A082284 -id:A082284 - cp's OEIS frontend

A265751 Square array A(row,col): A(row,0) = row and for col >= 1, if A082284(row) is 0, then A(row,col) = 0, otherwise A(row,col) = A(A082284(row),col-1).

0, 1, 1, 3, 3, 2, 5, 5, 6, 3, 7, 7, 9, 5, 4, 0, 0, 11, 7, 8, 5, 0, 0, 13, 0, 0, 7, 6, 0, 0, 0, 0, 0, 0, 9, 7, 0, 0, 0, 0, 0, 0, 11, 0, 8, 0, 0, 0, 0, 0, 0, 13, 0, 0, 9, 0, 0, 0, 0, 0, 0, 0, 0, 0, 11, 10, 0, 0, 0, 0, 0, 0, 0, 0, 0, 13, 14, 11, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 20, 13, 12, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 18, 13, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 22, 0, 14

Offset: 0

Antti Karttunen, Dec 21 2015

The square array A(row>=0, col>=0) is read by downwards antidiagonals as: A(0,0), A(0,1), A(1,0), A(0,2), A(1,1), A(2,0), A(0,3), A(1,2), A(2,1), A(3,0), ...

Each row n lists all the nodes in A263267-tree that one encounters when one starts from node with number n and always chooses the smallest possible child of it [given by A082284(n)], and then the smallest possible child of that child, etc, until a leaf-child (one of the terms of A045765) is encountered, after which the rest of the row contains only zeros.

			The top left corner of the array:
   0,  1,  3,  5,  7,  0,  0,  0,  0
   1,  3,  5,  7,  0,  0,  0,  0,  0
   2,  6,  9, 11, 13,  0,  0,  0,  0
   3,  5,  7,  0,  0,  0,  0,  0,  0
   4,  8,  0,  0,  0,  0,  0,  0,  0
   5,  7,  0,  0,  0,  0,  0,  0,  0
   6,  9, 11, 13,  0,  0,  0,  0,  0
   7,  0,  0,  0,  0,  0,  0,  0,  0
   8,  0,  0,  0,  0,  0,  0,  0,  0
   9, 11, 13,  0,  0,  0,  0,  0,  0
  10, 14, 20,  0,  0,  0,  0,  0,  0
  11, 13,  0,  0,  0,  0,  0,  0,  0
  12, 18, 22, 25,  0,  0,  0,  0,  0
  13,  0,  0,  0,  0,  0,  0,  0,  0
  14, 20,  0,  0,  0,  0,  0,  0,  0
  15, 17, 19,  0,  0,  0,  0,  0,  0
  16, 24,  0,  0,  0,  0,  0,  0,  0
  17, 19,  0,  0,  0,  0,  0,  0,  0
  18, 22, 25,  0,  0,  0,  0,  0,  0
  19,  0,  0,  0,  0,  0,  0,  0,  0
  20,  0,  0,  0,  0,  0,  0,  0,  0
  21, 23, 27, 29, 31, 35, 37,  0,  0
  22, 25,  0,  0,  0,  0,  0,  0,  0
  23, 27, 29, 31, 35, 37,  0,  0,  0
  ...
Starting from n = 21, we get the following chain: 21 -> 23 -> 27 -> 29 -> 31 -> 35 -> 37, with A082284 iterated 6 times before the final nonzero term 37 (for which A060990(37) = A082284(37) = 0) is encountered. Thus the row 21 of array contains terms 21, 23, 27, 29, 31, 35, 37, followed by an infinite number of zeros.

Cf. also A000005, A045765, A060990.
Column 0: A001477, Column 1: A082284.
Cf. A266111 (number of significant terms on each row, without the trailing zeros).
Cf. A266116 (the rightmost term before trailing zeros).
See also array A263271 constructed in the same way, but obtained by following always the largest child A262686, instead of the smallest child A082284.
Cf. also tree A263267 (and its illustration).

