A037097 -id:A037097 - cp's OEIS frontend

A136386 Quotient obtained when A037097(n) is considered as a GF(2)[X]-polynomial and it is divided by (x + 1) ^ A000225(n-1) (= A051179(n-2)).

4, 8, 352, 3728, 7269662752, 761166466256046848, 390022035611646394530728097023856870592, 91600670557117582933643002658167825054614175029432880501373395030525438396928, 13417853484388319477475698658536993288839029124735549539652836318808118017743106800015257954250357092148394821846783842030516713870361254572407216621548672

Offset: 3

Antti Karttunen, Dec 29 2007

A. Karttunen, Table of n, a(n) for n = 3..13
A. Karttunen, C program for computing this sequence

A136387 shows the same sequence in binary base. Cf. A037096, A037097, A136380, A136382, A136384.

A268389 a(n) = greatest k such that polynomial (X+1)^k divides the polynomial (in polynomial ring GF(2)[X]) that is encoded in the binary expansion of n. (See the comments for details).

Original entry on oeis.org

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

Offset: 1

Antti Karttunen, Feb 10 2016

a(n) gives the number of iterations of map k -> A006068(k)/2 that are required (when starting from k = n) until k is an odd number.
A001317 gives the record positions and particularly, A001317(n) gives the first occurrence of n in this sequence.
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) = the number of times polynomial X+1 (encoded by 3, "11" in binary) divides the polynomial encoded by n.

			For n = 5 ("101" in binary) which encodes polynomial x^2 + 1, we see that it can be factored over GF(2) as (X+1)(X+1), and thus a(5) = 2.
For n = 8 ("1000" in binary) which encodes polynomial x^3, we see that it is not divisible in ring GF(2)[X] by polynomial X+1, thus a(8) = 0.
For n = 9 ("1001" in binary) which encodes polynomial x^3 + 1, we see that it can be factored over GF(2) as (X+1)(X^2 + X + 1), and thus a(9) = 1.

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences operating on GF(2)[X]-polynomials

Cf. A001317, A006068, A007949, A048720, A048723, A048724, A057889, A268384, A235042, A268670.
Cf. A268669 (quotient left after (X+1)^a(n) has been divided out).
Cf. A268395 (partial sums).
Cf. A000069 (positions of zeros), A268679 (this sequence without zeros).
Cf. also A037096, A037097, A136386.

f[n_] := Which[n == 1, 0, OddQ@ #, 0, EvenQ@ #, 1 + f[#/2]] &@ Fold[BitXor, n, Quotient[n, 2^Range[BitLength@ n - 1]]]; Array[f, {120}] (* Michael De Vlieger, Feb 12 2016, after Jan Mangaldan at A006068 *)

a(n) = {my(b = binary(n), p = Pol(binary(n))*Mod(1,2), k = poldegree(p)); while (type(p/(x+1)^k*Mod(1,2)) != "t_POL", k--); k;} \\ Michel Marcus, Feb 12 2016

;; This employs the given recurrence and uses memoization-macro definec:
(definec (A268389 n) (if (odd? (A006068 n)) 0 (+ 1 (A268389 (/ (A006068 n) 2)))))
(define (A268389 n) (let loop ((n n) (s 0)) (let ((k (A006068 n))) (if (odd? k) s (loop (/ k 2) (+ 1 s)))))) ;; Computed in a loop, no memoization.

If A006068(n) is odd, then a(n) = 0, otherwise a(n) = 1 + a(A006068(n)/2).
Other identities. For all n >= 0:
a(A001317(n)) = n. [The sequence works as a left inverse of A001317.]
A048720(A268669(n),A048723(3,a(n))) = A048720(A268669(n),A001317(a(n))) = n.
A048724^a(n) (A268669(n)) = n. [The a(n)-th fold application (power) of A048724 when applied to A268669(n) gives n back.]
a(n) = A007949(A235042(n)).
a(A057889(n)) = a(n).

A037095 "Sloping binary representation" of powers of 3 (A000244), slope = -1.

Original entry on oeis.org

1, 1, 3, 1, 3, 9, 11, 17, 19, 25, 123, 65, 195, 169, 171, 753, 435, 249, 2267, 4065, 8163, 841, 843, 31313, 29651, 39769, 38331, 30081, 160643, 49769, 53867, 563377, 700659, 1611961, 760731, 1207073, 5668771, 5566345, 11844619, 8699025, 10386067, 55868313

Offset: 0

Antti Karttunen, Jan 28 1999

			When powers of 3 are written in binary (see A004656), under each other as:
  000000000001 (1)
  000000000011 (3)
  000000001001 (9)
  000000011011 (27)
  000001010001 (81)
  000011110011 (243)
  001011011001 (729)
  100010001011 (2187)
and one collects their bits from the column-0 to NW-direction (from the least to the most significant end), one gets 1 (1), 01 (1), 011 (3), 0001 (1), 00011 (3), 001001 (9), etc. (See A105033 for similar transformation done on nonnegative integers, A001477).

