

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.


23



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
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OFFSET

1,6


COMMENTS

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


LINKS



FORMULA

a(n) = 1 iff n is a power of two (A000079) or a Fermat Prime (A019434).
a(p) = a(p1) 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


EXAMPLE

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.
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)


MATHEMATICA

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]]


PROG

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


CROSSREFS



KEYWORD



AUTHOR



STATUS

approved



