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A356428 a(0) = a(1) = 0; for n > 1, a(n) is the number of distinct gpf(x)'s in the iterations x -> x - gpf(x) starting at n and ending at 0, where gpf = A006530.

%I A356428 #23 Aug 09 2022 15:33:36
%S A356428 0,0,1,1,1,1,1,1,2,1,1,1,1,1,1,1,2,1,2,1,1,1,1,1,2,1,1,2,1,1,1,1,2,1,
%T A356428 1,1,2,1,1,1,2,1,1,1,1,2,1,1,3,1,2,1,1,1,2,1,1,1,1,1,2,1,1,1,2,1,1,1,
%U A356428 1,1,1,1,2,1,1,2,1,1,1,1,2,2,1,1,2,1,1,1,1,1,2,1
%N A356428 a(0) = a(1) = 0; for n > 1, a(n) is the number of distinct gpf(x)'s in the iterations x -> x - gpf(x) starting at n and ending at 0, where gpf = A006530.
%C A356428 Conjecture: sequence is unbounded. Since a(n) - a(n-gpf(n)) = 0 or 1 (see the formula below), this would imply that every number occurs in this sequence. But it seems that the bigger terms appear rather late: 6 does not appear until a(6664), and 7 does not appear until a(135450) (see A356429).
%C A356428 The last largest prime p in this iteration is found when p^2 > x in this iteration. - _David A. Corneth_, Aug 09 2022
%H A356428 Jianing Song, <a href="/A356428/b356428.txt">Table of n, a(n) for n = 0..10000</a>
%F A356428 For n > 1, let p = gpf(n), then a(n) = 1+a(n-p) if p = n or gpf(n-p) > p; otherwise a(n) = a(n-p).
%e A356428 In the following examples the numbers produced by the iterations are listed together with their GPFs.
%e A356428 48 (3) -> 45 (5) -> 40 (5) -> 35 (7) -> ... -> 7 (7) -> 0, the distinct gpf(x)'s are 3, 5, and 7, so a(48) = 3.
%e A356428 96 (3) -> 93 (31) -> 62 (31) -> 31 (31) -> 0, the distinct gpf(x)'s are 3 and 31, so a(96) = 2.
%e A356428 320 (5) -> 315 (7) -> 308 (11) -> 297 (11) -> 286 (13) -> 273 (13) -> 260 (13) -> 247 (19) -> ... -> 19 (19) -> 0, the distinct gpf(x)'s are 5, 7, 11, 13, and 19, so a(320) = 5.
%e A356428 In the above computation for a(320) the calculation can stop at 247 (19) as all largest prime factors in positive x are 19. - _David A. Corneth_, Aug 09 2022
%o A356428 (PARI) gpf(n) = vecmax(factor(n)[, 1]);
%o A356428 a(n) = if(n>1, my(s=n, k=0, p); while(s, p=gpf(s); s-=p; k+=(s==0)||(gpf(s)>p)); k, 0)
%o A356428 (PARI) a(n) = {if(n <= 1, return(0)); my(cn = n, maxpr, pr = List()); while(cn > 1, maxpr = h(cn); listput(pr, maxpr); cn-=maxpr; if(maxpr^2 > cn, return(#Set(pr)))); #Set(pr)}
%o A356428 h(n) = {my(f = factor(n)); f[#f~, 1]} \\ _David A. Corneth_, Aug 08 2022
%o A356428 (Python)
%o A356428 from sympy import factorint
%o A356428 def gpf(n): return 1 if n == 1 else max(factorint(n))
%o A356428 def a(n):
%o A356428     s = set()
%o A356428     while n != 0: g = gpf(n); s.add(g); n = n - g
%o A356428     return len(s - {1})
%o A356428 print([a(n) for n in range(92)]) # _Michael S. Branicky_, Aug 08 2022
%Y A356428 Cf. A006530 (gpf), A076563, A309892, A356429.
%K A356428 nonn
%O A356428 0,9
%A A356428 _Jianing Song_, Aug 07 2022