A333794 a(1) = 1, for n > 1, a(n) = n + a(n-A052126(n)).

1, 3, 6, 7, 12, 13, 20, 15, 22, 25, 36, 27, 40, 41, 42, 31, 48, 45, 64, 51, 66, 73, 96, 55, 76, 81, 72, 83, 112, 85, 116, 63, 118, 97, 120, 91, 128, 129, 130, 103, 144, 133, 176, 147, 136, 193, 240, 111, 182, 153, 162, 163, 216, 145, 208, 167, 202, 225, 284, 171, 232, 233, 208, 127, 236, 237, 304, 195, 306, 241, 312, 183, 256, 257

Offset

1,2

Comments

Conjecturally, also the largest path sum when iterating from n to 1 with nondeterministic map k -> k - k/p, where p is any prime factor of k.

Links

Antti Karttunen, Table of n, a(n) for n = 1..16384

Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Michael De Vlieger, Graph montage of k -> k - k/p, with prime p|k for 2 <= k <= 211, red line showing path of greatest sum, blue the path of least sum (cf. A333790), and purple where the two paths coincide, with other paths in gray.

Formula

a(1) = 1; and for n > 1, a(n) = n + a(A171462(n)) = n + a(n-A052126(n)).

a(n) = A073934(n) + A333793(n).

a(n) = n + Max a(n - n/p), for p prime and dividing n. [Conjectured, holds at least up to n=2^24]

For all n >= 1, A333790(n) <= a(n) <= A332904(n).

For all n >= 1, a(n) >= A332993(n). [Apparently, have to check!]

Example

For n=119, the graph obtained is this:
              119
             _/\_
            /    \
          102    112
         _/|\_    | \_
       _/  |  \_  |   \_
      /    |    \ |     \
    51     68    96     56
    /|   _/ |   _/|   _/ |
   / | _/   | _/  | _/   |
  /  |/     |/    |/     |
(48) 34    64     48    28
     |\_    |    _/|   _/|
     |  \_  |  _/  | _/  |
     |    \_|_/    |/    |
    17     32     24    14
      \_    |    _/|   _/|
        \_  |  _/  | _/  |
          \_|_/    |/    |
           16      12    7
            |    _/|    _/
            |  _/  |  _/
            |_/    |_/
            8     _6
            |  __/ |
            |_/    |
            4      3
             \     /
              \_ _/
                2
                |
                1.
If we always subtract A052126(n) (i.e., n divided by its largest prime divisor), i.e., iterate with A171462 (starting from 119), we obtain 119-(119/17) = 112 -> 112-(112/7) -> 96-(96/3) -> 64-(64/2) -> 32-(32/2) -> 16-(16/2) -> 8-(8/2) -> 4-(4/2) -> 2-(2/2) -> 1, with sum 119+112+96+64+32+16+8+4+2+1 = 554, thus a(119) = 554. This happens also to be maximal sum of any path in above diagram.

Mathematica

Array[Total@ NestWhileList[# - #/FactorInteger[#][[-1, 1]] &, #, # > 1 &] &, 74] (* Michael De Vlieger, Apr 14 2020 *)

Prog

(PARI) A333794(n) = if(1==n, n, n + A333794(n-(n/vecmax(factor(n)[, 1]))));

Crossrefs

Cf. A052126, A073934, A171462, A332994, A333790, A333793.

Sequence in context: A295897 A032849 A038591 * A349214 A182181 A138038

Adjacent sequences: A333791 A333792 A333793 * A333795 A333796 A333797

Keyword

nonn

Author

Antti Karttunen, Apr 05 2020

Status

approved