A268389 -id:A268389 - cp's OEIS frontend

A001317 Sierpiński's triangle (Pascal's triangle mod 2) converted to decimal.

1, 3, 5, 15, 17, 51, 85, 255, 257, 771, 1285, 3855, 4369, 13107, 21845, 65535, 65537, 196611, 327685, 983055, 1114129, 3342387, 5570645, 16711935, 16843009, 50529027, 84215045, 252645135, 286331153, 858993459, 1431655765, 4294967295, 4294967297, 12884901891, 21474836485, 64424509455, 73014444049, 219043332147, 365072220245, 1095216660735, 1103806595329, 3311419785987

Offset: 0

N. J. A. Sloane

The members are all palindromic in binary, i.e., a subset of A006995. - Ralf Stephan, Sep 28 2004
J. H. Conway writes (in Math Forum): at least the first 31 numbers give odd-sided constructible polygons. See also A047999. - M. Dauchez (mdzzdm(AT)yahoo.fr), Sep 19 2005 [This observation was also made in 1982 by N. L. White (see letter). - N. J. A. Sloane, Jun 15 2015]
Decimal number generated by the binary bits of the n-th generation of the Rule 60 elementary cellular automaton. Thus: 1; 0, 1, 1; 0, 0, 1, 0, 1; 0, 0, 0, 1, 1, 1, 1; 0, 0, 0, 0, 1, 0, 0, 0, 1; ... . - Eric W. Weisstein, Apr 08 2006
Limit_{n->oo} log(a(n))/n = log(2). - Bret Mulvey, May 17 2008
Equals row sums of triangle A166548; e.g., 17 = (2 + 4 + 6 + 4 + 1). - Gary W. Adamson, Oct 16 2009
Equals row sums of triangle A166555. - Gary W. Adamson, Oct 17 2009
For n >= 1, all terms are in A001969. - Vladimir Shevelev, Oct 25 2010
Let n,m >= 0 be such that no carries occur when adding them. Then a(n+m) = a(n)*a(m). - Vladimir Shevelev, Nov 28 2010
Let phi_a(n) be the number of a(k) <= a(n) and respectively prime to a(n) (i.e., totient function over {a(n)}). Then, for n >= 1, phi_a(n) = 2^v(n), where v(n) is the number of 0's in the binary representation of n. - Vladimir Shevelev, Nov 29 2010
Trisection of this sequence gives rows of A008287 mod 2 converted to decimal. See also A177897, A177960. - Vladimir Shevelev, Jan 02 2011
Converting the rows of the powers of the k-nomial (k = 2^e where e >= 1) term-wise to binary and reading the concatenation as binary number gives every (k-1)st term of this sequence. Similarly with powers p^k of any prime. It might be interesting to study how this fails for powers of composites. - Joerg Arndt, Jan 07 2011
This sequence appears in Pascal's triangle mod 2 in another way, too. If we write it as
  1111111...
  10101010...
  11001100...
  10001000...
we get (taking the period part in each row):
  .(1) (base 2) = 1
  .(10) = 2/3
  .(1100) = 12/15 = 4/5
  .(1000) = 8/15
The k-th row, treated as a binary fraction, seems to be equal to 2^k / a(k). - Katarzyna Matylla, Mar 12 2011
From Daniel Forgues, Jun 16-18 2011: (Start)
Since there are 5 known Fermat primes, there are 32 products of distinct Fermat primes (thus there are 31 constructible odd-sided polygons, since a polygon has at least 3 sides). a(0)=1 (empty product) and a(1) to a(31) are those 31 non-products of distinct Fermat primes.
It can be proved by induction that all terms of this sequence are products of distinct Fermat numbers (A000215):
a(0)=1 (empty product) are products of distinct Fermat numbers in { };
a(2^n+k) = a(k) * (2^(2^n)+1) = a(k) * F_n, n >= 0, 0 <= k <= 2^n - 1.
Thus for n >= 1, 0 <= k <= 2^n - 1, and
  a(k) = Product_{i=0..n-1} F_i^(alpha_i), alpha_i in {0, 1},
this implies
  a(2^n+k) = Product_{i=0..n-1} F_i^(alpha_i) * F_n, alpha_i in {0, 1}.
(Cf. OEIS Wiki links below.)  (End)
The bits in the binary expansion of a(n) give the coefficients of the n-th power of polynomial (X+1) in ring GF(2)[X]. E.g., 3 ("11" in binary) stands for (X+1)^1, 5 ("101" in binary) stands for (X+1)^2 = (X^2 + 1), and so on. - Antti Karttunen, Feb 10 2016

