cp's OEIS Frontend

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.