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A332992 Maximum outdegree in the graph formed by a subset of numbers in range 1 .. n with edge relation k -> k - k/p, where p can be any of the prime factors of k. 4
0, 1, 1, 1, 1, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 1, 3, 2, 3, 2, 2, 2, 2, 2, 2, 3, 3, 2, 3, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 1, 3, 3, 3, 2, 3, 3, 3, 2, 2, 2, 3, 2, 3, 3, 3, 2, 2, 2, 2, 3, 2, 3, 3, 2, 2, 3, 3, 2, 3, 2, 3, 2, 2, 3, 3, 2, 2, 3, 3, 2, 3 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,6

COMMENTS

Maximum number of distinct prime factors of any one integer encountered on all possible paths from n to 1 when iterating with nondeterministic map k -> k - k/p, where p can be any of the prime factors of k.

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..65539

FORMULA

a(n) = max(A001221(n), {Max a(n - n/p), for p prime and dividing n}).

For all odd primes p, a(p) = a(p-1).

For all n >= 0, a(A002110(n)) = n.

EXAMPLE

For n=15 we have five alternative paths from 15 to 1: {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}. These form a lattice illustrated below:

        15

       / \

      /   \

    10     12

    / \   / \

   /   \ /   \

  5     8     6

   \__  |  __/|

      \_|_/   |

        4     3

         \   /

          \ /

           2

           |

           1

With edges going from 15 towards 1, the maximum outdegree is 2, which occurs at nodes 15, 12, 10 and 6, therefore a(15) = 2.

MATHEMATICA

With[{s = Nest[Function[{a, n}, Append[a, Join @@ Table[Flatten@ Prepend[#, n] & /@ a[[n - n/p]], {p, FactorInteger[n][[All, 1]]}]]] @@ {#, Length@ # + 1} &, {{{1}}}, 105]}, Array[If[# == 1, 0, Max@ Tally[#][[All, -1]] &@ Union[Join @@ Map[Partition[#, 2, 1] &, s[[#]] ]][[All, 1]] ] &, Length@ s]] (* Michael De Vlieger, May 02 2020 *)

PROG

(PARI)

up_to = 105;

A332992list(up_to) = { my(v=vector(up_to)); v[1] = 0; for(n=2, up_to, v[n] = max(omega(n), vecmax(apply(p -> v[n-n/p], factor(n)[, 1]~)))); (v); };

v332992 = A332992list(up_to);

A332992(n) = v332992[n];

CROSSREFS

Cf. A001221, A064097, A332809, A332999 (max. indegree), A333123, A334111, A334144, A334184.

Cf. A002110 (positions of records and the first occurrence of each n).

Sequence in context: A275992 A271824 A253589 * A023568 A081753 A332999

Adjacent sequences:  A332989 A332990 A332991 * A332993 A332994 A332995

KEYWORD

nonn

AUTHOR

Antti Karttunen, Apr 04 2020

STATUS

approved

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Last modified July 8 00:31 EDT 2020. Contains 335502 sequences. (Running on oeis4.)