(define (A265751 n) (A265751bi (A002262 n) (A025581 n)))
(define (A265751bi row col) (cond ((zero? col) row) ((A082284 row) => (lambda (lad) (if (zero? lad) lad (A265751bi lad (- col 1)))))))
;; Alternatively:
(define (A265751bi row col) (cond ((zero? col) row) ((and (zero? row) (= 1 col)) 1) ((zero? (A265751bi row (- col 1))) 0) (else (A082284 (A265751bi row (- col 1))))))

A(row,0) = row and for col >= 1, if A082284(row) is 0, then A(row,col) = 0, otherwise A(row,col) = A(A082284(row),col-1).

A(0,0) = 0, A(0,1) = 1; if col = 0, A(row,0) = row; and for col > 0, if A(row,col-1) = 0, then A(row,col) = 0, otherwise A(row,col) = A082284(A(row,col-1)).

A266110 If A082284(n) = 0, a(n) = 0, otherwise a(n) = 1 + 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).

Original entry on oeis.org

4, 3, 4, 2, 1, 1, 3, 0, 0, 2, 2, 1, 3, 0, 1, 2, 1, 1, 2, 0, 0, 6, 1, 5, 0, 0, 2, 4, 0, 3, 4, 2, 1, 0, 3, 1, 0, 0, 2, 2, 0, 1, 2, 0, 1, 8, 1, 7, 1, 0, 0, 6, 0, 5, 3, 0, 0, 4, 2, 3, 7, 2, 1, 0, 0, 1, 0, 0, 0, 4, 1, 3, 6, 2, 0, 0, 8, 1, 4, 0, 1, 7, 3, 6, 5, 0, 2, 5, 0, 4, 12, 5, 1, 3, 11, 4, 4, 3, 0, 2, 0, 1, 10, 0, 3, 2, 9, 1, 0, 0, 1, 1, 0, 0, 8, 2, 0, 0, 7, 1, 2, 6

Offset: 0

Antti Karttunen, Dec 21 2015

nonn

Starting from n, search for a smallest number k such that k - d(k) = n, and if found such a number, replace n with k and repeat the procedure. When eventually such k is no longer found, then (new) n must be one of the terms of A045765. The number of times the procedure can be repeated before that happens is the value of a(n). Sequence A266116 gives the stopping value of n.

			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) = 6.

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

One less than A266111.
Cf. A000005, A060990, A082284, A266116.
Cf. A045765 (positions of zeros).
Cf. tree A263267 (and its illustration).
Cf. also A264970.

A266111 If A082284(n) = 0, a(n) = 1, otherwise a(n) = 1 + 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).

Original entry on oeis.org

5, 4, 5, 3, 2, 2, 4, 1, 1, 3, 3, 2, 4, 1, 2, 3, 2, 2, 3, 1, 1, 7, 2, 6, 1, 1, 3, 5, 1, 4, 5, 3, 2, 1, 4, 2, 1, 1, 3, 3, 1, 2, 3, 1, 2, 9, 2, 8, 2, 1, 1, 7, 1, 6, 4, 1, 1, 5, 3, 4, 8, 3, 2, 1, 1, 2, 1, 1, 1, 5, 2, 4, 7, 3, 1, 1, 9, 2, 5, 1, 2, 8, 4, 7, 6, 1, 3, 6, 1, 5, 13, 6, 2, 4, 12, 5, 5, 4, 1, 3, 1, 2, 11, 1, 4, 3, 10, 2, 1, 1, 2, 2, 1, 1, 9, 3, 1, 1, 8, 2, 3, 7

Offset: 0

Antti Karttunen, Dec 21 2015

nonn

			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. Here we count the terms (not steps) in whole chain, thus a(21) = 7.

Antti Karttunen, Table of n, a(n) for n = 0..10000

One more than A266110.
Number of significant terms on row n of A265751 (without its trailing zeros).
Cf. tree A263267 (and its illustration).
Cf. A000005, A060990, A082284.
Cf. also A264971.

If A060990(n) = 0, a(n) = 1, otherwise a(n) = 1 + a(A082284(n)).
Other identities. For all n >= 0:
a(n) = 1 + A266110(n).

