A319770 Let f(1) = 1, f(2) = 1 + i (where i denotes the imaginary unit), f(3) = 2 + i, and for n > 2, f(n+1) is the Gaussian integer in the first quadrant (with positive real part and nonnegative imaginary part) with least modulus and sharing at least one prime factor with f(n-1) but none with f(n) (in case of a tie, minimize the imaginary part); a(n) = the square of the modulus of f(n).

1, 2, 5, 4, 25, 8, 5, 10, 25, 16, 45, 20, 9, 10, 45, 20, 25, 18, 65, 26, 13, 13, 26, 65, 32, 65, 34, 85, 17, 17, 68, 85, 36, 65, 34, 85, 40, 125, 50, 145, 52, 29, 40, 145, 50, 85, 52, 125, 58, 125, 58, 125, 29, 50, 261, 64, 81, 68, 117, 74, 117, 37, 72, 185

Offset

1,2

Comments

The real and imaginary parts of f are respectively given by A319771 and A319772.

The sequence f is a complex variant of the Yellowstone permutation (A098550).

Apparently, the sequence f runs through all Gaussian integers in the first quadrant.

See A319561 for a similar sequence.

Links

Rémy Sigrist, Table of n, a(n) for n = 1..10000

Rémy Sigrist, PARI program for A319770

Rémy Sigrist, Representation of f(n) in the complex plane for n = 1..100000

Formula

a(n) = A319771(n)^2 + A319772(n)^2.

Example

The first terms, alongside f(n) and gcd(f(n), f(n+2)), are:
  n  a(n)    f(n)       gcd(f(n), f(n+2))
  -- ----    ----       -----------------
   1    1    1          1
   2    2    1 + i      1 + i
   3    5    2 + i      2 + i
   4    4    2          2
   5   25    5          1 + 2*i
   6    8    2 + 2*i    1 + i
   7    5    1 + 2*i    1 + 2*i
   8   10    1 + 3*i    1 + i
   9   25    4 + 3*i    1 + 2*i
  10   16    4          2
  11   45    3 + 6*i    3
  12   20    4 + 2*i    1 + i
  13    9    3          3
  14   10    3 + i      3 + i
  15   45    6 + 3*i    2 + i

Prog

(PARI) See Links section.

Crossrefs

Cf. A098550, A319561, A319771, A319772.

Sequence in context: A349475 A111364 A128202 * A229789 A357240 A307131

Adjacent sequences: A319767 A319768 A319769 * A319771 A319772 A319773

Keyword

nonn

Author

Rémy Sigrist, Sep 27 2018

Status

approved