A280500 -id:A280500 - 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

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.

A048720 Multiplication table {0..i} X {0..j} of binary polynomials (polynomials over GF(2)) interpreted as binary vectors, then written in base 10; or, binary multiplication without carries.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 2, 2, 0, 0, 3, 4, 3, 0, 0, 4, 6, 6, 4, 0, 0, 5, 8, 5, 8, 5, 0, 0, 6, 10, 12, 12, 10, 6, 0, 0, 7, 12, 15, 16, 15, 12, 7, 0, 0, 8, 14, 10, 20, 20, 10, 14, 8, 0, 0, 9, 16, 9, 24, 17, 24, 9, 16, 9, 0, 0, 10, 18, 24, 28, 30, 30, 28, 24, 18, 10, 0, 0, 11, 20, 27, 32, 27, 20, 27, 32, 27, 20, 11, 0

Offset: 0

Antti Karttunen, Apr 26 1999

Essentially same as A091257 but computed starting from offset 0 instead of 1.
Each polynomial in GF(2)[X] is encoded as the number whose binary representation is given by the coefficients of the polynomial, e.g., 13 = 2^3 + 2^2 + 2^0 = 1101_2 encodes 1*X^3 + 1*X^2 + 0*X^1 + 1*X^0 = X^3 + X^2 + X^0. - Antti Karttunen and Peter Munn, Jan 22 2021
To listen to this sequence, I find instrument 99 (crystal) works well with the other parameters defaulted. - Peter Munn, Nov 01 2022

			Top left corner of array:
  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0 ...
  0  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 ...
  0  2  4  6  8 10 12 14 16 18 20 22 24 26 28 30 ...
  0  3  6  5 12 15 10  9 24 27 30 29 20 23 18 17 ...
  ...
From _Antti Karttunen_ and _Peter Munn_, Jan 23 2021: (Start)
Multiplying 10 (= 1010_2) and 11 (= 1011_2), in binary results in:
     1011
  *  1010
  -------
   c1011
  1011
  -------
  1101110  (110 in decimal),
and we see that there is a carry-bit (marked c) affecting the result.
In carryless binary multiplication, the second part of the process (in which the intermediate results are summed) looks like this:
    1011
  1011
  -------
  1001110  (78 in decimal).
(End)

Alois P. Heinz, Antidiagonals n = 0..200, flattened
Antti Karttunen, Scheme-program for computing this sequence.
N. J. A. Sloane, Transforms: Maple implementation of binary eXclusive OR (XORnos).
Index entries for sequences related to carryless arithmetic
Index entries for sequences related to polynomials in ring GF(2)[X]

Cf. A051776 (Nim-product), A091257 (subtable).
Carryless multiplication in other bases: A325820 (3), A059692 (10).
Ordinary {0..i} * {0..j} multiplication table: A004247 and its differences from this: A061858 (which lists further sequences related to presence/absence of carry in binary multiplication).
Carryless product of the prime factors of n: A234741.
Binary irreducible polynomials ("X-primes"): A014580, factorization table: A256170, table of "X-powers": A048723, powers of 3: A001317, rearranged subtable with distinct terms (comparable to A054582): A277820.
See A014580 for further sequences related to the difference between factorization into GF(2)[X] irreducibles and ordinary prime factorization of the integer encoding.
Row/column 3: A048724 (even bisection of A003188), 5: A048725, 6: A048726, 7: A048727; main diagonal: A000695.
Associated additive operation: A003987.
Equivalent sequences, as compared with standard integer multiplication: A048631 (factorials), A091242 (composites), A091255 (gcd), A091256 (lcm), A280500 (division).
See A091202 (and its variants) and A278233 for maps from/to ordinary multiplication.
See A115871, A115872 and A277320 for tables related to cross-domain congruences.

trinv := n -> floor((1+sqrt(1+8*n))/2); # Gives integral inverses of the triangular numbers
# Binary multiplication of nn and mm, but without carries (use XOR instead of ADD):
Xmult := proc(nn,mm) local n,m,s; n := nn; m := mm; s := 0; while (n > 0) do if(1 = (n mod 2)) then s := XORnos(s,m); fi; n := floor(n/2); # Shift n right one bit. m := m*2; # Shift m left one bit. od; RETURN(s); end;

trinv[n_] := Floor[(1 + Sqrt[1 + 8*n])/2];
Xmult[nn_, mm_] := Module[{n = nn, m = mm, s = 0}, While[n > 0, If[1 == Mod[n, 2], s = BitXor[s, m]]; n = Floor[n/2]; m = m*2]; Return[s]];
a[n_] := Xmult[(trinv[n] - 1)*((1/2)*trinv[n] + 1) - n, n - (trinv[n]*(trinv[n] - 1))/2];
Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Mar 16 2015, updated Mar 06 2016 after Maple *)

up_to = 104;
A048720sq(b,c) = fromdigits(Vec(Pol(binary(b))*Pol(binary(c)))%2, 2);
A048720list(up_to) = { my(v = vector(1+up_to), i=0); for(a=0, oo, for(col=0, a, i++; if(i > up_to, return(v)); v[i] = A048720sq(col, a-col))); (v); };
v048720 = A048720list(up_to);
A048720(n) = v048720[1+n]; \\ Antti Karttunen, Feb 15 2021

a(n) = Xmult( (((trinv(n)-1)*(((1/2)*trinv(n))+1))-n), (n-((trinv(n)*(trinv(n)-1))/2)) );
T(2b, c)=T(c, 2b)=T(b, 2c)=2T(b, c); T(2b+1, c)=T(c, 2b+1)=2T(b, c) XOR c - Henry Bottomley, Mar 16 2001
For n >= 0, A003188(2n) = T(n, 3); A003188(2n+1) = T(n, 3) XOR 1, where XOR is the bitwise exclusive-or operator, A003987. - Peter Munn, Feb 11 2021

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

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.

A106449 Square array (P(x) XOR P(y))/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 all calculations are done in polynomial ring GF(2)[X], with the result converted back to a binary number, and then expressed in decimal. Array is symmetric, and is read by antidiagonals.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, May 21 2005

Array is read by antidiagonals, with row x and column y ranging as: (x,y) = (1,1), (1,2), (2,1), (1,3), (2,2), (3,1), ...
"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).
This is GF(2)[X] analog of A106448. In the definition XOR means addition in polynomial ring GF(2)[X], that is, a carryless binary addition, A003987.

			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 :  0,  3,  2,  5,  4,  7,  6,  9,  8, 11, 10, 13, 12, 15, 14, 17, 16, ...
   2 :  3,  0,  1,  3,  7,  2,  5,  5, 11,  4,  9,  7, 15,  6, 13,  9, 19, ...
   3 :  2,  1,  0,  7,  2,  3,  4, 11,  6,  7,  8,  5, 14, 13,  4, 19, 14, ...
   4 :  5,  3,  7,  0,  1,  1,  3,  3, 13,  7, 15,  2,  9,  5, 11,  5, 21, ...
   5 :  4,  7,  2,  1,  0,  1,  2, 13,  4,  3, 14,  7,  8, 11,  2, 21,  4, ...
   6 :  7,  2,  3,  1,  1,  0,  1,  7,  5,  2, 13,  3, 11,  4,  7, 11, 13, ...
   7 :  6,  5,  4,  3,  2,  1,  0, 15,  2, 13, 12, 11, 10,  3,  8, 23, 22, ...
   8 :  9,  5, 11,  3, 13,  7, 15,  0,  1,  1,  3,  1,  5,  3,  7,  3, 25, ...
   9 :  8, 11,  6, 13,  4,  5,  2,  1,  0,  1,  2,  3,  4,  1,  2, 25,  8, ...
  10 : 11,  4,  7,  7,  3,  2, 13,  1,  1,  0,  1,  1,  7,  2,  1, 13,  7, ...
  11 : 10,  9,  8, 15, 14, 13, 12,  3,  2,  1,  0,  7,  6,  5,  4, 27, 26, ...
  12 : 13,  7,  5,  2,  7,  3, 11,  1,  3,  1,  7,  0,  1,  1,  1,  7, 11, ...
  13 : 12, 15, 14,  9,  8, 11, 10,  5,  4,  7,  6,  1,  0,  3,  2, 29, 28, ...
  14 : 15,  6, 13,  5, 11,  4,  3,  3,  1,  2,  5,  1,  3,  0,  1, 15, 31, ...
  15 : 14, 13,  4, 11,  2,  7,  8,  7,  2,  1,  4,  1,  2,  1,  0, 31,  2, ...
  16 : 17,  9, 19,  5  21, 11, 23,  3, 25, 13, 27,  7, 29, 15, 31,  0,  1, ...
  17 : 16, 19, 14, 21,  4, 13, 22, 25,  8,  7, 26, 11, 28, 31,  2,  1,  0, ...

Row 1: A004442 (without its initial term), row 2: A106450 (without its initial term).

Cf. A003987, A048720, A091255, A106448, A280500.

up_to = 105;
A106449sq(a,b) = { my(Pa=Pol(binary(a))*Mod(1, 2), Pb=Pol(binary(b))*Mod(1, 2)); fromdigits(Vec(lift((Pa+Pb)/gcd(Pa,Pb))),2); }; \\ Note that XOR is just + in GF(2)[X] world.
A106449list(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] = A106449sq(col,(a-(col-1))))); (v); };
v106449 = A106449list(up_to);
A106449(n) = v106449[n]; \\ Antti Karttunen, Oct 21 2019

A(x, y) = A280500(A003987(x, y), A091255(x, y)), that is, A003987(x, y) = A048720(A(x, y), A091255(x, y)).

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

Antti Karttunen, Jan 09 2017

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.

			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).

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

(define (A280499 n) (A280500bi (A002024 n) (A002260 n))) ;; Code for A280500bi given in A280500.

T(n,k) = the unique d such that A048720(d,k) = n, provided that such d exists, otherwise zero.

A280494 Rows of triangular table A280499 read in reverse order.

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Jan 09 2017

This is GF(2)[X] analog of A127013, using "carryless division in base-2" instead of ordinary division.

			The first 17 rows of the triangle:
1
1 2
1 0 3
1 0 2 4
1 0 3 0 5
1 0 0 2 3 6
1 0 0 0 0 0 7
1 0 0 0 2 0 4 8
1 0 3 0 0 0 7 0 9
1 0 0 0 3 2 0 6 5 10
1 0 0 0 0 0 0 0 0 0 11
1 0 0 0 0 0 2 0 3 4 6 12
1 0 0 0 0 0 0 0 0 0 0 0 13
1 0 0 0 0 0 0 2 0 0 0 0 7 14
1 0 0 0 0 0 0 0 0 0 3 0 5 0 15
1 0 0 0 0 0 0 0 2 0 0 0 4 0 8 16
1 0 3 0 0 0 0 0 0 0 0 0 5 0 15 0 17

Cf. A048720, A127013, A280499, A280500, A280493 (the row sums).

(define (A280494 n) (A280500bi (A002024 n) (A004736 n))) ;; Code for A280500bi given in A280500.

A379236 Numbers k such that x=(sigma(k) XOR 2*k) divides k in carryless binary arithmetic, when the binary expansions of k and x are interpreted as polynomials in ring GF(2)[X].

Original entry on oeis.org

10, 12, 18, 20, 24, 40, 56, 88, 104, 116, 136, 184, 196, 224, 312, 368, 428, 464, 520, 528, 650, 672, 760, 884, 992, 1472, 1504, 1888, 1952, 2528, 3424, 3724, 4832, 5312, 6464, 7136, 9112, 11096, 11288, 11744, 13216, 15352, 15376, 15872, 15968, 16256, 17816, 17964, 22616, 24448, 26728, 28544, 29296, 30592, 30656

Offset: 1

Antti Karttunen, Jan 05 2025

nonn

Among the first 484 terms, there are no odd numbers, the only squares are 196, 15376, 1032256, and 18 is the only twice square.

			196 is a term as sigma(196) = 399, 2*196 XOR 399 = 7 is not zero, and A048720(7, 89) = 399.

Cf. A000203, A000396, A003987, A048720, A280500, A318467.
Cf. A379234 (subsequence).
Cf. also A097498 (= A153501 U A271816).

divides_in_GF2X(a,b) = { my(Pa=Pol(binary(a))*Mod(1, 2), Pb=Pol(binary(b))*Mod(1, 2)); !lift(Pa % Pb); };
is_A379236(n) = { my(s=sigma(n), x=bitxor(2*n, s)); (x && divides_in_GF2X(n, x)); };

{k such that k = A048720(A318467(k), x) for some x > 0}.

{k not in A000396 such that A280500(k, A318467(k)) > 0}.

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

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

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A048720 Multiplication table {0..i} X {0..j} of binary polynomials (polynomials over GF(2)) interpreted as binary vectors, then written in base 10; or, binary multiplication without carries.

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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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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.

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