A207670 -id:A207670 - cp's OEIS frontend

A263273 Bijective base-3 reverse: a(0) = 0; for n >= 1, a(n) = A030102(A038502(n)) * A038500(n).

0, 1, 2, 3, 4, 7, 6, 5, 8, 9, 10, 19, 12, 13, 22, 21, 16, 25, 18, 11, 20, 15, 14, 23, 24, 17, 26, 27, 28, 55, 30, 37, 64, 57, 46, 73, 36, 31, 58, 39, 40, 67, 66, 49, 76, 63, 34, 61, 48, 43, 70, 75, 52, 79, 54, 29, 56, 33, 38, 65, 60, 47, 74, 45, 32, 59, 42, 41, 68, 69, 50, 77, 72, 35, 62, 51, 44, 71, 78, 53, 80, 81

Offset: 0

Antti Karttunen, Dec 05 2015

Here the base-3 reverse has been adjusted so that the maximal suffix of trailing zeros (in base-3 representation A007089) stays where it is at the right side, and only the section from the most significant digit to the least significant nonzero digit is reversed, thus making this sequence a self-inverse permutation of nonnegative integers.
Because successive powers of 3 and 9 modulo 2, 4 and 8 are always either constant 1, 1, 1, ... or alternating 1, -1, 1, -1, ... it implies similar simple divisibility rules for 2, 4 and 8 in base 3 as e.g. 3, 9 and 11 have in decimal base (see the Wikipedia-link). As these rules do not depend on which direction they are applied from, it means that this bijection preserves the fact whether a number is divisible by 2, 4 or 8, or whether it is not. Thus natural numbers are divided to several subsets, each of which is closed with respect to this bijection. See the Crossrefs section for permutations obtained from these sections.
When polynomials over GF(3) are encoded as natural numbers (coefficients presented with the digits of the base-3 expansion of n), this bijection works as a multiplicative automorphism of the ring GF(3)[X]. This follows from the fact that as there are no carries involved, the multiplication (and thus also the division) of such polynomials could be as well performed by temporarily reversing all factors (like they were seen through mirror). This implies also that the sequences A207669 and A207670 are closed with respect to this bijection.

			For n = 15, A007089(15) = 120. Reversing this so that the trailing zero stays at the right yields 210 = A007089(21), thus a(15) = 21 and vice versa, a(21) = 15.

Antti Karttunen, Table of n, a(n) for n = 0..6561
Wikipedia, Divisibility rule
Index entries for sequences that are permutations of the natural numbers

Bisections: A264983, A264984.
Cf. A000035, A007089, A010873, A030102, A038500, A038502, A207669, A207670.
Cf. also A057889, A264965, A264966, A264980.
Permutations induced by various sections: A263272 (a(2n)/2), A264974 (a(4n)/4), A264978 (a(8n)/8), A264985, A264989.
Cf. also A004488, A140263, A140264, A246207, A246208 (other base-3 related permutations).

r[n_] := FromDigits[Reverse[IntegerDigits[n, 3]], 3]; b[n_] := n/ 3^IntegerExponent[n, 3]; c[n_] := n/b[n]; a[0]=0; a[n_] := r[b[n]]*c[n]; Table[a[n], {n, 0, 80}] (* Jean-François Alcover, Dec 29 2015 *)

from sympy import factorint
from sympy.ntheory.factor_ import digits
from operator import mul
def a030102(n): return 0 if n==0 else int(''.join(map(str, digits(n, 3)[1:][::-1])), 3)
def a038502(n):
    f=factorint(n)
    return 1 if n==1 else reduce(mul, [1 if i==3 else i**f[i] for i in f])
def a038500(n): return n/a038502(n)
def a(n): return 0 if n==0 else a030102(a038502(n))*a038500(n) # Indranil Ghosh, May 22 2017

(define (A263273 n) (if (zero? n) n (* (A030102 (A038502 n)) (A038500 n))))

a(0) = 0; for n >= 1, a(n) = A030102(A038502(n)) * A038500(n).
Other identities. For all n >= 0:
a(3*n) = 3*a(n).
A000035(a(n)) = A000035(n). [This permutation preserves the parity of n.]
A010873(a(n)) = 0 if and only if A010873(n) = 0. [See the comments section.]

A325820 Multiplication table for carryless product i X j in base 3 for i >= 0 and j >= 0, read by antidiagonals.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 2, 2, 0, 0, 3, 1, 3, 0, 0, 4, 6, 6, 4, 0, 0, 5, 8, 9, 8, 5, 0, 0, 6, 7, 12, 12, 7, 6, 0, 0, 7, 3, 15, 16, 15, 3, 7, 0, 0, 8, 5, 18, 11, 11, 18, 5, 8, 0, 0, 9, 4, 21, 24, 13, 24, 21, 4, 9, 0, 0, 10, 18, 24, 19, 21, 21, 19, 24, 18, 10, 0, 0, 11, 20, 27, 23, 26, 9, 26, 23, 27, 20, 11, 0, 0, 12, 19, 30, 36, 19, 15, 15, 19, 36, 30, 19, 12, 0

Offset: 0

Antti Karttunen, May 22 2019

			The array begins as:
  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, ...
  0,  2,  1,  6,  8,  7,  3,  5,  4,  18,  20,  19,  24, ...
  0,  3,  6,  9, 12, 15, 18, 21, 24,  27,  30,  33,  36, ...
  0,  4,  8, 12, 16, 11, 24, 19, 23,  36,  40,  44,  48, ...
  0,  5,  7, 15, 11, 13, 21, 26, 19,  45,  50,  52,  33, ...
  0,  6,  3, 18, 24, 21,  9, 15, 12,  54,  60,  57,  72, ...
  0,  7,  5, 21, 19, 26, 15, 13, 11,  63,  70,  68,  57, ...
  0,  8,  4, 24, 23, 19, 12, 11, 16,  72,  80,  76,  69, ...
  0,  9, 18, 27, 36, 45, 54, 63, 72,  81,  90,  99, 108, ...
  0, 10, 20, 30, 40, 50, 60, 70, 80,  90, 100,  83, 120, ...
  0, 11, 19, 33, 44, 52, 57, 68, 76,  99,  83,  91, 132, ...
  0, 12, 24, 36, 48, 33, 72, 57, 69, 108, 120, 132, 144, ...
  etc.
A(2,2) = 2*2 mod 3 = 1.

