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A333000 Sum of integers (with multiplicity) encountered on all possible paths from n to 1 when iterating with nondeterministic map k -> k- k/p, where p is any prime factor of k.

%I A333000 #9 Apr 12 2020 21:44:48
%S A333000 1,3,6,7,12,25,39,15,43,47,69,76,115,185,198,31,48,209,304,138,604,
%T A333000 317,432,203,213,500,344,640,901,899,1271,63,1777,179,2274,736,1069,
%U A333000 1572,1860,361,525,3156,4360,1074,2580,2150,2808,506,4528,924,1042,1630,2266,1836,2878,1930,5004,4165,5522,3026,4307,6343,7638,127,6801
%N A333000 Sum of integers (with multiplicity) encountered on all possible paths from n to 1 when iterating with nondeterministic map k -> k- k/p, where p is any prime factor of k.
%H A333000 Antti Karttunen, <a href="/A333000/b333000.txt">Table of n, a(n) for n = 1..20000</a>
%F A333000 a(n) = n*A333123(n) + Sum_{p prime and dividing n} a(n - n/p).
%F A333000 For all n, a(n) >= A332904(n).
%e A333000 a(12): we have three alternative paths: {12, 8, 4, 2, 1}, {12, 6, 4, 2, 1} or {12, 6, 3, 2, 1}, therefore a(12) = (12+8+4+2+1) + (12+6+4+2+1) + (12+6+3+2+1) = 27+25+24 = 76
%e A333000 For n=15 we have five alternative paths from 15 to 1 (illustrated below): therefore a(15) = (15+10+5+4+2+1) + (15+10+8+4+2+1) + (15+12+8+4+2+1) + (15+12+6+4+2+1) + (15+12+6+3+2+1) = 37+40+42+40+39 = 198.
%e A333000         15
%e A333000        / \
%e A333000       /   \
%e A333000     10     12
%e A333000     / \   / \
%e A333000    /   \ /   \
%e A333000   5     8     6
%e A333000    \__  |  __/|
%e A333000       \_|_/   |
%e A333000         4     3
%e A333000          \   /
%e A333000           \ /
%e A333000            2
%e A333000            |
%e A333000            1.
%o A333000 (PARI)
%o A333000 up_to = 20000;
%o A333000 A333000list(up_to) = { my(u=vector(up_to), v=vector(up_to)); u[1] = v[1] = 1; for(n=2,up_to, u[n] = vecsum(apply(p -> u[n-n/p], factor(n)[, 1]~)); v[n] = (u[n]*n)+vecsum(apply(p -> v[n-n/p], factor(n)[, 1]~))); (v); };
%o A333000 v333000 = A333000list(up_to);
%o A333000 A333000(n) = v333000[n];
%Y A333000 Cf. A332904, A333001, A333123.
%K A333000 nonn
%O A333000 1,2
%A A333000 _Antti Karttunen_, Apr 06 2020