			Given a(5)=51, a(6)=85 since a(5) XOR 2*a(5) = 51 XOR 102 = 85.
From _Daniel Forgues_, Jun 18 2011: (Start)
  a(0) = 1 (empty product);
  a(1) = 3 = 1 * F_0 = a(2^0+0) = a(0) * F_0;
  a(2) = 5 = 1 * F_1 = a(2^1+0) = a(0) * F_1;
  a(3) = 15 = 3 * 5 = F_0 * F_1 = a(2^1+1) = a(1) * F_1;
  a(4) = 17 = 1 * F_2 = a(2^2+0) = a(0) * F_2;
  a(5) = 51 = 3 * 17 = F_0 * F_2 = a(2^2+1) = a(1) * F_2;
  a(6) = 85 = 5 * 17 = F_1 * F_2 = a(2^2+2) = a(2) * F_2;
  a(7) = 255 = 3 * 5 * 17 = F_0 * F_1 * F_2 = a(2^2+3) = a(3) * F_2;
  ... (End)

Jean-Paul Allouche and Jeffrey Shallit, Automatic sequences, Cambridge University Press, 2003, p. 113.
Henry Wadsworth Gould, Exponential Binomial Coefficient Series, Tech. Rep. 4, Math. Dept., West Virginia Univ., Morgantown, WV, Sept. 1961.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, pages 136-137.

Amiram Eldar, Table of n, a(n) for n = 0..3321 (terms 0..300 from T. D. Noe, terms 301..1023 from Tilman Piesk)
Gary W. Adamson, Letter to N. J. A. Sloane, Apr 29 1994, and MS "Algorithm for the Chinese Remainder Theorem".
Gary W. Adamson and N. J. A. Sloane, Correspondence, May 1994, including Adamson's MSS "Algorithm for Generating nth Row of Pascal's Triangle, mod 2, from n", and "The Tower of Hanoi Wheel".
Cristian Cobeli and Alexandru Zaharescu, Promenade around Pascal Triangle-Number Motives, Bull. Math. Soc. Sci. Math. Roumanie, Tome 56(104), No. 1 (2013), pp. 73-98; alternative link.
Richard K. Guy, The Second Strong Law of Small Numbers, Math. Mag, Vol. 63, No. 1 (1990), pp. 3-20. [Annotated scanned copy]
Richard K. Guy and N. J. A. Sloane, Correspondence, 1988.
Denton Hewgill, A relationship between Pascal's triangle and Fermat numbers, Fib. Quart., Vol. 15, No. 2 (1977), pp. 183-184.
A. J. Macfarlane, Generating functions for integer sequences defined by the evolution of cellular automata with even rule numbers, Fig. 4.
Dr. Math, Regular polygon formulas.
P. Mathonet, M. Rigo, M. Stipulanti and N. Zénaïdi, On digital sequences associated with Pascal's triangle, arXiv:2201.06636 [math.NT], 2022.
OEIS Wiki, Constructible odd-sided polygons and Sierpinski's triangle.
Vladimir Shevelev, On Stephan's conjectures concerning Pascal triangle modulo 2, J. Alg. Number Theory, Vol. 7, No. 1 (2012) pp. 11-29, preprint, arXiv:1011.6083 [math.NT], 2010-2012.
John Frederick Sweeney, Clifford Clock and the Moolakaprithi Cube, 2014. See p. 26. - _N. J. A. Sloane_, Mar 20 2014
Eric Weisstein's World of Mathematics, Rule 60 and Rule 102.
Neil L. White, Letter to N. J. A. Sloane, Apr 15 1982.
R. G. Wilson, V, Letter to N. J. A. Sloane, Aug. 1993.
Index entries for sequences related to cellular automata
Index entries for sequences operating on polynomials in ring GF(2)[X]

