178908 gives another, maybe a more consistent analog.

Examples

			9 ("1001" in binary) encodes polynomial X^3 + 1, which is factored over GF(2) as (X+1)(X^2 + X + 1), where polynomial X + 1 is encoded by 3 ("11" in binary), and polynomial X^2 + X + 1 by 7 ("111" in binary), and furthermore (like all polynomials) it is also divisible by 1 and itself, thus a(9) = 1 + 3 + 7 + 9 = 20.

Links

Crossrefs

Row sums of triangles A280494 and A280499.
Cf. A000225, A000203, A178908.
Cf. A014580 (gives the positions where a(n) = n+1).

Programs

  • Scheme
    (define (A280493 n) (let loop ((k n) (s 0)) (if (zero? k) s (loop (- k 1) (+ s (if (= k (A091255bi n k)) k 0))))))
;; A091255bi implements the 2-argument GF(2)[X] GCD-function (A091255) which is used for checking that k is a divisor of n.
;; Another version:
(define (A280493 n) (add A280494 (+ 1 (A000217 (- n 1))) (A000217 n)))
(define (add intfun lowlim uplim) (let sumloop ((i lowlim) (res 0)) (cond ((> i uplim) res) (else (sumloop (+ 1 i) (+ res (intfun i)))))))

Formula

For all n >= 0, a(2^n) = A000203(2^n) = A178908(2^n) = A000225(1+n).

A280499 Triangular table for division in ring GF(2)[X]: T(n,k) = n/k, or 0 if k is not a divisor of n, where the binary expansion of each number defines the corresponding (0,1)-polynomial.

Original entry on oeis.org

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

Views

Author

Antti Karttunen, Jan 09 2017

Keywords

Comments

This is GF(2)[X] analog of A126988, using "carryless division in base-2" instead of ordinary division.
The triangular table T(n,k), n=1.., k=1..n is read by rows: T(1,1), T(2,1), T(2,2), T(3,1), T(3,2), T(3,3), etc.

Examples

			The first 17 rows of the triangle:
   1
   2 1
   3 0 1
   4 2 0 1
   5 0 3 0 1
   6 3 2 0 0 1
   7 0 0 0 0 0 1
   8 4 0 2 0 0 0 1
   9 0 7 0 0 0 3 0 1
  10 5 6 0 2 3 0 0 0 1
  11 0 0 0 0 0 0 0 0 0 1
  12 6 4 3 0 2 0 0 0 0 0 1
  13 0 0 0 0 0 0 0 0 0 0 0 1
  14 7 0 0 0 0 2 0 0 0 0 0 0 1
  15 0 5 0 3 0 0 0 0 0 0 0 0 0 1
  16 8 0 4 0 0 0 2 0 0 0 0 0 0 0 1
  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), so it is divisible only by itself and 1, and thus T(7,1) = 7, T(7,k) = 0 for k=2..6 and T(7,7) = 1.
9 ("1001" in binary) encodes polynomial X^3 + 1, which is factored over GF(2) as (X+1)(X^2 + X + 1), and thus T(9,3) = 7 and T(9,7) = 3 because the polynomial X + 1 is encoded by 3 ("11" in binary).

Links

Crossrefs

Lower triangular region of square array A280500.
Transpose: A280494.
Cf. A014580, A048720, A126988, A178908, A280500, A280493 (the row sums).

Programs

Formula

T(n,k) = the unique d such that A048720(d,k) = n, provided that such d exists, otherwise zero.
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.