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%I A327969 #137 Apr 25 2022 08:03:24
%S A327969 0,1,2,2,5,2,3,2,6,4,3,2,5,2,5,6,6,2,5,2,7,4,3,2
%N A327969 The length of a shortest path from n to zero when using the transitions x -> A003415(x) and x -> A276086(x), or -1 if no zero can ever be reached from n.
%C A327969 The terms of this sequence are currently known only up to n=23, with the value of a(24) still being uncertain. For the tentative values of the later terms, see sequence A328324 which gives upper bounds for these terms, many of which are very likely also exact values for them.
%C A327969 As A051903(A003415(n)) >= A051903(n)-1, it means that it takes always at least A051903(n) steps to a prime if iterating solely with A003415.
%C A327969 Some known values and upper bounds from n=24 onward:
%C A327969 a(24) <= 11.
%C A327969 a(25) = 4.
%C A327969 a(26) = 7.
%C A327969 a(27) <= 22.
%C A327969 a(33) = 4.
%C A327969 a(39) = 4.
%C A327969 a(40) = 5.
%C A327969 a(42) = 3.
%C A327969 a(44) <= 10.
%C A327969 a(45) = 5.
%C A327969 a(46) = 5.
%C A327969 a(48) = 9.
%C A327969 a(49) = 6.
%C A327969 a(50) = 6.
%C A327969 a(55) = 7.
%C A327969 a(74) = 5.
%C A327969 a(77) = 6.
%C A327969 a(80) <= 18.
%C A327969 a(111) = 6.
%C A327969 a(112) = 8.
%C A327969 a(125) <= 9.
%C A327969 a(240) = 7.
%C A327969 a(625) <= 10.
%C A327969 a(875) = 8.
%C A327969 From _Antti Karttunen_, Feb 20 2022: (Start)
%C A327969 a(2556) <= 20.
%C A327969 a(5005) <= 19.
%C A327969 What is the value of a(128), and is A328324(128) well-defined?
%C A327969 When I created this sequence, I conjectured that by applying two simple arithmetic operations "arithmetic derivative" (A003415) and "primorial base exp-function" (A276086) in some combination, and starting from any positive integer, we could always reach zero (via a prime and 1).
%C A327969 At the first sight it seems almost certain that the conjecture holds, as it is always possible at every step to choose from two options (which very rarely meet, see A351088), leading to an exponentially growing search tree, and also because A276086 always jumps out of any dead-end path with p^p-factors (dead-end from the arithmetic derivative's point of view). However, it should be realized that one can reach the terms of either A157037 or A327978 with a single step of A003415 only from squarefree numbers (or respectively, cubefree numbers that are not multiples of 4, see A328234), and in general, because A003415 decreases the maximal exponent of the prime factorization (A051903) at most by one, if the maximal exponent in the prime factorization of n is large, there is a correspondingly long path to traverse if we take only A003415-steps in the iteration, and any step could always lead with certain probability to a p^p-number. Note that the antiderivatives of primorials with a square factor seem quite rare, see A351029.
%C A327969 And although taking a A276086-step will always land us to a p^p-free number (which a priori is not in the obvious dead-end path of A003415, although of course it might eventually lead to one), it (in most cases) also increases the magnitude of number considerably, that tends to make the escape even harder. Particularly, in the majority of cases A276086 increases the maximal exponent (which in the preimage is A328114, "maximal digit value used when n is written in primorial base"), so there will be even a longer journey down to squarefree numbers when using A003415. See the sequences A351067 and A351071 for the diminishing ratios suggesting rapidly diminishing chances of successfully reaching zero from larger terms of A276086. Also, the asymptotic density of A276156 is zero, even though A351073 may contain a few larger values.
%C A327969 On the other hand, if we could prove that by (for example) continuing upwards with any p^p-path of A003415 we could eventually reach with a near certainty a region of numbers with low values of A328114 (i.e., numbers with smallish digits in primorial base, like A276156), then the situation might change (see also A351089). However, a few empirical runs seemed to indicate otherwise.
%C A327969 For all of the above reasons, I now conjecture that there are natural numbers from which it is not possible to reach zero with any combination of steps. For example 128 or 5^5 = 3125.
%C A327969 (End)
%F A327969 a(0) = 0, a(p^p) = 1 + a(A276086(p^p)) for primes p, and for other numbers, a(n) = 1+min(a(A003415(n)), a(A276086(n))).
%F A327969 a(p) = 2 for all primes p.
%F A327969 For all n, a(n) <= A328324(n).
