A340716 Lexicographically earliest sequence of positive integers with as many distinct values as possible such that for any n > 0, a(n + pi(n)) = a(n) (where pi(n) = A000720(n) corresponds to the number of prime numbers <= n).

1, 2, 2, 3, 2, 3, 4, 2, 3, 5, 4, 2, 3, 5, 6, 4, 2, 7, 3, 5, 6, 4, 8, 2, 7, 9, 3, 5, 6, 4, 10, 8, 2, 7, 9, 3, 5, 11, 6, 4, 12, 10, 8, 2, 7, 9, 3, 13, 5, 11, 6, 4, 14, 12, 10, 15, 8, 2, 7, 9, 16, 3, 13, 5, 11, 6, 4, 17, 14, 12, 10, 15, 8, 2, 18, 7, 9, 19, 16, 3

Offset: 1

Author

Rémy Sigrist, Jan 17 2021

Keywords

• nonn
• look

Comments

The condition "with as many distinct values as possible" means here that for any distinct m and n, provided the orbits of m and n under the map x -> x + pi(x) do not merge, then a(m) <> a(n).

This sequence has similarities with A003602 (A003602(2*n) = A003602(n)) and with A163491 (A163491(n+ceiling(n/2)) = A163491(n)).

Examples

			The first terms, alongside n + pi(n), are:
  n   a(n)  n + pi(n)
  --  ----  ---------
   1     1          1
   2     2          3
   3     2          5
   4     3          6
   5     2          8
   6     3          9
   7     4         11
   8     2         12
   9     3         13
  10     5         14
  11     4         16
  12     2         17
		

Links

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

Crossrefs

See A003602, A163491 and A340717 for similar sequences.

Cf. A000720, A095117.

Programs

PARI

u=0; for (n=1, #a=vector(80), if (a[n]==0, a[n]=u++); print1 (a[n]", "); m=n+primepi(n); if (m<=#a, a[m]=a[n]))

Formula

a(n) = 2 iff n belongs to A061535.

a(A095116(n)) = n + 1.