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A333794
a(1) = 1, for n > 1, a(n) = n + a(n-A052126(n)).
8
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
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
KEYWORD
nonn
AUTHOR
Antti Karttunen, Apr 05 2020
STATUS
approved