A369407 A variant of A008336 based on polynomials over GF(2) (see Comments for precise definition).

1, 2, 6, 24, 120, 20, 108, 864, 96, 960, 6720, 624, 6192, 37152, 491232, 30702, 1806, 127, 1905, 27348, 486596, 25102, 1890, 19760, 456624, 5581280, 439712, 21624, 451032, 5199760, 123954032, 3966529024, 123317760, 3850804224, 127210628096, 4070965504

Offset: 1

Author

Rémy Sigrist, Jan 22 2024

Keywords

• nonn
• base

Comments

Let P(m) denote the polynomial over GF(2) whose coefficients are encoded in the binary expansion of the nonnegative integer m.

Let b(1) = 1 and for any n > 0, if P(n) divides b(n) then b(n+1) = b(n) / P(n), otherwise b(n+1) = b(n) * P(n).

For any n > 0, a(n) is the unique number v such that P(v) = b(n).

Examples

			The first terms, alongside the corresponding polynomials, are:
  n   a(n)  b(n)                   P(n)
  --  ----  ---------------------  -----------
   1     1  1                      1
   2     2  X                      X
   3     6  X^2 + X                X + 1
   4    24  X^4 + X^3              X^2
   5   120  X^6 + X^5 + X^4 + X^3  X^2 + 1
   6    20  X^4 + X^2              X^2 + X
   7   108  X^6 + X^5 + X^3 + X^2  X^2 + X + 1
   8   864  X^9 + X^8 + X^6 + X^5  X^3
   9    96  X^6 + X^5              X^3 + 1
  10   960  X^9 + X^8 + X^7 + X^6  X^3 + X
		

Links

• Rémy Sigrist, Table of n, a(n) for n = 1..3842
• Index entries for sequences operating on GF(2)[X]-polynomials

Crossrefs

Cf. A008336.

Programs

PARI

P(n) = Mod(1, 2) * Pol(binary(n))
P_1(p) = fromdigits(lift(Vec(p)), 2)
{ b = 1; for (n = 1, 36, p = P(n); if (b % p==0, b \= p, b *= p); print1 (P_1(b)", ");); }