A106448 -id:A106448 - cp's OEIS frontend

A206284 Numbers that match irreducible polynomials over the nonnegative integers.

3, 6, 9, 10, 12, 18, 20, 22, 24, 27, 28, 30, 36, 40, 42, 44, 46, 48, 50, 52, 54, 56, 60, 66, 68, 70, 72, 76, 80, 81, 88, 92, 96, 98, 100, 102, 104, 108, 112, 114, 116, 118, 120, 124, 126, 130, 132, 136, 140, 144, 148, 150, 152, 154, 160, 162, 164, 168, 170

Offset: 1

Clark Kimberling, Feb 05 2012

nonn

Starting with 1, which encodes 0-polynomial, each integer m encodes (or "matches") a polynomial p(m,x) with nonnegative integer coefficients determined by the prime factorization of m. Write m = prime(1)^e(1) * prime(2)^e(2) * ... * prime(k)^e(k); then p(m,x) = e(1) + e(2)x + e(3)x^2 + ... + e(k)x^k.
Identities:
  p(m*n,x) = p(m,x) + p(n,x),
  p(m*n,x) = p(gcd(m,n),x) + p(lcm(m,n),x),
  p(m+n,x) = p(gcd(m,n),x) + p((m+n)/gcd(m,n),x), so that if A003057 is read as a square matrix, then
  p(A003057,x) = p(A003989,x) + p(A106448,x).
Apart from powers of 3, all terms are even. - Charles R Greathouse IV, Feb 11 2012
Contains 2*p^m and p*2^m if p is an odd prime and m is in A052485. - Robert Israel, Oct 09 2016

			Polynomials having nonnegative integer coefficients are matched to the positive integers as follows:
   m    p(m,x)    irreducible
  ---------------------------
   1    0         no
   2    1         no
   3    x         yes
   4    2         no
   5    x^2       no
   6    1+x       yes
   7    x^3       no
   8    3         no
   9    2x        yes
  10    1+x^2     yes

Antti Karttunen, Table of n, a(n) for n = 1..10566

Cf. A052485, A206285 (complement), A206296.
Positions of ones in A277322.
Terms of A277318 form a proper subset of this sequence. Cf. also A277316.
Other sequences about factorization in the same polynomial ring: A206442, A284010.
Polynomial multiplication using the same encoding: A297845.

P:= n -> add(f[2]*x^(numtheory:-pi(f[1])-1), f =  ifactors(n)[2]):
select(irreduc @ P, [$1..200]); # Robert Israel, Oct 09 2016

b[n_] := Table[x^k, {k, 0, n}];
f[n_] := f[n] = FactorInteger[n]; z = 400;
t[n_, m_, k_] := If[PrimeQ[f[n][[m, 1]]] && f[n][[m, 1]]
== Prime[k], f[n][[m, 2]], 0];
u = Table[Apply[Plus,
    Table[Table[t[n, m, k], {k, 1, PrimePi[n]}], {m, 1,
      Length[f[n]]}]], {n, 1, z}];
p[n_, x_] := u[[n]].b[-1 + Length[u[[n]]]]
Table[p[n, x], {n, 1, z/4}]
v = {}; Do[n++; If[IrreduciblePolynomialQ[p[n, x]],
AppendTo[v, n]], {n, z/2}]; v  (* A206284 *)
Complement[Range[200], v]      (* A206285 *)

is(n)=my(f=factor(n));polisirreducible(sum(i=1, #f[,1], f[i,2]*'x^primepi(f[i,1]-1))) \\ Charles R Greathouse IV, Feb 12 2012

Introductory comments edited by Antti Karttunen, Oct 09 2016 and Peter Munn, Aug 13 2022

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)).

A277227 Triangular array T read by rows: T(n,k) gives the additive orders k modulo n, for k = 0,1, ..., n-1.

Original entry on oeis.org

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

Offset: 1

Wolfdieter Lang, Oct 20 2016

As a sequence A054531(n) = a(n+1), n >= 1.
As a triangular array this is the row reversed version of A054531.
The additive order of an element x of a group (G, +) is the least positive integer j with j*x := x + x + ... + x (j summands) = 0.
Equals A106448 when the first column (k = 0) of ones is removed. - Georg Fischer, Jul 26 2023

			The triangle begins:
n\k 0  1  2  3  4  5  6  7  8  9 10 11 ...
1:  1
2:  1  2
3:  1  3  3
4:  1  4  2  4
5:  1  5  5  5  5
6:  1  6  3  2  3  6
7:  1  7  7  7  7  7  7
8:  1  8  4  8  2  8  4  8
9:  1  9  9  3  9  9  3  9  9
10: 1 10  5 10  5  2  5 10  5 10
11: 1 11 11 11 11 11 11 11 11 11 11
12: 1 12  6  4  3 12  2 12  3  4  6 12
...
T(n, 0) = 1*0 = 0 = 0 (mod n), and n/GCD(n,0) = n/n = 1.
T(4, 2) = 2 because 2 + 2 = 4 = 0 (mod 4) and 2 is not 0 (mod 4).
T(4, 2) = n/GCD(2, 4) = 4/2 = 2.

Indranil Ghosh, Rows 1..100 of triangle, flattened

Cf. A054531, A106448.

T(n, k) = order of the elements k  of the finite abelian group (Z/(n Z), +), for k = 0, 1, ..., n-1.
T(n, k) = n/GCD(n, k), n >= 1, k = 0, 1, ..., n-1.
T(n, k) = A054531(n, n-k), n >=1, k = 0, 1, ..., n-1.

A355860 Triangle read by rows: T(n,k) = nk/(n + k) if n+k divides nk, otherwise T(n,k) = 0; n >= 1, k >= 1.

Original entry on oeis.org

0, 0, 1, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 4, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 7

Offset: 1

Ctibor O. Zizka, Jul 19 2022

			The triangle begins:
  0;
  0, 1;
  0, 0, 0;
  0, 0, 0, 2;
  0, 0, 0, 0, 0;
  0, 0, 2, 0, 0, 3;
and so on.

Antti Karttunen, Table of n, a(n) for n = 1..22155; the first 210 rows of the triangle

Cf. also A106448.

T[n_, k_] := If[Divisible[n*k, n + k], n*k/(n + k), 0]; Table[T[n, k], {n, 1, 12}, {k, 1, n}] // Flatten (* Amiram Eldar, Jul 20 2022 *)

up_to = 105;
A355860tr(n,k) = ((q->if(1==denominator(q),q,0))((n*k)/(n+k)));
A355860list(up_to) = { my(v = vector(up_to), i=0); for(n=1,oo, for(k=1,n, if(i++ > up_to, return(v)); v[i] = A355860tr(n,k))); (v); };
v355860 = A355860list(up_to);
A355860(n) = v355860[n]; \\ Antti Karttunen, Jan 16 2025

Data section extended up to a(105) by Antti Karttunen, Jan 16 2025

cp's OEIS Frontend

A206284 Numbers that match irreducible polynomials over the nonnegative integers.

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A277227 Triangular array T read by rows: T(n,k) gives the additive orders k modulo n, for k = 0,1, ..., n-1.

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A355860 Triangle read by rows: T(n,k) = n*k/(n + k) if n+k divides n*k, otherwise T(n,k) = 0; n >= 1, k >= 1.

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A355860 Triangle read by rows: T(n,k) = nk/(n + k) if n+k divides nk, otherwise T(n,k) = 0; n >= 1, k >= 1.