A333785 -id:A333785 - cp's OEIS frontend

A333123 Consider the mapping k -> (k - (k/p)), where p is any of k's prime factors. a(n) is the number of different possible paths from n to 1.

1, 1, 1, 1, 1, 2, 2, 1, 2, 2, 2, 3, 3, 5, 5, 1, 1, 5, 5, 3, 10, 5, 5, 4, 3, 7, 5, 9, 9, 12, 12, 1, 17, 2, 21, 9, 9, 14, 16, 4, 4, 28, 28, 9, 21, 14, 14, 5, 28, 7, 7, 12, 12, 14, 16, 14, 28, 23, 23, 21, 21, 33, 42, 1, 33, 47, 47, 3, 61, 56, 56, 14, 14, 23, 28, 28, 103, 42, 42, 5

Offset: 1

Ali Sada and Robert G. Wilson v, Mar 09 2020

The iteration always terminates at 1, regardless of the prime factor chosen at each step.
Although there may exist multiple paths to 1, their path lengths (A064097) are the same! See A064097 for a proof. Note that this behavior does not hold if we allow any divisor of k.
First occurrence of k or 0 if no such value exists: 1, 6, 12, 24, 14, 96, 26, 85, 28, 21, 578, 30, 194, 38, 164, 39, 33, 104, 1538, 112, 35, 328, 58, 166, ..., .
Records: 1, 2, 3, 5, 10, 12, 17, 21, 28, 33, 42, 47, 61, 103, 168, ..., .
Record indices: 1, 6, 12, 14, 21, 30, 33, 35, 42, 62, 63, 66, 69, ..., .
When viewed as a graded poset, the paths of the said graph are the chains of the corresponding poset. This poset is also a lattice (see Ewan Delanoy's answer to Peter Kagey's question at the Mathematics Stack Exchange link). - Antti Karttunen, May 09 2020

			a(1): {1}, therefore a(1) = 1;
a(6): {6, 4, 2, 1} or {6, 3, 2, 1}, therefore a(6) = 2;
a(12): {12, 8, 4, 2, 1}, {12, 6, 4, 2, 1} or {12, 6, 3, 2, 1}, therefore a(12) = 3;
a(14): {14, 12, 8, 4, 2, 1}, {14, 12, 6, 4, 2, 1}, {14, 12, 6, 3, 2, 1}, {14, 7, 6, 4, 2, 1} or {14, 7, 6, 3, 2, 1}, therefore a(14) = 5.
From _Antti Karttunen_, Apr 05 2020: (Start)
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}, therefore a(15) = 5. These form a graph illustrated below:
        15
       / \
      /   \
    10     12
    / \   / \
   /   \ /   \
  5     8     6
   \_   |  __/|
     \__|_/   |
        4     3
         \   /
          \ /
           2
           |
           1
(End)

Antti Karttunen, Table of n, a(n) for n = 1..20000
Peter Kagey, Mathematics Stack Exchange, Does a graded poset on the positive integers generated from subtracting factors define a lattice?

Cf. A064097, A332809 (size of the lattice), A332810.
Cf. A332904 (sum of distinct integers present in such a graph/lattice), A333000 (sum over all paths), A333001, A333785.
Cf. A332992 (max. outdegree), A332999 (max. indegree), A334144 (max. rank level).
Cf. A334230, A334231 (meet and join).
Partial sums of A332903.
Cf. also tables A334111, A334184.

a[n_] := Sum[a[n - n/p], {p, First@# & /@ FactorInteger@n}]; a[1] = 1; (* after PARI coding by Rémy Sigrist *) Array[a, 70]
(* view the various paths *)
f[n_] := Block[{i, j, k, p, q, mtx = {{n}}}, Label[start]; If[mtx[[1, -1]] != 1, j = Length@ mtx;  While[j > 0, k = mtx[[j, -1]]; p = First@# & /@ FactorInteger@k; q = k - k/# & /@ p; pl = Length@p; If[pl > 1, Do[mtx = Insert[mtx, mtx[[j]], j], {pl - 1}]]; i = 1;  While[i < 1 + pl, mtx[[j + i - 1]] = Join[mtx[[j + i - 1]], {q[[i]]}]; i++]; j--]; Goto[start], mtx]]

for (n=1, #a=vector(80), print1 (a[n]=if (n==1, 1, vecsum(apply(p -> a[n-n/p], factor(n)[,1]~)))", ")) \\ Rémy Sigrist, Mar 11 2020

a(n) = 1 iff n is a power of two (A000079) or a Fermat Prime (A019434).
a(p) = a(p-1) if p is prime.
a(n) = Sum_{p prime and dividing n} a(n - n/p) for any n > 1. - Rémy Sigrist, Mar 11 2020

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.

Original entry on oeis.org

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

Antti Karttunen, Apr 06 2020

nonn

			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.

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

Cf. A333000, A333123.
Cf. A333002/A333003 (average as exact rational, numerator/denominator in lowest terms), A333785 (where the average is an integer).
Cf. A333790 (smallest path sum), A333794 (conjectured largest path sum).

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 *)

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];

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

A333003 Denominator of the average path sum when iterating from n to 1 with nondeterministic map k -> k - k/p, where p is any prime factor of k.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 2, 1, 2, 2, 2, 3, 3, 1, 5, 1, 1, 5, 5, 1, 5, 5, 5, 4, 1, 7, 5, 9, 9, 12, 12, 1, 17, 2, 7, 9, 9, 7, 4, 4, 4, 7, 7, 3, 7, 7, 7, 5, 7, 1, 7, 6, 6, 7, 8, 7, 7, 23, 23, 21, 21, 33, 7, 1, 11, 47, 47, 1, 61, 28, 28, 7, 7, 23, 14, 2, 103, 3, 3, 5, 7, 1, 1, 1, 4, 21, 79, 7, 7, 7, 7, 7, 89, 7, 14, 2, 2, 21, 103, 1, 1, 16, 16, 18, 84

Offset: 1

Antti Karttunen, Apr 06 2020

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

See A333002 for numerator.

Cf. A333000, A333001, A333123, A333785 (positions of ones).

Map[Denominator@ 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}}}, 104] (* Michael De Vlieger, Apr 15 2020 *)

up_to = 20000;
A333003list(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, denominator(v[n]/u[n])); };
v333003 = A333003list(up_to);
A333003(n) = v333003[n];

a(n) = denominator(A333000(n)/A333123(n)).

cp's OEIS Frontend

A333123 Consider the mapping k -> (k - (k/p)), where p is any of k's prime factors. a(n) is the number of different possible paths from n to 1.

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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.

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A333003 Denominator of the average path sum when iterating from n to 1 with nondeterministic map k -> k - k/p, where p is any prime factor of k.

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Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula