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(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, 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 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Antti Karttunen, Apr 05 2020
STATUS
approved