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a(1) = 1, for n > 1, a(n) = n + a(n-A052126(n)).
9

%I #27 Oct 05 2021 21:07:15

%S 1,3,6,7,12,13,20,15,22,25,36,27,40,41,42,31,48,45,64,51,66,73,96,55,

%T 76,81,72,83,112,85,116,63,118,97,120,91,128,129,130,103,144,133,176,

%U 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

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

%C 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.

%H Antti Karttunen, <a href="/A333794/b333794.txt">Table of n, a(n) for n = 1..16384</a>

%H Antti Karttunen, <a href="/A333794/a333794.txt">Data supplement: n, a(n) computed for n = 1..65537</a>

%H Michael De Vlieger, <a href="/A333794/a333794.png">Graph montage</a> 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.

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

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

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

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

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

%e For n=119, the graph obtained is this:

%e 119

%e _/\_

%e / \

%e 102 112

%e _/|\_ | \_

%e _/ | \_ | \_

%e / | \ | \

%e 51 68 96 56

%e /| _/ | _/| _/ |

%e / | _/ | _/ | _/ |

%e / |/ |/ |/ |

%e (48) 34 64 48 28

%e |\_ | _/| _/|

%e | \_ | _/ | _/ |

%e | \_|_/ |/ |

%e 17 32 24 14

%e \_ | _/| _/|

%e \_ | _/ | _/ |

%e \_|_/ |/ |

%e 16 12 7

%e | _/| _/

%e | _/ | _/

%e |_/ |_/

%e 8 _6

%e | __/ |

%e |_/ |

%e 4 3

%e \ /

%e \_ _/

%e 2

%e |

%e 1.

%e 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.

%t Array[Total@ NestWhileList[# - #/FactorInteger[#][[-1, 1]] &, #, # > 1 &] &, 74] (* _Michael De Vlieger_, Apr 14 2020 *)

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

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

%K nonn

%O 1,2

%A _Antti Karttunen_, Apr 05 2020