Alois P. Heinz, Table of n, a(n) for n = 0..2000

Cf. A105033, A000244, A037093, A037094, A037096, A037097, A339601.

A037095:= n-> add(bit_n(3^(n-i), i)*(2^i), i=0..n):
bit_n := (x, n) -> `mod`(floor(x/(2^n)), 2):
seq(A037095(n), n=0..41);
# second Maple program:
b:= proc(n) option remember; `if`(n=0, 1, (p->
       expand((p-(p mod 2))*x/2)+3^n)(b(n-1)))
    end:
a:= n-> subs(x=2, b(n) mod 2):
seq(a(n), n=0..42);  # Alois P. Heinz, Dec 10 2020

A339601(n) = { my(m=1, s=0); while(n>=m, s += bitand(m,n); m <<= 1; n \= 3); (s); };
A037095(n) = A339601(3^n); \\ Antti Karttunen, Dec 09 2020

BINSLOPE(f) = n -> sum(i=0,n,bitand(2^(n-i),f(i))); \\ General transformation for these kinds of sequences.
A037095 = BINSLOPE(n -> 3^n); \\ And its application to A000244. - Antti Karttunen, Dec 09 2020

a(n) = A339601(A000244(n)). - Antti Karttunen, Dec 09 2020

Entry revised Dec 29 2007

More terms from Sean A. Irvine, Dec 08 2020

A037096 Periodic vertical binary vectors computed for powers of 3: a(n) = Sum_{k=0 .. (2^n)-1} (floor((3^k)/(2^n)) mod 2) * 2^k.

Original entry on oeis.org

1, 2, 0, 204, 30840, 3743473440, 400814250895866480, 192435610587299441243182587501623263200, 2911899996313975217187797869354128351340558818020188112521784134070351919360

Offset: 0

Antti Karttunen, Jan 29 1999

This sequence can be also computed with a recurrence that does not explicitly refer to 3^n. See the C program.

Conjecture: For n >= 3, each term a(n), when considered as a GF(2)[X] polynomial, is divisible by the GF(2)[X] polynomial (x + 1) ^ A055010(n-1). If this holds, then for n >= 3, a(n) = A048720(A136386(n), A048723(3,A055010(n-1))).

			When powers of 3 are written in binary (see A004656), under each other as:
  000000000001 (1)
  000000000011 (3)
  000000001001 (9)
  000000011011 (27)
  000001010001 (81)
  000011110011 (243)
  001011011001 (729)
  100010001011 (2187)
it can be seen that the bits in the n-th column from the right can be arranged in periods of 2^n: 1, 2, 4, 8, ... This sequence is formed from those bits: 1, is binary for 1, thus a(0) = 1. 01, reversed is 10, which is binary for 2, thus a(1) = 2, 0000 is binary for 0, thus a(2)=0, 000110011, reversed is 11001100 = A007088(204), thus a(3) = 204.

S. Wolfram, A New Kind of Science, Wolfram Media Inc., (2002), p. 119.

Antti Karttunen, Table of n, a(n) for n = 0..11
Antti Karttunen, C program for computing this sequence.
S. Wolfram, A New Kind of Science, Wolfram Media Inc., (2002), p. 119.

Cf. A036284, A037095, A037097, A136386 for related sequences.
Cf. A000079, A000215, A000225, A000244, A004656, A007088, A048720, A048723, A055010.
Cf. also A004642, A265209, A265210 (for 2^n written in base 3).

a(n) := sum( 'bit_n(3^i, n)*(2^i)', 'i'=0..(2^(n))-1);
bit_n := (x, n) -> `mod`(floor(x/(2^n)), 2);

a(n) = Sum_{k=0 .. A000225(n)} (floor(A000244(k)/(2^n)) mod 2) * 2^k.
Other identities and observations:
For n >= 2, a(n) = A000215(n-1)*A037097(n) = A048720(A037097(n), A048723(3, A000079(n-1))).

Entry revised by Antti Karttunen, Dec 29 2007

Name changed and the example corrected by Antti Karttunen, Dec 05 2015

cp's OEIS Frontend

A136386 Quotient obtained when A037097(n) is considered as a GF(2)[X]-polynomial and it is divided by (x + 1) ^ A000225(n-1) (= A051179(n-2)).

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A268389 a(n) = greatest k such that polynomial (X+1)^k divides the polynomial (in polynomial ring GF(2)[X]) that is encoded in the binary expansion of n. (See the comments for details).

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A037095 "Sloping binary representation" of powers of 3 (A000244), slope = -1.

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A037096 Periodic vertical binary vectors computed for powers of 3: a(n) = Sum_{k=0 .. (2^n)-1} (floor((3^k)/(2^n)) mod 2) * 2^k.

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