A266116 The last nonzero term on each row of A265751.

7, 7, 13, 7, 8, 7, 13, 7, 8, 13, 20, 13, 25, 13, 20, 19, 24, 19, 25, 19, 20, 37, 25, 37, 24, 25, 40, 37, 28, 37, 50, 37, 40, 33, 50, 37, 36, 37, 50, 43, 40, 43, 49, 43, 50, 67, 49, 67, 56, 49, 50, 67, 52, 67, 68, 55, 56, 67, 68, 67, 136, 67, 68, 63, 64, 67, 66, 67, 68, 79, 74, 79, 136, 79, 74, 75, 103, 79, 98, 79, 88, 103, 98, 103, 136, 85

Offset: 0

Author

Antti Karttunen, Dec 21 2015

Keywords

• nonn

Comments

Starting from j = n, search for a smallest number k such that k - d(k) = j, and if found such a number, replace j with k and repeat the procedure. When eventually such k is no longer found, then the (last such) j must be one of the terms of A045765, and it is set as the value of a(n).

Examples

			Starting from n = 21, we get the following chain: 21 -> 23 -> 27 -> 29 -> 31 -> 35 -> 37, with A082284 iterated 6 times before the final term 37 (for which A060990(37) = A082284(37) = 0) is encountered. Thus a(21) = 37.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 0..124340

Crossrefs

Cf. A000005, A045765, A060990, A082284, A265751.

Cf. A266110 (gives the number of iterations of A082284 needed before a(n) is found).

Cf. also tree A263267 (and its illustration).

Programs

Scheme

(definec (A266116 n) (cond ((A082284 n) => (lambda (lad) (if (zero? lad) n (A266116 lad))))))
;; Alternatively:
(define (A266116 n) (A265751bi n (A266110 n))) ;; Code for A265751bi given in A265751.

Formula

a(n) = A265751(n, A266110(n)).

If A060990(n) = 0, a(n) = n, otherwise a(n) = a(A082284(n)), where A082284(n) = smallest number k such that k - d(k) = n, or 0 if no such number exists, and d(n) = the number of divisors of n (A000005).

Other identities and observations. For all n >= 0:

a(n) >= n.

A060990(a(n)) = 0. [All terms are in A045765.]