A060990 Number of solutions to x - d(x) = n, where d(n) is the number of divisors of n (A000005).

Original entry on oeis.org

2, 2, 1, 1, 1, 1, 3, 0, 0, 1, 1, 3, 1, 0, 1, 1, 1, 2, 1, 0, 0, 1, 4, 1, 0, 0, 1, 2, 0, 2, 1, 1, 1, 0, 2, 2, 0, 0, 2, 2, 0, 1, 1, 0, 1, 1, 3, 1, 2, 0, 0, 2, 0, 1, 1, 0, 0, 3, 2, 1, 1, 1, 2, 0, 0, 2, 0, 0, 0, 2, 4, 1, 1, 1, 0, 0, 1, 1, 2, 0, 1, 2, 1, 1, 1, 0, 1, 2, 0, 1, 1, 2, 1, 1, 1, 1, 2, 1, 0, 1, 0, 1, 3, 0, 1, 1

Offset: 0

Labos Elemer, May 11 2001

nonn

If x-d(x) is never equal to n, then n is in A045765 and a(n) = 0.

Number of solutions to A049820(x) = n. - Jaroslav Krizek, Feb 09 2014

			a(11) = 3 because three numbers satisfy equation x-d(x)=11, namely {13,15,16} with {2,4,5} divisors respectively.

Antti Karttunen, Table of n, a(n) for n = 0..110880

Cf. A000005, A002183, A049820, A049816, A082284, A155043, A236561, A236565, A259934, A261100, A262507, A262513, A262686.
Cf. A045765 (positions of zeros), A236562 (positions of nonzeros), A262511 (positions of ones).
Cf. A263087 (computed for squares).

lim = 105; s = Table[n - DivisorSigma[0, n], {n, 2 lim + 3}]; Length@ Position[s, #] & /@ Range[0, lim] (* Michael De Vlieger, Sep 29 2015, after Wesley Ivan Hurt at A049820 *)

allocatemem(123456789);
uplim = 2162160; \\ = A002182(41).
v060990 = vector(uplim);
for(n=3, uplim, v060990[n-numdiv(n)]++);
A060990 = n -> if(!n,2,v060990[n]);
uplim2 = 110880; \\ = A002182(30).
for(n=0, uplim2, write("b060990.txt", n, " ", A060990(n)));
\\ Antti Karttunen, Sep 25 2015

(define (A060990 n) (if (zero? n) 2 (add (lambda (k) (if (= (A049820 k) n) 1 0)) n (+ n (A002183 (+ 2 (A261100 n)))))))
;; Auxiliary function add implements sum_{i=lowlim..uplim} intfun(i)
(define (add intfun lowlim uplim) (let sumloop ((i lowlim) (res 0)) (cond ((> i uplim) res) (else (sumloop (1+ i) (+ res (intfun i)))))))
;; Proof-of-concept code for the given formula, by Antti Karttunen, Sep 25 2015

a(0) = 2; for n >= 1, a(n) = Sum_{k = n .. n+A002183(2+A261100(n))} [A049820(k) = n]. (Here [...] denotes the Iverson bracket, resulting 1 when A049820(k) is n and 0 otherwise.) - Antti Karttunen, Sep 25 2015, corrected Oct 12 2015.
a(n) = Sum_{k = A082284(n) .. A262686(n)} [A049820(k) = n] (when tacitly assuming that A049820(0) = 0.) - Antti Karttunen, Oct 12 2015
Other identities and observations. For all n >= 0:
a(A045765(n)) = 0. a(A236562(n)) > 0. - Jaroslav Krizek, Feb 09 2014

Offset corrected by Jaroslav Krizek, Feb 09 2014

A263267 Breadth-first traversal of the tree defined by the edge-relation A049820(child) = parent.

