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These numbers are one step nearer (than those of A262509) to the root (zero) of the tree where the parent-child relation is given by A049820(child) = parent. Like the terms of A262509, they are also vertices in the infinite trunk of that tree. Cf. A259934.

a(n) = largest k for which A155043(k) < A155043(A262509(n)).

If a(n) > A262509(n) then it must be a leaf (see comments in A262909 for why). Particularly, we have A045765(40722) = 124340, A045765(8191770) = 24684000.

Terms of sequence (together with the corresponding values in A262508) give particularly clean values for the boundaries that are used for example in the C++-program which computes A262896.

a(n) = largest k such that A155043(k+A262509(n)) < A262508(n).

There might occur also negative terms, but no zeros.

For all terms a(n) > 0, a(n)+A262509(n) = A263081(n) is by necessity one of the leaves (A045765) in the tree generated by edge-relation A049820(child) = parent. See also comments in A262908.

For all nonzero terms a(n), A263083(n) = a(n) + A262509(n) and A155043(A263083(n)) < A155043(A262509(n)) because at each A262509(n) the "distance to zero", A155043 obtains a unique value A262508(n), thus no A049820-iteration trajectory starting from any k larger than A262509(n) and using a greater or equal number of steps to reach zero may bypass A262509(n) [i.e., without going through A262509(n)], because then A262508(n) would not be unique anymore. See also comments in A262909.

When a(n) > A262509(n), then a(n) is the "farthest immediate bypasser" of A262509(n) [the n-th "constriction point" in the tree generated by edge-relation A049820(child) = parent], bypassing it in the single A049820-step. In contrast, A263081(n) gives the farthest node (by necessity a leaf-node) which bypasses A262509(n) in multiple A049820-steps.

Sequence b(n) = A155043(A262509(n)) - A155043(a(n)) = A262508(n) - A155043(a(n)) gives the following terms: 395, 396, 354, 363, 364, 399, 390, 419, 422, 420, 421, 442, 430, 437, 460, 456, 498, 511, 512, 513, 515, 516, 506, 509, 533, 543, 564, 565, 557, 558, 591, 608, 612, 613, 614, 617, 617, 655, 3240, 3241, 3236, 3239, 3291, 3346, 3350, 3373, 3451, 3455, 2, 3598, 3637, 3605, 3674, 3688, 3689, 3748, 3749, 3792, 3793, 3794, 3800, 3803, 3858, 3843, 3902, 3947, 3985, 3986, ... which tells how many steps shorter trajectory there is to zero (using A049820) for those bypassers than for the constriction points themselves.

Equivalently, satisfies the property: A000005(a(n)) = a(n)-a(n-1). The first differences a(n)-a(n-1) are given in A259935.

V. S. Guba (2015) proved that such an infinite sequence exists. Numerical evidence suggests that it may also be unique -- is it? All terms below 10^10 are defined uniquely.

If the current definition does not uniquely define the sequence, the "lexicographically earliest" condition may be added to make the sequence well-defined.

From Vladimir Shevelev, Jul 21 2015: (Start)

If a(k), a(k+1), a(k+2) is an arithmetic progression, then a(k+1) is in A175304.

Indeed, by the definition of this sequence, a(n)-a(n-1) = d(a(n)), for all n>=1, where d(n) = A000005(n). Hence, have a(k+1) - a(k) = a(k+2) - a(k+1) = d(a(k+1)) = d(a(k+2)). So a(k+1) + d(a(k+2)) = a(k+2) and a(k+1) + d(a(k+1)) = a(k+2).

Therefore, d(a(k+1) + d(a(k+1))) = d(a(k+2))= d(a(k+1)), i.e., a(k+1) is in A175304. Thus, if there are infinitely many pairs of the same consecutive terms of A259935, then A175304 is infinite (see there my conjecture). (End)

From Antti Karttunen, Nov 27 2015: (Start)

If multiple apparently infinite branches would occur at some point of computing, then even if the "lexicographically earliest" condition were then added to the definition, it would not help us much (when computing the sequence), as we would still not know which of the said branches were truly infinite. [See also Max Alekseyev's latter Jul 9 2015 posting on SeqFan-list, where he notes the same thing.] Note that many of the derived sequences tacitly assume that the uniqueness-conjecture is true. See also comments at A262693 and A262896.

One sufficient (but not a necessary) condition for the uniqueness of this sequence is that the sequence A262509 has infinite number of terms. Please see further comments there.

The graph of sequence exhibits two markedly different slopes, depending on whether it is on the "fast lane" of A049820 (even numbers) or the "slow lane" [odd numbers, for example when traversing the 1356 odd terms from 123871 to 113569 at range a(9859) .. a(8504)]. See A263086/A263085 for the "average cumulative speed difference" between the lanes. In general, slow and fast lane stay separate, except when they terminate into one of the squares (A262514) that work as "exchange ramps", forcing the parity (and thus the speed) to change. In average, the odd squares are slightly better than the even squares in attracting lanes going towards smaller numbers (compare A263253 to A263252). The cumulative effect of this bias is that the odd terms are much rarer in this sequence than the even terms (compare A263278 to A262516).

(End)

From Antti Karttunen, Sep 23 2015: (Start)

Number of steps needed to reach zero when starting from k = n and repeatedly applying the map that replaces k by k - d(k), where d(k) is the number of divisors of k (A000005).

The original name was: a(n) = 1 + a(n-sigma_0(n)), a(0)=0, sigma_0(n) number of divisors of n.

(End)

Numbers n for which there exists exactly one natural number x from which one can reach zero in n steps by setting first k = x and then repeatedly applying the map where k is replaced with k - A000005(k). See A262509 for the corresponding x's and implications concerning A259934.

Starting offset is zero, because a(0) = 0 is a special case in this sequence.

Number of perfect squares (A000290) encountered before zero is reached when starting from k = n and repeatedly applying the map that replaces k by k - d(k), where d(k) is the number of divisors of k (A000005). This count includes n itself if it is a square, but excludes the final zero.

Also number of times the parity (of numbers encountered) changes until zero is reached when iterating A049820. This count includes also the last parity change 1 - d(1) -> 0 if coming to zero through 1.

There is a lower bound for this sequence that grows without limit if and only if either (1) A259934 is indeed the unique sequence (satisfying its given condition) and it contains an infinite number of squares (see A262514), or (2) more generally, if each one of all (hypothetically multiple) infinite branches of the tree (defined by parent-child relation A049820(child) = parent) contains an infinite number of squares. See also comments in A262509.

a(n) is the largest leaf-node among the finite subtrees branching from node n in the infinite trunk (A259934) of the tree generated by edge-relation A049820(child) = parent, and A259934(n) itself if it is one of the nonbranching nodes (A262897).

Note that without (so far undetected) regularity in A262509, there is no a priori upper bound for the value of a(n), and for some n this might not even be finite, if it happens that contrary to its conjectured nature, A259934 is not the unique infinite component, but just the lexicographically earliest instance of multiple infinite branches of the tree. In that case we might consider this sequence to be well-defined only up to the least such node branching to multiple infinite components, or alternatively, we might mark the nonfinite values at those points with -1.