A333001 The average path sum (floored down) when iterating from n to 1 with nondeterministic map k -> k - k/p, where p is any prime factor of k.

1, 3, 6, 7, 12, 12, 19, 15, 21, 23, 34, 25, 38, 37, 39, 31, 48, 41, 60, 46, 60, 63, 86, 50, 71, 71, 68, 71, 100, 74, 105, 63, 104, 89, 108, 81, 118, 112, 116, 90, 131, 112, 155, 119, 122, 153, 200, 101, 161, 132, 148, 135, 188, 131, 179, 137, 178, 181, 240, 144, 205, 192, 181, 127, 206, 191, 258, 170, 251, 199, 270, 160, 233, 218, 216

Offset

1,2

Links

Antti Karttunen, Table of n, a(n) for n = 1..20000

Formula

a(n) = floor(A333000(n)/A333123(n)) = floor(A333002(n)/A333003(n)).

Example

a(12): we have three alternative paths: {12, 8, 4, 2, 1}, {12, 6, 4, 2, 1} or {12, 6, 3, 2, 1}, with path sums 27, 25, 24, whose average is 76/3 = 25.333..., therefore a(12) = 25.
For n=15 we have five alternative paths from 15 to 1 (illustrated below) with path sums 37, 40, 42, 40, 39, whose average is 198/5 = 39.6, therefore a(15) = 39.
        15
       / \
      /   \
    10     12
    / \   / \
   /   \ /   \
  5     8     6
   \_   |  __/|
     \__|_/   |
        4     3
         \   /
          \ /
           2
           |
           1.

Mathematica

Map[Floor@ Mean[Total /@ #] &, #] &@ Nest[Function[{a, n}, Append[a, Join @@ Table[Flatten@ Prepend[#, n] & /@ a[[n - n/p]], {p, FactorInteger[n][[All, 1]]}]]] @@ {#, Length@ # + 1} &, {{{1}}}, 74] (* Michael De Vlieger, Apr 15 2020 *)

Prog

(PARI)
up_to = 20000;
A333001list(up_to) = { my(u=vector(up_to), v=vector(up_to)); u[1] = v[1] = 1; for(n=2, up_to, my(ps=factor(n)[, 1]~); u[n] = vecsum(apply(p -> u[n-n/p], ps)); v[n] = (u[n]*n)+vecsum(apply(p -> v[n-n/p], ps))); vector(up_to, n, floor(v[n]/u[n])); };
v333001 = A333001list(up_to);
A333001(n) = v333001[n];

Crossrefs

Cf. A333000, A333123.

Cf. A333002/A333003 (average as exact rational, numerator/denominator in lowest terms), A333785 (where the average is integer).

Cf. A333790 (smallest path sum), A333794 (conjectured largest path sum).

Sequence in context: A209246 A073934 A333790 * A092150 A028802 A141742

Adjacent sequences: A332998 A332999 A333000 * A333002 A333003 A333004

Keyword

nonn

Author

Antti Karttunen, Apr 06 2020

Status

approved