Original entry on oeis.org

0, 1, 2, 3, 4, 6, 5, 8, 9, 10, 12, 7, 11, 14, 18, 13, 15, 16, 20, 22, 17, 24, 25, 26, 28, 30, 19, 21, 32, 34, 23, 40, 38, 42, 27, 44, 48, 46, 29, 36, 50, 56, 60, 49, 52, 54, 31, 33, 72, 58, 35, 84, 62, 66, 37, 39, 96, 68, 70, 41, 45, 104, 108, 74, 76, 78, 80, 43, 47, 120, 81, 82, 90, 88, 51, 128, 132, 83, 85, 86, 94, 53, 55, 136, 140, 87, 92, 102

Offset: 0

Antti Karttunen, Nov 27 2015

It is conjectured that the terms of A259934 trace the only infinite path in this tree.

After the root (0), the tree narrows next time to the width of just one node at level A262508(1) = 9236, with vertex 119143.

			Rows 0 - 21 of the table. The lines show the nodes of the tree connected by the edge-relation A049820(child) = parent:
0;
| \
1, 2;
| \  \
3, 4, 6;____
|  |  | \   \
5, 8, 9, 10, 12;
|     |   |   |
7, _ 11, 14, 18;
  /  | \   \   \
13, 15, 16, 20, 22;____
     |  |      / | \   \
    17, 24, 25, 26, 28, 30;
     | \         |      |
    19, 21,     32,     34;
         |       |      | \
        23,     40,    38, 42;____
         |              | \       \
        27,            44, 48,     46;____
         | \            |   | \    |  \   \
        29, 36,        50, 56, 60, 49, 52, 54;
         | \                   |           |
        31, 33,                72,         58;
         |                     |           |  \
        35,                    84,         62, 66;
         | \                   |           |  \
        37, 39,                96,         68, 70;_______
            |  \               |  \           / |  \     \
            41, 45,           104, 108,     74, 76, 78,   80;
            |   |              |                |   |  \    \
            43, 47,           120,             _81, 82, 90, 88;
                |              |  \           / |   |   |
                51,           128, 132,     83, 85, 86, 94;
                 | \            | \          |       |   |
                53, 55        136, 140      87,     92, 102;______
                 |                           | \     |    |  \    \
                57,_                        89, 91, 98, 106,  110, 112;
               / |  \                       /   / \       |     |
             59, 63, 64,                  93, 95, 100,   114,   116;
              |                            |   |          |  \
             61,                          99, 97,       _118, 126;
              |                            |   |       /  |  \
             65,                         101, 105,  121, 122, 124;
(See also _Michael De Vlieger_'s poster in the Links section.)

Antti Karttunen, Table of n, a(n) for n = 0..10425; levels 0 .. 1001 of the tree
Michael De Vlieger, Poster illustrating A259934 and A263267
Index entries for sequences that are permutations of the natural numbers

Inverse permutation: A263268.
Cf. A000005, A049820, A060990, A082284, A155043, A259934, A262508.
Cf. A262507 (number of terms on row/level n), A263260 (total number of terms in levels 0 .. n).
Cf. A264988 (the left edge), this differs from A261089 (the least term on each level) for the first time at level 69.
Cf. A263269 (the right edge).
Cf. A262686 (maximum term on the level n).
Cf. A045765 (the leaves of the tree).
Cf. also permutations A263265 (obtained from this table by sorting each row into ascending order), A263266.
Cf. also arrays A265751 and A263271.
Differs from A263265 for the first time at n=31, where a(31) = 40, while A263265(31) = 38.
Cf. also A088975.

uplim = 125753; \\ = A263260(10001).
checklimit = 1440; \\ Hard limit 1440 good for at least up to A002182(67) = 1102701600 as A002183(67) = 1440.
v263267 = vector(uplim);
A263267 = n -> if(!n,n,v263267[n]);
z = 0; for(n=0, uplim, t = A263267(n); write("b263267.txt", n, " ", t); for(k=t+1, t+checklimit, if((k-numdiv(k)) == t, z++; if(z <= uplim, v263267[z] = k))));

