A332809 Number of distinct integers encountered on possible paths from n to 1 when iterating the nondeterministic map k -> k - k/p, where p is any of the prime factors of k.

1, 2, 3, 3, 4, 5, 6, 4, 6, 6, 7, 7, 8, 9, 10, 5, 6, 9, 10, 8, 12, 10, 11, 9, 9, 11, 10, 12, 13, 14, 15, 6, 17, 8, 18, 12, 13, 14, 15, 10, 11, 17, 18, 13, 18, 15, 16, 11, 18, 12, 14, 14, 15, 14, 16, 15, 17, 17, 18, 18, 19, 20, 20, 7, 22, 23, 24, 10, 26, 24, 25, 15, 16, 17, 21, 18, 30, 20, 21, 12, 15, 14, 15, 22, 16, 24, 25, 16

Offset: 1

Author

Antti Karttunen, Apr 04 2020

Keywords

• nonn
• look

Comments

The count includes also n itself, and the final 1 when it is distinct from n.

a(n) >= A000005(n) because all divisors of n can be found in the union of those paths. - Antti Karttunen, Apr 19 2020

Examples

			a(1): {1}, therefore a(1) = 1;
a(6): we have two alternative paths: {6, 4, 2, 1} or {6, 3, 2, 1}, with numbers [1, 2, 3, 4, 6] present, therefore a(6) = 5;
a(12): we have three alternative paths: {12, 8, 4, 2, 1}, {12, 6, 4, 2, 1} or {12, 6, 3, 2, 1}, with numbers [1, 2, 3, 4, 6, 8, 12] present, therefore a(12) = 7;
a(14): we have five alternative paths: {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}, with numbers [1, 2, 3, 4, 6, 7, 8, 12, 14] present in at least one of the paths, therefore a(14) = 9.
		

Links

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

Crossrefs

Cf. A064097, A332810, A333123, A334230, A334231, A333786 (first occurrence of each n), A334112.

Partial sums of A332902.

See A332904 for the sum.

Programs

Mathematica

a[n_] := Block[{lst = {{n}}}, While[lst[[-1]] != {1}, lst = Join[ lst, {Union[ Flatten[# - #/(First@# & /@ FactorInteger@#) & /@ lst[[-1]]]]}]]; Length@ Union@ Flatten@ lst]; Array[a, 75] (* Robert G. Wilson v, Apr 06 2020 *)

PARI

up_to = 105;
A332809list(up_to) = { my(v=vector(up_to)); v[1] = Set([1]); for(n=2,up_to, my(f=factor(n)[, 1]~, s=Set([n])); for(i=1,#f,s = setunion(s,v[n-(n/f[i])])); v[n] = s); apply(length,v); }
v332809 = A332809list(up_to);
A332809(n) = v332809[n];

Python

from sympy import factorint
from functools import cache
@cache
def b(n): return {n}.union(*(b(n - n//p) for p in factorint(n)))
def a(n): return len(b(n))
print([a(n) for n in range(1, 89)]) # Michael S. Branicky, Aug 13 2022

Formula

a(p) = 1 + a(p-1) for all primes p.

a(n) = n - A332810(n).

a(n) = A334112(n) + A000005(n). - Antti Karttunen, May 09 2020