A393813 Among all the numbers k such that 0 <= k < n and {phi * k} < {phi * n}, a(n) is the one maximizing {phi * k}, where {} denotes the fractional part and phi = A001622.

0, 0, 1, 2, 0, 1, 2, 3, 4, 5, 6, 7, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39

Offset

1,4

Comments

Every nonnegative integer appears infinitely many times in the sequence.

Links

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

Formula

Empirically: a(n) = 0 iff n belongs to A001519, a(n) = 1 + a(n-1) otherwise.

Example

The first terms are:
  n   {phi*n}  a(n)
  --  -------  ----
   0  0.       N/A
   1  0.61...     0
   2  0.23...     0
   3  0.85...     1
   4  0.47...     2
   5  0.09...     0
   6  0.70...     1
   7  0.32...     2
   8  0.94...     3
   9  0.56...     4
  10  0.18...     5

Prog

(PARI) a(n, phi = (1+sqrt(5))/2) = { my (f = frac(phi * n), mx = -oo, v = -oo); for (k = 0, n-1, my (g = frac(phi * k)); if (mx < g && g < f, mx = g; v = k; ); ); return (v); }

Crossrefs

Cf. A001519, A001622, A019587, A395009.

Sequence in context: A332663 A227179 A115218 * A023858 A011118 A354773

Adjacent sequences: A393810 A393811 A393812 * A393814 A393815 A393816

Keyword

nonn

Author

Rémy Sigrist, Apr 10 2026

Status

approved