# After David Eppstein's Python-code for A088975.
def A263267():
  '''Breadth-first reading of irregular tree defined by the edge-relation A049820(child) = parent'''
  yield 0
  for x in A263267():
    for k in [x+1 .. 2*(x+1)]:
      if ((k - sloane.A000005(k)) == x): yield k
def take(n,g):
  '''Returns a list composed of the next n elements returned by generator g.'''
  return [next(g) for _ in range(n)]
take(120, A263267())

;; This version creates the list of terms incrementally, using append! function that physically modifies the list at the same time as it is traversed. Otherwise the idea is essentially the same as with Python/Sage-program above:
(define (A263267list_up_to_n_terms_at_least n) (let ((terms-produced (list 0))) (let loop ((startp terms-produced) (endp terms-produced) (k (- n 1))) (cond ((<= k 0) terms-produced) (else (let ((children (children-of-n-in-A049820-tree (car startp)))) (cond ((null? children) (loop (cdr startp) endp k)) (else (begin (append! endp children) (loop (cdr startp) children (- k (length children))))))))))))
(define (children-of-n-in-A049820-tree n) (let loop ((k (A262686 n)) (children (list))) (cond ((<= k n) children) ((= (A049820 k) n) (loop (- k 1) (cons k children))) (else (loop (- k 1) children)))))

A262686 a(n) = largest number k such that k - d(k) = n, or 0 if no such number exists, where d(n) = the number of divisors of n (A000005).

Original entry on oeis.org

2, 4, 6, 5, 8, 7, 12, 0, 0, 11, 14, 16, 18, 0, 20, 17, 24, 21, 22, 0, 0, 23, 30, 27, 0, 0, 32, 36, 0, 33, 34, 35, 40, 0, 42, 39, 0, 0, 48, 45, 0, 43, 46, 0, 50, 47, 54, 51, 60, 0, 0, 55, 0, 57, 58, 0, 0, 64, 66, 61, 72, 65, 70, 0, 0, 69, 0, 0, 0, 75, 80, 73, 84, 77, 0, 0, 81, 79, 90, 0, 88, 85, 86, 87, 96, 0, 92, 91, 0, 93, 94, 100, 98, 99, 102, 97, 108, 105, 0, 101

Offset: 0

Antti Karttunen, Sep 28 2015

nonn

Antti Karttunen, Table of n, a(n) for n = 0..10080

Cf. A000005, A002183, A049820, A060990, A261100, A262512.

Cf. also A082284 (the smallest such number), A262511 (positions where these are equal and nonzero).

Table[k = 2 n + 3; While[Nor[k - DivisorSigma[0, k] == n, k == 0], k--]; k, {n, 0, 99}] (* Michael De Vlieger, Sep 29 2015 *)

(definec (A262686 n) (if (zero? n) 2 (let ((u (+ n (A002183 (+ 2 (A261100 n)))))) (let loop ((k u)) (cond ((= (A049820 k) n) k) ((< k n) 0) (else (loop (- k 1))))))))

A262697 a(n)=0 if n is in A259934, otherwise number of nodes (including leaves and the node n itself) in that finite subtree whose root is n and edge-relation is defined by A049820(child) = parent.

Original entry on oeis.org

0, 6, 0, 3, 2, 2, 0, 1, 1, 38, 3, 37, 0, 1, 2, 33, 2, 32, 0, 1, 1, 30, 0, 29, 1, 1, 3, 28, 1, 26, 0, 24, 2, 1, 0, 23, 1, 1, 16, 21, 1, 2, 0, 1, 2, 18, 0, 17, 13, 1, 1, 16, 1, 14, 0, 1, 1, 13, 0, 10, 11, 9, 0, 1, 1, 8, 1, 1, 1, 6, 0, 4, 10, 3, 1, 1, 23, 2, 0, 1, 2, 22, 4, 20, 9, 1, 3, 19, 1, 5, 0, 13, 2, 4, 0, 11, 8, 10, 1, 3, 1, 2, 0, 1, 6, 9, 0, 8, 1, 1, 2, 6, 1, 1, 0, 3, 1, 1, 0, 2, 5, 0, 12, 1

Offset: 0

Antti Karttunen, Oct 04 2015

nonn

			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, together {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 both 7 and 8 are terms of A045765. Thus a(1) = 6.
For n=9, its transitive closure is {9, 11, 13, 15, 16, 17, 19, 21, 23, 24, 27, 29, 31, 33, 35, 36, 37, 39, 41, 43, 45, 47, 51, 53, 55, 57, 59, 61, 63, 64, 65, 67, 69, 71, 73, 75, 77, 79}, containing 38 terms, thus a(9) = 38.