%F A327969 Let A stand the transition x -> A003415(x), and B stand for x -> A276086(x). The following sequences give some constant upper limits, because it is guaranteed that the combination given in brackets (the leftmost A or B is applied first) will always lead to a prime:
%F A327969 For all n, a(A157037(n)) = 3.  [A]
%F A327969 For n > 1, a(A002110(n)) = 3.  [B]
%F A327969 For all n, a(A192192(n)) <= 4. [AA]
%F A327969 For all n, a(A327978(n)) = 4.  [AB]
%F A327969 For all n, a(A328233(n)) <= 4. [BA]
%F A327969 For all n, a(A143293(n)) <= 4. [BB]
%F A327969 For all n, a(A328239(n)) <= 5. [AAA]
%F A327969 For all n, a(A328240(n)) <= 5. [BAA]
%F A327969 For all n, a(A328243(n)) <= 5. [ABB]
%F A327969 For all n, a(A328313(n)) <= 5. [BBB]
%F A327969 For all n, a(A328249(n)) <= 6. [BAAA]
%F A327969 For all k in A046099, a(k) >= 4, and if A328114(k) > 1, then certainly a(k) > 4.
%e A327969 Let -A> stand for an application of A003415 and -B> for an application of A276086, then, we have for example:
%e A327969 a(8) = 6 as we have 8 -A>  12 -B>  25 -A> 10 -A>  7 -A> 1 -A> 0, six transitions in total (and there are no shorter paths).
%e A327969 a(15) = 6 as we have 15 -B> 150 -A> 185 -A> 42 -A> 41 -A> 1 -A> 0, six transitions in total (and there are no shorter paths).
%e A327969 a(20) = 7, as 20 -B> 375 -A> 350 -A> 365 -A> 78 -A> 71 -A> 1 -A> 0, and there are no shorter paths.
%e A327969 For n=112, we know that a(112) cannot be larger than eight, as A328099^(8)(112) = 0, so we have a path of length 8 as 112 -A> 240 -B> 77 -A> 18 -A> 21 -A> 10 -A> 7 -A> 1 -A> 0. Checking all 32 combinations of the paths of lengths of 5 starting from 112 shows that none of them or their prefixes ends with a prime, thus there cannot be any shorter path, and indeed a(112) = 8.
%e A327969 a(24) <= 11 as A328099^(11)(24) = 0, i.e., we have 24 -A> 44 -A> 48 -A> 112 -A> 240 -B> 77 -A> 18 -A> 21 -A> 10 -A> 7 -A> 1 -A> 0. On the other hand, 24 -B> 625 -B> 17794411250 -A> 41620434625 -A> 58507928150 -A> 86090357185 -A> 54113940517 -A> 19982203325 -A> 12038411230 -A> 8426887871 -A> 1 -A> 0, thus offering another path of length 11.
%o A327969 (PARI)
%o A327969 A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
%o A327969 A276086(n) = { my(m=1, p=2); while(n, m *= (p^(n%p)); n = n\p; p = nextprime(1+p)); (m); };
%o A327969 A327969(n,searchlim=0) = if(!n,n,my(xs=Set([n]),newxs,a,b,u); for(k=1,oo, print("n=", n, " k=", k, " xs=", xs); newxs=Set([]); for(i=1,#xs,u = xs[i]; a = A003415(u); if(0==a, return(k)); if(isprime(a), return(k+2)); b = A276086(u); if(isprime(b), return(k+1+(u>2))); newxs = setunion([a],newxs); if(!searchlim || (b<=searchlim),newxs = setunion([b],newxs))); xs = newxs));
%Y A327969 Cf. A003415, A046099, A051674, A051903, A068346, A276086, A276087, A327859, A327860, A328099, A328112, A328114, A328116, A328307.
%Y A327969 Cf. A328324 (a sequence giving upper bounds, computed with restricted search space).
%Y A327969 Sequences for whose terms k, value a(k) has a guaranteed constant upper bound: A000040, A002110, A143293, A157037, A192192, A327978, A328232, A328233, A328239, A328240, A328243, A328249, A328313.
%Y A327969 Sequences for whose terms k, it is guaranteed that a(k) has finite value > 0, even if not bound by a constant: A099308, A328116.
%Y A327969 Cf. also A256750, A327966, A328110, A351029, A351088, A351067, A351071, A351073, A351089 and A351255, A351256, A351257, A351258, A351261.
%K A327969 nonn,hard,more
%O A327969 0,3
%A A327969 _Antti Karttunen_, Oct 07 2019

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