Cf. A169999 (the main diagonal).
Row/Column 0: A000004, Row/Column 1: A001477, Row/Column 2: A004488, Row/Column 3: A008585, Row/Column 4: A242399, Row/Column 9: A008591.
Cf. A007089, A207669, A207670, A263273, A325825.
Cf. A325821 (same table without the zero row and column).
Cf. A048720 (binary), A059692 (decimal), A004247 (full multiply).

up_to = 105;
A325820sq(b, c) = fromdigits(Vec(Pol(digits(b,3))*Pol(digits(c,3)))%3, 3);
A325820list(up_to) = { my(v = vector(up_to), i=0); for(a=0,oo, for(col=0,a, if(i++ > up_to, return(v)); v[i] = A325820sq(a-col,col))); (v); };
v325820 = A325820list(up_to);
A325820(n) = v325820[1+n];

A207669 Numbers that match polynomials irreducible (mod 3), with coefficients in {0,1,2}.

Original entry on oeis.org

3, 4, 5, 6, 7, 8, 10, 14, 17, 20, 22, 25, 34, 35, 38, 41, 43, 46, 49, 53, 58, 59, 65, 67, 71, 73, 77, 79, 86, 89, 92, 94, 97, 101, 110, 115, 118, 121, 125, 134, 137, 139, 145, 149, 151, 158, 166, 169, 172, 181, 185, 188, 190, 197, 205, 209, 212, 214, 217

Offset: 1

Clark Kimberling, Feb 26 2012

nonn

For such polynomials irreducible over the field of rational numbers, see A207966, which also describes the enumeration of all the nonzero polynomials whose coefficients are all in {0,1,2}.

			Polynomials having coefficients in {0,1,2} are
enumerated by the positive integers as follows:
n ... p[n,x] .. irreducible (mod 3)
1 ... 1 ....... no
2 ... 2 ....... no
3 ... x ....... yes
4 ... x+1 ..... yes
5 ... x+2 ..... yes
6 ... 2x ...... yes
7 ... 2x+1 .... yes
8 ... 2x+2 .... yes
9 ... x^2 ..... no
10 .. x^2+1 ... yes
11 .. x^2+2 ... no
The least n for which p(n,x) is irreducible over the
rationals but not modulo 3 is 13; the factorization of
p(13,x) is (x+1)(x+2) (mod 3).

Cf. A207670 (complement), A207671 (ternary).

t = Table[IntegerDigits[n, 3], {n, 1, 1000}];
b[n_] := Reverse[Table[x^k, {k, 0, n}]]
p[n_, x_] := t[[n]].b[-1 + Length[t[[n]]]]
Table[p[n, x], {n, 1, 15}]
u = {}; Do[n++;
If[IrreduciblePolynomialQ[p[n, x], Modulus -> 3],
  AppendTo[u, n]], {n, 1, 400}]
u                           (* A207669 *)
Complement[Range[200], %]   (* A207670 *)
b[n_] := FromDigits[IntegerDigits[u, 3][[n]]]
Table[b[n], {n, 1, 50}]     (* A207671 *)

A207671 Numbers in ternary representation that match nonzero polynomials with all coefficients in {0,1,2} that are irreducible modulo 3.

Original entry on oeis.org

10, 11, 12, 20, 21, 22, 101, 112, 122, 202, 211, 221, 1021, 1022, 1102, 1112, 1121, 1201, 1211, 1222, 2011, 2012, 2102, 2111, 2122, 2201, 2212, 2221, 10012, 10022, 10102, 10111, 10121, 10202, 11002, 11021, 11101, 11111, 11122, 11222

Offset: 1

Clark Kimberling, Feb 26 2012

For a discussion and examples in base-10 representation, see A207670. For the analogous sequence in base 2, see A206073.

			(See the Example section of A207669.)

Cf. A207669, A206073.

t = Table[IntegerDigits[n, 3], {n, 1, 1000}];
b[n_] := Reverse[Table[x^k, {k, 0, n}]]
p[n_, x_] := t[[n]].b[-1 + Length[t[[n]]]]
Table[p[n, x], {n, 1, 15}]
u = {}; Do[n++;
If[IrreduciblePolynomialQ[p[n, x], Modulus -> 3],
  AppendTo[u, n]], {n, 1, 400}]
u                           (* A207669 *)
Complement[Range[200], %]   (* A207670 *)
b[n_] := FromDigits[IntegerDigits[u, 3][[n]]]
Table[b[n], {n, 1, 50}]     (* A207671 *)

cp's OEIS Frontend

A263273 Bijective base-3 reverse: a(0) = 0; for n >= 1, a(n) = A030102(A038502(n)) * A038500(n).

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A325820 Multiplication table for carryless product i X j in base 3 for i >= 0 and j >= 0, read by antidiagonals.

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A207669 Numbers that match polynomials irreducible (mod 3), with coefficients in {0,1,2}.

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Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A207671 Numbers in ternary representation that match nonzero polynomials with all coefficients in {0,1,2} that are irreducible modulo 3.

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Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica