A319561 Let f(1) = 1, f(2) = 1 + i (where i denotes the imaginary unit), and for n > 1, f(n+1) is the Gaussian integer in the first quadrant (with positive real part and nonnegative imaginary part) with least modulus and sharing a prime factor with f(n) (in case of a tie, minimize the imaginary part); a(n) = the square of the modulus of f(n).

1, 2, 4, 8, 10, 5, 20, 10, 5, 20, 16, 18, 9, 36, 26, 13, 52, 26, 13, 52, 32, 34, 17, 68, 34, 17, 68, 40, 25, 25, 45, 45, 25, 40, 50, 50, 50, 58, 29, 116, 58, 29, 116, 64, 72, 74, 37, 148, 74, 37, 148, 80, 65, 65, 65, 65, 85, 85, 85, 80, 82, 41, 164, 82, 41

Offset

1,2

Comments

The real and imaginary parts of f are respectively given by A319562 and A319563.

The sequence f is a complex variant of the EKG sequence (A064413).

This sequence and A064413 have similar graphical features.

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

Links

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

Rémy Sigrist, PARI program for A319561

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

Formula

a(n) = A319562(n)^2 + A319563(n)^2.

Example

The first terms, alongside f(n) and gcd(f(n), f(n+1)), are:
  n  a(n)    f(n)       gcd(f(n), f(n+1))
  -- ----    ----       -----------------
   1    1    1          1
   2    2    1 + i      1 + i
   3    4    2          2
   4    8    2 + 2*i    1 + i
   5   10    3 + i      1 + 2*i
   6    5    1 + 2*i    1 + 2*i
   7   20    2 + 4*i    1 + i
   8   10    1 + 3*i    2 + i
   9    5    2 + i      2 + i
  10   20    4 + 2*i    2
  11   16    4          1 + i
  12   18    3 + 3*i    3
  13    9    3          3
  14   36    6          1 + i
  15   26    5 + i      2 + 3*i

Prog

(PARI) See Links section.

Crossrefs

Cf. A064413, A319562, A319563.

Sequence in context: A028984 A294369 A302907 * A153181 A256624 A247324

Adjacent sequences: A319558 A319559 A319560 * A319562 A319563 A319564

Keyword

nonn

Author

Rémy Sigrist, Sep 23 2018

Status

approved