Cf. A000079, A000215 (Fermat numbers), A047999, A048720, A054432, A173019, A177882, A177897, A177960, A193231, A230116, A247282, A249184, A268391.
Cf. A038183 (odd bisection, 1D Cellular Automata Rule 90).
Iterates of A048724 (starting from 1).
Row 3 of A048723.
Positions of records in A268389.
Positions of ones in A268669 and A268384 (characteristic function).
Not the same as A045544 nor as A053576.
Cf. A045544.

a001317 = foldr (\u v-> 2*v + u) 0 . map toInteger . a047999_row
-- Reinhard Zumkeller, Nov 24 2012
(Scheme, with memoization-macro definec, two variants)
(definec (A001317 n) (if (zero? n) 1 (A048724 (A001317 (- n 1)))))
(definec (A001317 n) (if (zero? n) 1 (A048720bi 3 (A001317 (- n 1))))) ;; Where A048720bi implements the dyadic function given in A048720.
;; Antti Karttunen, Feb 10 2016

[&+[(Binomial(n, i) mod 2)*2^i: i in [0..n]]: n in [0..41]]; // Vincenzo Librandi, Feb 12 2016

A001317 := proc(n) local k; add((binomial(n,k) mod 2)*2^k, k=0..n); end;

a[n_] := Nest[ BitXor[#, BitShiftLeft[#, 1]] &, 1, n]; Array[a, 42, 0] (* Joel Madigan (dochoncho(AT)gmail.com), Dec 03 2007 *)
NestList[BitXor[#,2#]&,1,50] (* Harvey P. Dale, Aug 02 2021 *)

PARI
```
a(n)=sum(i=0,n,(binomial(n,i)%2)*2^i)
```

a=1; for(n=0, 66, print1(a,", "); a=bitxor(a,a<<1) ); \\ Joerg Arndt, Mar 27 2013

A001317(n,a=1)={for(k=1,n,a=bitxor(a,a<<1));a} \\ M. F. Hasler, Jun 06 2016

a(n) = subst(lift(Mod(1+'x,2)^n), 'x, 2); \\ Gheorghe Coserea, Nov 09 2017

from sympy import binomial
def a(n): return sum([(binomial(n, i)%2)*2**i for i in range(n + 1)]) # Indranil Ghosh, Apr 11 2017

def A001317(n): return int(''.join(str(int(not(~n&k))) for k in range(n+1)),2) # Chai Wah Wu, Feb 04 2022

a(n+1) = a(n) XOR 2*a(n), where XOR is binary exclusive OR operator. - Paul D. Hanna, Apr 27 2003
a(n) = Product_{e(j, n) = 1} (2^(2^j) + 1), where e(j, n) is the j-th least significant digit in the binary representation of n (Roberts: see Allouche & Shallit). - Benoit Cloitre, Jun 08 2004
a(2*n+1) = 3*a(2*n). Proof: Since a(n) = Product_{k in K} (1 + 2^(2^k)), where K is the set of integers such that n = Sum_{k in K} 2^k, clearly K(2*n+1) = K(2*n) union {0}, hence a(2*n+1) = (1+2^(2^0))*a(2*n) = 3*a(2*n). - Emmanuel Ferrand and Ralf Stephan, Sep 28 2004
a(32*n) = 3 ^ (32 * n * log(2) / log(3)) + 1. - Bret Mulvey, May 17 2008
For n >= 1, A000120(a(n)) = 2^A000120(n). - Vladimir Shevelev, Oct 25 2010
a(2^n) = A000215(n); a(2^n-1) = a(2^n)-2; for n >= 1, m >= 0,
a(2^(n-1)-1)*a(2^n*m + 2^(n-1)) = 3*a(2^(n-1))*a(2^n*m + 2^(n-1)-2). - Vladimir Shevelev, Nov 28 2010
Sum_{k>=0} 1/a(k) = Product_{n>=0} (1 + 1/F_n), where F_n=A000215(n);
Sum_{k>=0} (-1)^(m(k))/a(k) = 1/2, where {m(n)} is Thue-Morse sequence (A010060).
If F_n is defined by F_n(z) = z^(2^n) + 1 and a(n) by (1/2)*Sum_{i>=0}(1-(-1)^{binomial(n,i)})*z^i, then, for z > 1, the latter two identities hold as well with the replacement 1/2 in the right hand side of the 2nd one by 1-1/z. - Vladimir Shevelev, Nov 29 2010
G.f.: Product_{k>=0} ( 1 + z^(2^k) + (2*z)^(2^k) ). - conjectured by Shamil Shakirov, proved by Vladimir Shevelev
a(n) = A000225(n+1) - A219843(n). - Reinhard Zumkeller, Nov 30 2012
From Antti Karttunen, Feb 10 2016: (Start)
a(0) = 1, and for n > 1, a(n) = A048720(3, a(n-1)) = A048724(a(n-1)).
a(n) = A048723(3,n).
a(n) = A193231(A000079(n)).
For all n >= 0: A268389(a(n)) = n.
(End)

A277820 Square array: A(r,1) = A065621(r); for c > 1, A(r,c) = A048724(A(r,c-1)), read by descending antidiagonals as A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), etc.

Original entry on oeis.org

1, 3, 2, 5, 6, 7, 15, 10, 9, 4, 17, 30, 27, 12, 13, 51, 34, 45, 20, 23, 14, 85, 102, 119, 60, 57, 18, 11, 255, 170, 153, 68, 75, 54, 29, 8, 257, 510, 427, 204, 221, 90, 39, 24, 25, 771, 514, 765, 340, 359, 238, 105, 40, 43, 26, 1285, 1542, 1799, 1020, 937, 306, 187, 120, 125, 46, 31, 3855, 2570, 2313, 1028, 1275, 854, 461, 136, 135, 114, 33, 28

Offset: 1

Antti Karttunen, Nov 01 2016

For all n >= 1, A277818 (= A268389(n)+1) gives the (one-based) index of the column where n is located in this array, while A268671(n) gives the (one-based) index of the row where it is on.

This array is obtained when one selects from A277320 the columns 1, 3, 5, 15, 17, 51, ..., i.e., those with an index A001317(k).

			The top left corner of the array:
   1,  3,   5,  15,  17,   51,   85,  255,   257,   771,  1285,  3855
   2,  6,  10,  30,  34,  102,  170,  510,   514,  1542,  2570,  7710
   7,  9,  27,  45, 119,  153,  427,  765,  1799,  2313,  6939, 11565
   4, 12,  20,  60,  68,  204,  340, 1020,  1028,  3084,  5140, 15420
  13, 23,  57,  75, 221,  359,  937, 1275,  3341,  5911, 14649, 19275
  14, 18,  54,  90, 238,  306,  854, 1530,  3598,  4626, 13878, 23130
  11, 29,  39, 105, 187,  461,  599, 1785,  2827,  7453, 10023, 26985
   8, 24,  40, 120, 136,  408,  680, 2040,  2056,  6168, 10280, 30840
  25, 43, 125, 135, 393,  667, 1965, 2295,  6425, 11051, 32125, 34695
  26, 46, 114, 150, 442,  718, 1874, 2550,  6682, 11822, 29298, 38550
  31, 33,  99, 165, 495,  561, 1619, 2805,  7967,  8481, 25443, 42405
  28, 36, 108, 180, 476,  612, 1708, 3060,  7196,  9252, 27756, 46260
  21, 63,  65, 195, 325,  975, 1105, 3315,  5397, 16191, 16705, 50115
  22, 58,  78, 210, 374,  922, 1198, 3570,  5654, 14906, 20046, 53970
  19, 53,  95, 225, 291,  869, 1455, 3825,  4883, 13621, 24415, 57825
  16, 48,  80, 240, 272,  816, 1360, 4080,  4112, 12336, 20560, 61680
  49, 83, 245, 287, 801, 1379, 4005, 4335, 12593, 21331, 62965, 73247
  50, 86, 250, 270, 786, 1334, 3930, 4590, 12850, 22102, 64250, 69390
  55, 89, 235, 317, 839, 1481, 3675, 4845, 14135, 22873, 60395, 80957

