A091257 -id:A091257 - cp's OEIS frontend

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.

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

A003991 Multiplication table read by antidiagonals: T(i,j) = i*j, i>=1, j>=1.

Original entry on oeis.org

1, 2, 2, 3, 4, 3, 4, 6, 6, 4, 5, 8, 9, 8, 5, 6, 10, 12, 12, 10, 6, 7, 12, 15, 16, 15, 12, 7, 8, 14, 18, 20, 20, 18, 14, 8, 9, 16, 21, 24, 25, 24, 21, 16, 9, 10, 18, 24, 28, 30, 30, 28, 24, 18, 10, 11, 20, 27, 32, 35, 36, 35, 32, 27, 20, 11, 12, 22, 30, 36, 40, 42, 42, 40, 36, 30, 22, 12

Offset: 1

Marc LeBrun

Or, triangle X(n,m) = T(n-m+1,m) read by rows, in which row n gives the numbers n*1, (n-1)*2, (n-2)*3, ..., 2*(n-1), 1*n.
Radius of incircle of Pythagorean triangle with sides a=(n+1)^2-m^2, b=2*(n+1)*m and c=(n+1)^2+m^2. - Floor van Lamoen, Aug 16 2001
A permutation of A061017. - Matthew Vandermast, Feb 28 2003
In the proof of countability of rational numbers they are arranged in a square array. a(n) = p*q where p/q is the corresponding rational number as read from the array. - Amarnath Murthy, May 29 2003
Permanent of upper right n X n corner is A000442. - Marc LeBrun, Dec 11 2003
Row 12 gives total number of partridges, turtle doves, ... and drummers drumming that you have received at the end of the Twelve Days of Christmas song. - Alonso del Arte, Jun 17 2005
Consider a particle with spin S (a half-integer) and 2S+1 quantum states |m>, m = -S,-S+1,...,S-1,S. Then the matrix element  = sqrt((S+m+1)(S-m)) of the spin-raising operator is the square-root of the triangular (tabl) element T(r,o) of this sequence in row r = 2S, and at offset o=2(S+m). T(r,o) is also the intensity || of the transition between the states |m> and |m+1>. For example, the five transitions between the 6 states of a spin S=5/2 particle have relative intensities 5,8,9,8,5. The total intensity of all spin 5/2 transitions (relative to spin 1/2) is 35, which is the tetrahedral number A000292(5). - Stanislav Sykora, May 26 2012
Sum_{k=0..2n-2} (-1)^k*a(A000124(2n-2)+k) = n. See A098359. - Charlie Marion, Apr 22 2013
T(n, k) is also the (k-1)-superdiagonal sum of an n X n Toeplitz matrix M(n) whose first row consists of successive positive integer numbers 1, ..., n. - Stefano Spezia, Jul 12 2019
From Eric Lengyel, Jun 28 2023: (Start)
X(n, m+1) is the number of degrees of freedom that an m-dimensional flat geometry (point, line, plane, etc.) has when embedded in an n-dimensional Euclidean space.
X(n+1, m+1) is the number of degrees of freedom that an m-ball has when embedded in an n-dimensional Euclidean space. (End)
T(n, k) is also the average number of steps it takes a person to fall off a board of length n+k, if the person starts a random walk at k. - Ruediger Jehn, May 12 2025

			The array T starts in row n=1 with columns m>=1 as:
   1   2   3   4   5   6   7   8   9  10  11  12  13  14  15
   2   4   6   8  10  12  14  16  18  20  22  24  26  28  30
   3   6   9  12  15  18  21  24  27  30  33  36  39  42  45
   4   8  12  16  20  24  28  32  36  40  44  48  52  56  60
   5  10  15  20  25  30  35  40  45  50  55  60  65  70  75
   6  12  18  24  30  36  42  48  54  60  66  72  78  84  90
   7  14  21  28  35  42  49  56  63  70  77  84  91  98 105
   8  16  24  32  40  48  56  64  72  80  88  96 104 112 120
   9  18  27  36  45  54  63  72  81  90  99 108 117 126 135
  10  20  30  40  50  60  70  80  90 100 110 120 130 140 150
The triangle X(n, m) begins
   n\m  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 ...
   1:   1
   2:   2  2
   3:   3  4  3
   4:   4  6  6  4
   5:   5  8  9  8  5
   6:   6 10 12 12 10  6
   7:   7 12 15 16 15 12  7
   8:   8 14 18 20 20 18 14  8
   9:   9 16 21 24 25 24 21 16  9
  10:  10 18 24 28 30 30 28 24 18 10
  11:  11 20 27 32 35 36 35 32 27 20 11
  12:  12 22 30 36 40 42 42 40 36 30 22 12
  13:  13 24 33 40 45 48 49 48 45 40 33 24 13
  14:  14 26 36 44 50 54 56 56 54 50 44 36 26 14
  15:  15 28 39 48 55 60 63 64 63 60 55 48 39 28 15
  ... Formatted by _Wolfdieter Lang_, Dec 02 2014

