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approved

editing

proposed

Antti Karttunen, <a href="/A262522/b262522.txt">Table of n, a(n) for n = 0..17724</a>

If n is itself in A045765, we iterate 0 times, and then thus a(n) = n.

a(n)=0 if n is in A259934, otherwise the largest term in A045765 from which one can reach n by iterating A049820. If n is itself in A045765, we iterate 0 zero or more times, and then a(n) = n.

If n is itself in A045765, we iterate 0 times, and then a(n) = n.

For n=1, its transitive closure (as defined by edge-relation A049820(child) = parent) is the union of {1} itself together with all its descendants: {1, 3, 4, 5, 7, 8}. We see that there are no other nodes in a subtree whose root is 1, because A049820(3) = 3 - d(3) = 1, A049820(4) = 1, A049820(5) = 3, A049820(7) = 5, A049820(8) = 4 and of these only 7 and 8 are terms of A045765. The largest term (which by necessity is always a term of A045765) is here 8, thus a(1) = 8. Note however that it is not always the largest leaf from which starts the longest path leading back to n. (In this case it is 7 instead 7, of 8, see the example in A262695).

a(A262511(n)) = a(A262512(n)) = a(A082284(A262511(n))).

Other identities. For all n >= 1:

For any n in A262511 but not in A259934, a(n) = a(A082284(n)).

a(A262511(n)) = a(A082284(A262511(n))).

Cf. A000005, A045765, A049820, A060990, A082284, A259934, A262511, A262512, A262686, A262693.

Other identities:

For any n in A262511 but not in A259934, a(n) = a(A082284(n)).

For n=1, its transitive closure (as defined by edge-relation A049820(child) = parent) is the union of {1} itself together with all its descendants: {1, 3, 4, 5, 7, 8}. We see that there are no other nodes in a subtree whose root is 1, because A049820(3) = 3 - d(3) = 1, A049820(4) = 1, A049820(5) = 3, A049820(7) = 5, A049820(8) = 4 and of these only 7 and 8 are terms of A045765. The largest term (which by necessity is always a term of A045765) is here 8, thus a(1) = 8. Note however that it is not always the leaf from which starts the longest path leading back to n. In this case it is instead 7, see the example in A262695.