Inverse permutation: A277821.
Transpose: A277819.
Cf. A048724, A065621.
Row 1: A001317.
Column 1: A065621, column 2: A277823, column 3: A277825.
Cf. A268389, A277818, A268671.
Cf. also A193231, A277320, A277810, A276618 (A099884), A248513.
Other related tables or permutations: A277880, A277901.

(define (A277820 n) (A277820bi (A002260 n) (A004736 n)))
(define (A277820bi row col) (if (= 1 col) (A065621 row) (A048724 (A277820bi row (- col 1)))))

A(r,1) = A065621(r); for c > 1, A(r,c) = A048724(A(r,c-1)).
A(r,c) = A048675(A277810(r,c)).
As a composition of other permutations:
a(n) = A277901(A277880(n)).

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

Antti Karttunen, Jan 09 2017

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.

			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).

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

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

(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.

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).

A305421 GF(2)[X] factorization prime shift towards larger terms.

Original entry on oeis.org

1, 3, 7, 5, 21, 9, 11, 15, 49, 63, 13, 27, 19, 29, 107, 17, 273, 83, 25, 65, 69, 23, 121, 45, 31, 53, 151, 39, 35, 189, 37, 51, 251, 819, 173, 245, 41, 43, 233, 195, 47, 207, 93, 57, 997, 139, 55, 119, 127, 33, 1911, 95, 79, 441, 59, 105, 367, 101, 61, 455, 67, 111, 475, 85, 1281, 269, 73, 1365, 81, 503, 457, 287, 87, 123, 1549, 125, 179, 315

Offset: 1

Antti Karttunen, Jun 07 2018

nonn

Permutation of the odd numbers, A005408.

Let a x b stand for the carryless binary multiplication of positive integers a and b, that is, the result of operation A048720(a,b). With n having a unique factorization as A014580(i) x A014580(j) x ... x A014580(k), 1 <= i <= j <= ... <= k, a(n) = A014580(1+i) x A014580(1+j) x ... x A014580(1+k).

			For n = 12, which by its binary representation '1100' corresponds with (0,1)-polynomial x^3 + x^2, which over GF(2)[X] is factored as (x)(x)(x+1), i.e., 12 = A048720(2,A048720(2,3)) = A048720(A014580(1), A048720(A014580(1),A014580(2))), we then form a(12) as A048720(A014580(2), A048720(A014580(2),A014580(3))) = A048720(3,A048720(3,7)) = 27. Note that x, x+1 and x^2 + x + 1 are the three smallest irreducible (0,1)-polynomials when factored over GF(2)[X], and their binary representations 2, 3 and 7 are the three initial terms of A014580.

Antti Karttunen, Table of n, a(n) for n = 1..21845
Index entries for sequences related to polynomials in ring GF(2)[X]

Cf. A305422 (a left inverse).
Cf. A001317, A014580, A091225, A193231, A268389, A305420, A305423.
Cf. also A003961, A300841.