J. H. Conway and R. K. Guy, The Book of Numbers, Copernicus Press, NY, 1996, p. 46.
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, pages 5-6.

T. D. Noe, Rows n = 1..100 of triangle, flattened
Iva Kodrnja and Helena Koncul, Number of Polynomials Vanishing on a Basis of S_m(Gamma_0(N)), arXiv:2405.10747 [math.NT], 2024. See p. 10.
G. W. Leibniz, Dissertatio de arte combinatoria, 1666, Leipzig. (in Latin. This triangle appears on p. 208, page 44 of the PDF file).
Abdelkader Necer, Séries formelles et produit de Hadamard, Journal de théorie des nombres de Bordeaux, 9 no. 2 (1997), p. 319-335.
Boris Putievskiy, Transformations Integer Sequences And Pairing Functions arXiv:1212.2732 [math.CO], 2012.
N. J. A. Sloane, "A Handbook of Integer Sequences" Fifty Years Later, arXiv:2301.03149 [math.NT], 2023, p. 5.

Main diagonal gives squares A000290. Antidiagonal sums are tetrahedral numbers A000292. See A004247 for another version.
Cf. A003989, A003990, A003056, A049581, A000442, A027424, A002260, A033638, A059895, A059896, A059897, A002620.
Cf. also A051776, A067138, A091257, A325821, A331590, A341520, A350066.

/* As triangle */ [[k*(n-k+1): k in [1..n]]: n in [1..15]]; // Vincenzo Librandi, Jul 12 2019

seq(seq(i*(n-i),i=1..n-1),n=2..10); # Robert Israel, Dec 14 2015

Table[(x + 1 - y) y, {x, 13}, {y, x}] // Flatten (* Robert G. Wilson v, Oct 06 2007 *)
f[n_] := Table[SeriesCoefficient[E^(x + y) (1+ x - y +x*y-y^2), {x, 0, i}, {y, 0, j}]*i!*j!, {i, n, n}, {j, 0, n}]; Flatten[Array[f, 11,0]] (* Stefano Spezia, Jul 12 2019 *)

PARI
```
A003991(n,k) = if(k<1 || n<1,0,k*n)
```

Rectangular array: T(n, m) = n*m, n>=1, m>= 1.
Triangle X(n, m) = T(n-m+1, m) = (n-m+1)*m.
Sum_{i=1..n} Sum_{j=1..n} a(n) = A000537(n) [Sum of first n cubes; or n-th triangular number squared.] Determinant of all n X n contiguous subarrays of A003991 is 0. - Gerald McGarvey, Sep 26 2004
G.f. as rectangular array: x*y/((1 - x)^2*(1 - y)^2).
a(n) = i*j, where i=floor((1+sqrt(8n-7))/2), j=n-i*(i-1)/2. - Hieronymus Fischer, Aug 08 2007
As an infinite lower triangular matrix equals A000012 * A002260; where A000012 = (1; 1,1; 1,1,1; ...) and A002260 = (1; 1,2; 1,2,3; ...). - Gary W. Adamson, Oct 23 2007
As a linear array, the sequence is a(n) = A002260(n)*A004736(n) or a(n) = ((t*t+3*t+4)/2-n)*(n-(t*(t+1)/2)), where t=floor((-1+sqrt(8*n-7))/2). - Boris Putievskiy, Dec 17 2012
G.f. as linear array: (x - 3*x^2 + Sum_{k >= 0} ((k+2-x-(k+1)*x^2)*x^((k^2+3*k+4)/2)))/(1-x)^3. - Robert Israel, Dec 14 2015
E.g.f. as triangle: exp(x+y)*(1 + x - y + x*y - y^2). - Stefano Spezia, Jul 12 2019
a(n) = (1/2)*t + (n - 1/4)*t^2 - (1/4)*t^4 - n^2 + n, where t = floor(sqrt(2*n) + 1/2). - Ridouane Oudra, Nov 21 2020
a(n) = A003989(n) * A003990(n) = A059895(n) * A059896(n) = A059895(n)^2 * A059897(n). - Antti Karttunen, Dec 13 2021
T(n,k) = A002620(n+k) - A002620(n-k). - Michel Marcus, Jan 06 2023
T(n,k) = number of sums |x-y|+|y-z| = k, where x,y,z are in {1,2,...,n} and x < y < z. - Clark Kimberling, Jan 22 2024
E.g.f. as rectangular array: x*y*exp(x+y). - Stefano Spezia, Jun 27 2025

