A340717 -id:A340717 - cp's OEIS frontend

A340718 a(n) is the least k such that A340717(k) = n.

0, 1, 3, 5, 7, 9, 11, 15, 17, 19, 21, 23, 27, 31, 33, 35, 37, 39, 43, 45, 47, 51, 55, 63, 65, 67, 69, 71, 73, 75, 77, 79, 83, 85, 87, 91, 93, 95, 99, 103, 107, 111, 119, 127, 129, 131, 133, 135, 137, 139, 141, 143, 147, 149, 151, 153, 155, 157, 159, 163, 165

Offset: 0

Rémy Sigrist, Jan 17 2021

These are also the numbers k such that k <= A030101(k).
All terms, except a(0) = 0, are odd.
For any k > 0, there are A005418(k) terms with k+1 binary digits.
A000225 and A006995 are subsequences.

			k          | 0,  1,  2,  3,  4,  5,  6,  7,  8,  9, 10, 11, 12, 13, 14, 15, ...
A340717(k) | 0,  1,  1,  2,  1,  3,  2,  4,  1,  5,  3,  6,  2,  6,  4,  7, ...
n          | 0,  1,      2,      3,      4,      5,      6,              7, ...
a(n)       | 0,  1,      3,      5,      7,      9,     11,             15, ...

Rémy Sigrist, Table of n, a(n) for n = 0..8383

Cf. A000225, A005418, A006995, A030101, A340717.

is(n) = n <= fromdigits(Vecrev(binary(n)), 2)

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).

Original entry on oeis.org

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

Rémy Sigrist, Jan 17 2021

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)).

			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

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

See A003602, A163491 and A340717 for similar sequences.

Cf. A000720, A095117.

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]))

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

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

cp's OEIS Frontend

A340718 a(n) is the least k such that A340717(k) = n.

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