A091225(n) = polisirreducible(Pol(binary(n))*Mod(1, 2));
A305420(n) = { my(k=1+n); while(!A091225(k),k++); (k); };
A305421(n) = { my(f = subst(lift(factor(Pol(binary(n))*Mod(1, 2))),x,2)); for(i=1,#f~,f[i,1] = Pol(binary(A305420(f[i,1])))); fromdigits(Vec(factorback(f))%2,2); };

For all n >= 1:
A305422(a(n)) = n.
A268389(a(n)) = A007814(n).
a(A000079(n)) = A001317(n).

A305422 GF(2)[X] factorization prime shift towards smaller terms.

Original entry on oeis.org

1, 1, 2, 1, 4, 2, 3, 1, 6, 4, 7, 2, 11, 3, 8, 1, 16, 6, 13, 4, 5, 7, 22, 2, 19, 11, 12, 3, 14, 8, 25, 1, 50, 16, 29, 6, 31, 13, 28, 4, 37, 5, 38, 7, 24, 22, 41, 2, 9, 19, 32, 11, 26, 12, 47, 3, 44, 14, 55, 8, 59, 25, 10, 1, 20, 50, 61, 16, 21, 29, 118, 6, 67, 31, 88, 13, 110, 28, 53, 4, 69, 37, 18, 5, 64, 38, 73, 7, 94, 24, 87, 22, 43, 41, 52, 2, 91

Offset: 1

Antti Karttunen, Jun 07 2018

nonn

Let a x b stand for the carryless binary multiplication of positive integers a and b, that is, the result of operation A048720(a,b). With n having a unique factorization as f(i) x f(j) x ... x f(k), with 1 <= i <= j <= ... <= k, a(n) = f(i-1) x f(j-1) x ... x f(k-1), where f(0) = 1, and f(n) = A014580(n) for n >= 1.

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences related to polynomials in ring GF(2)[X]

Cf. A000079 (positions of ones), A014580, A091225, A268389, A305419, A305421, A305424 (odd bisection), A305425.

Cf. also A064989, A300840.

A091225(n) = polisirreducible(Pol(binary(n))*Mod(1, 2));
A305419(n) = if(n<3,1, my(k=n-1); while(k>1 && !A091225(k),k--); (k));
A305422(n) = { my(f = subst(lift(factor(Pol(binary(n))*Mod(1, 2))),x,2)); for(i=1,#f~,f[i,1] = Pol(binary(A305419(f[i,1])))); fromdigits(Vec(factorback(f))%2,2); };

For all n >= 1:
a(A305421(n)) = n.
a(A001317(n)) = A000079(n).
A007814(a(n)) = A268389(n).

A268669 a(n) = polynomial quotient (computed over GF(2), result is its binary encoding) that is left after all instances of polynomial (X+1) have been factored out of the polynomial that is encoded by the binary expansion of n. (See comments for details).

Original entry on oeis.org

1, 2, 1, 4, 1, 2, 7, 8, 7, 2, 11, 4, 13, 14, 1, 16, 1, 14, 19, 4, 21, 22, 13, 8, 25, 26, 7, 28, 11, 2, 31, 32, 31, 2, 35, 28, 37, 38, 11, 8, 41, 42, 25, 44, 7, 26, 47, 16, 49, 50, 1, 52, 19, 14, 55, 56, 13, 22, 59, 4, 61, 62, 21, 64, 21, 62, 67, 4, 69, 70, 61, 56, 73, 74, 13, 76, 59, 22, 79, 16, 81, 82, 49, 84, 1

Offset: 1

Antti Karttunen, Feb 10 2016

When polynomials over GF(2) are encoded in the binary representation of n in a natural way where each polynomial b(n)*X^n+...+b(0)*X^0 over GF(2) is represented by the binary number b(n)*2^n+...+b(0)*2^0 in N (each coefficient b(k) is either 0 or 1), then a(n) = representation of the polynomial that is left as a quotient when all X+1 polynomials (encoded by 3, "11" in binary) have been divided out.
Each a(n) is one of the "Garden of Eden" patterns of Rule-60 one-dimensional cellular automaton, a seed pattern which after A268389(n) generations yields the configuration encoded in binary expansion of n.
No terms of A001969 occur so all terms are odious (in A000069). Each odious number occurs an infinitely many times.

			For n = 5 ("101" in binary) which encodes polynomial x^2 + 1, we observe that it can be factored in ring GF(2)[X] as (X+1)(X+1), and thus a(5) = 1, because after dividing both instances of (X+1) off, we are left with the quotient polynomial 1 which is encoded by 1.
For n = 8 ("1000" in binary) which encodes polynomial x^3, we observe that it is not divisible in ring GF(2)[X] by polynomial X+1, thus a(8) = 8.
For n = 9 ("1001" in binary) which encodes polynomial x^3 + 1, we observe that it can be factored over GF(2) as (X+1)(X^2 + X + 1), and thus a(9) = 7, because the quotient polynomial X^2 + X + 1 is encoded by 7 ("111" in binary).

Antti Karttunen, Table of n, a(n) for n = 1..65537
Eric Weisstein's World of Mathematics, Rule 60
Index entries for sequences related to cellular automata
Index entries for sequences operating on GF(2)[X]-polynomials

Cf. A001317 (positions of ones).
Cf. A000069, A001969, A003188, A003987, A010060, A048720, A048723, A048724, A268384, A268670.
Cf. A268389 (the highest exponent for (X+1)).
Cf. also A136386.

f[n_] := If[OddQ@ #, n, f[#/2]] &@ Fold[BitXor, n, Quotient[n, 2^Range[BitLength@ n - 1]]]; Array[f, {85}] (* Michael De Vlieger, Feb 12 2016, after Jan Mangaldan at A006068 *)

a(n) = {p = Pol(binary(n))*Mod(1,2); q = (x+1)*Mod(1,2); while (type(r = p/q) == "t_POL", p = r); np = Polrev(vector(poldegree(p)+1, k, k--; lift(polcoeff(p, k)))); subst(np, x, 2);} \\ Michel Marcus, Feb 12 2016

;; This employs the given recurrence and uses memoization-macro definec:
(definec (A268669 n) (if (odd? (A006068 n)) n (A268669 (/ (A006068 n) 2))))
(define (A268669 n) (let loop ((n n)) (let ((k (A006068 n))) (if (odd? k) n (loop (/ k 2)))))) ;; Computed in a loop, no memoization.

If A006068(n) is odd, then a(n) = n, otherwise a(n) = a(A006068(n)/2).
Other identities and observations. For all n >= 1:
a(n) = A003188(A268670(n)).
A010060(a(n)) = 1. [All terms are odious.]
a(n) <= n.
More precisely, a(A000069(n)) = A000069(n) and a(A001969(n)) < A001969(n).
The equivalence of the following two formulas stems from the additive nature of Rule-60 cellular automaton. Or more plainly, because carryless binary multiplication A048720 distributes over carryless binary sum, XOR A003987:
A048724^A268389(n) (a(n)) = n. [Starting from k = a(n), and iterating map k -> A048724(k) exactly A268389(n) times yields n back.]
A048720(a(n),A048723(3,A268389(n))) = A048720(a(n),A001317(A268389(n))) = n.

cp's OEIS Frontend

A268395 Partial sums of A268389.

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A277818 Index of the column where n is located in array A277820: a(n) = 1 + A268389(n).

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A268679 a(n) = A268389(A001969(1+n)); A268389 without its zero terms.

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A001317 Sierpiński's triangle (Pascal's triangle mod 2) converted to decimal.

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A277820 Square array: A(r,1) = A065621(r); for c > 1, A(r,c) = A048724(A(r,c-1)), read by descending antidiagonals as A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), etc.

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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.

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A305421 GF(2)[X] factorization prime shift towards larger terms.

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