More terms from Michael Somos

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

A091220 Number of divisors of the n-th GF(2)[X]-polynomial.

Original entry on oeis.org

1, 2, 2, 3, 3, 4, 2, 4, 4, 6, 2, 6, 2, 4, 4, 5, 5, 8, 2, 9, 3, 4, 4, 8, 2, 4, 6, 6, 4, 8, 2, 6, 4, 10, 4, 12, 2, 4, 6, 12, 2, 6, 4, 6, 8, 8, 2, 10, 4, 4, 6, 6, 4, 12, 2, 8, 6, 8, 2, 12, 2, 4, 6, 7, 9, 8, 2, 15, 3, 8, 4, 16, 2, 4, 8, 6, 4, 12, 4, 15, 3, 4, 8, 9, 7, 8, 2, 8, 4, 16, 2, 12, 4, 4, 6, 12, 2

Offset: 1

Antti Karttunen, Jan 03 2004

nonn

a(n) = A000005(A091203(n)) = A000005(A091205(n)). Cf. A091257.

a(n)=local(p,fm,k);while(n>0,p+=Mod(n,2)*x^k;n\=2;k++);fm=factor(p);prod(k=1,matsize(fm)[1],fm[k,2]+1) \\ Franklin T. Adams-Watters, Jun 22 2010

A091256 Table of lcm(x,y) computed for polynomials over GF(2), where (x,y) runs as (1,1),(1,2),(2,1),(1,3),(2,2),(3,1),...

Original entry on oeis.org

1, 2, 2, 3, 2, 3, 4, 6, 6, 4, 5, 4, 3, 4, 5, 6, 10, 12, 12, 10, 6, 7, 6, 5, 4, 5, 6, 7, 8, 14, 6, 20, 20, 6, 14, 8, 9, 8, 9, 12, 5, 12, 9, 8, 9, 10, 18, 24, 28, 10, 10, 28, 24, 18, 10, 11, 10, 9, 8, 27, 6, 27, 8, 9, 10, 11, 12, 22, 10, 36, 40, 18, 18, 40, 36, 10, 22, 12, 13, 12, 29

Offset: 1

Antti Karttunen, Jan 03 2004

Analogous to A003990.

Cf. A091255, A091257.

A325821 Multiplication table for carryless product i X j in base 3 for i >= 1 and j >= 1, read by antidiagonals.

Original entry on oeis.org

1, 2, 2, 3, 1, 3, 4, 6, 6, 4, 5, 8, 9, 8, 5, 6, 7, 12, 12, 7, 6, 7, 3, 15, 16, 15, 3, 7, 8, 5, 18, 11, 11, 18, 5, 8, 9, 4, 21, 24, 13, 24, 21, 4, 9, 10, 18, 24, 19, 21, 21, 19, 24, 18, 10, 11, 20, 27, 23, 26, 9, 26, 23, 27, 20, 11, 12, 19, 30, 36, 19, 15, 15, 19, 36, 30, 19, 12, 13, 24, 33, 40, 45, 12, 13, 12, 45, 40, 33, 24, 13, 14, 26, 36, 44, 50, 54, 11, 11, 54, 50, 44, 36, 26, 14

Offset: 1

Antti Karttunen, May 22 2019

This is table A325820 without the zero row and column. See there for more comments.

			The array begins as:
   1,  2,  3,  4,  5,  6,  7,  8,   9,  10,  11,  12, ...
   2,  1,  6,  8,  7,  3,  5,  4,  18,  20,  19,  24, ...
   3,  6,  9, 12, 15, 18, 21, 24,  27,  30,  33,  36, ...
   4,  8, 12, 16, 11, 24, 19, 23,  36,  40,  44,  48, ...
   5,  7, 15, 11, 13, 21, 26, 19,  45,  50,  52,  33, ...
   6,  3, 18, 24, 21,  9, 15, 12,  54,  60,  57,  72, ...
   7,  5, 21, 19, 26, 15, 13, 11,  63,  70,  68,  57, ...
   8,  4, 24, 23, 19, 12, 11, 16,  72,  80,  76,  69, ...
   9, 18, 27, 36, 45, 54, 63, 72,  81,  90,  99, 108, ...
  10, 20, 30, 40, 50, 60, 70, 80,  90, 100,  83, 120, ...
  11, 19, 33, 44, 52, 57, 68, 76,  99,  83,  91, 132, ...
  12, 24, 36, 48, 33, 72, 57, 69, 108, 120, 132, 144, ...

Cf. A003991, A091257, A325820.

up_to = 105;
A325820sq(b, c) = fromdigits(Vec(Pol(digits(b,3))*Pol(digits(c,3)))%3, 3);
A325821list(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] = A325820sq(col,(a-(col-1))))); (v); };
v325821 = A325821list(up_to);
A325821(n) = v325821[n];

A346795 Irregular triangle T(n, k), n > 0, k = 1..A091220(n), read by rows; the n-th row gives, in ascending order, the distinct integers k such that A048720(k, m) = n for some m.

Original entry on oeis.org

1, 1, 2, 1, 3, 1, 2, 4, 1, 3, 5, 1, 2, 3, 6, 1, 7, 1, 2, 4, 8, 1, 3, 7, 9, 1, 2, 3, 5, 6, 10, 1, 11, 1, 2, 3, 4, 6, 12, 1, 13, 1, 2, 7, 14, 1, 3, 5, 15, 1, 2, 4, 8, 16, 1, 3, 5, 15, 17, 1, 2, 3, 6, 7, 9, 14, 18, 1, 19, 1, 2, 3, 4, 5, 6, 10, 12, 20, 1, 7, 21

Offset: 1

Rémy Sigrist, Sep 29 2021

The n-th row corresponds to the divisors of the n-th GF(2)[X]-polynomial.
The greatest value both in the n-th row and in the k-th row corresponds to A091255(n, k).
The index of the first row containing both n and k corresponds to A091256(n, k).

			The triangle starts:
      1:   [1]
      2:   [1, 2]
      3:   [1, 3]
      4:   [1, 2, 4]
      5:   [1, 3, 5]
      6:   [1, 2, 3, 6]
      7:   [1, 7]
      8:   [1, 2, 4, 8]
      9:   [1, 3, 7, 9]
     10:   [1, 2, 3, 5, 6, 10]
     11:   [1, 11]
     12:   [1, 2, 3, 4, 6, 12]
     13:   [1, 13]
     14:   [1, 2, 7, 14]
     15:   [1, 3, 5, 15]

Rémy Sigrist, Table of n, a(n) for n = 1..9228 (first 1024 rows flattened)
Rémy Sigrist, PARI program for A346795
Index entries for sequences operating on GF(2)[X]-polynomials

Cf. A048720, A091220, A091255, A091256, A091257, A178908, A280493, A348135.

PARI
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See Links section.
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T(n, 1) = 1.
T(n, A091220(n)) = n.
Sum_{k = 1..A091220(n)} T(n, k) = A280493(n).
T(n, 1) XOR ... XOR T(n, A091220(n)) = A178908(n) (where XOR denotes the bitwise XOR operator).

cp's OEIS Frontend

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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A003991 Multiplication table read by antidiagonals: T(i,j) = i*j, i>=1, j>=1.

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A091220 Number of divisors of the n-th GF(2)[X]-polynomial.

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A091256 Table of lcm(x,y) computed for polynomials over GF(2), where (x,y) runs as (1,1),(1,2),(2,1),(1,3),(2,2),(3,1),...

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A325821 Multiplication table for carryless product i X j in base 3 for i >= 1 and j >= 1, read by antidiagonals.

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PARI

A346795 Irregular triangle T(n, k), n > 0, k = 1..A091220(n), read by rows; the n-th row gives, in ascending order, the distinct integers k such that A048720(k, m) = n for some m.

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