Antti Karttunen, Table of n, a(n) for n = 0..17724

Cf. A000005, A045765, A049820, A060990, A082284, A259934, A262686, A262693.
Cf. A262679, A262522, A262695, A262696, A262890.
Cf. also A213727, A227643, A255327.

If A262693(n) = 1 [when n is in A259934],
  then a(n) = 0,
otherwise, if A060990(n) = 0 [when n is one of the leaves, A045765],
  then a(n) = 1,
otherwise:
  a(n) = 1 + Sum_{k = A082284(n) .. A262686(n)} [A049820(k) = n] * a(k).
(In the last clause [ ] stands for Iverson bracket, giving as its result 1 only when A049820(k) = n, and 0 otherwise).

A262695 a(n)=0 if n is in A259934, otherwise 1 + number of steps to reach the farthest leaf in that finite branch of the tree defined by edge-relation A049820(child) = parent.

Original entry on oeis.org

0, 4, 0, 3, 2, 2, 0, 1, 1, 24, 3, 23, 0, 1, 2, 22, 2, 21, 0, 1, 1, 20, 0, 19, 1, 1, 3, 18, 1, 17, 0, 16, 2, 1, 0, 15, 1, 1, 10, 14, 1, 2, 0, 1, 2, 13, 0, 12, 9, 1, 1, 11, 1, 10, 0, 1, 1, 9, 0, 8, 8, 7, 0, 1, 1, 6, 1, 1, 1, 5, 0, 4, 7, 3, 1, 1, 13, 2, 0, 1, 2, 12, 4, 11, 6, 1, 3, 10, 1, 5, 0, 9, 2, 4, 0, 8, 5, 7, 1, 3, 1, 2, 0, 1, 4, 6, 0, 5, 1, 1, 2, 4, 1, 1, 0, 3, 1, 1, 0, 2, 3

Offset: 0

Antti Karttunen, Oct 04 2015

nonn

			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 this 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 (leaves). Starting iterating from 7 with A049820, we get 7 -> 5, 5 -> 3, 3 -> 1, and starting from 8 we get 8 -> 4, 4 -> 1, of which the former path is longer (3 steps), thus a(1) = 3+1 = 4.
For n=9, its transitive closure is {9, 11, 13, 15, 16, 17, 19, 21, 23, 24, 27, 29, 31, 33, 35, 36, 37, 39, 41, 43, 45, 47, 51, 53, 55, 57, 59, 61, 63, 64, 65, 67, 69, 71, 73, 75, 77, 79}. In this case the longest path is obtained by starting iterating from the largest of these: 79 -> 77 -> 73 -> 71 -> 69 -> 65 -> 61 -> 59 -> 57 -> 53 -> 51 -> 47 -> 45 -> 39 -> 35 -> 31 -> 29 -> 27 -> 23 -> 21 -> 17 -> 15 -> 11 -> 9, which is 23 steps long, thus a(9) = 23+1 = 24.

Antti Karttunen, Table of n, a(n) for n = 0..17724

Cf. A000005, A045765, A049820, A060990, A082284, A259934, A262686, A262693.
Cf. A262522, A262696, A262697.
Cf. also A213725.

If A262693(n) = 1 [when n is in A259934],
  then a(n) = 0,
otherwise, if A060990(n) = 0 [when n is one of the leaves, A045765],
  then a(n) = 1,
otherwise:
  a(n) = 1 + Max_{k = A082284(n) .. A262686(n)} [A049820(k) = n] * a(k).
(In the last clause [ ] stands for Iverson bracket, giving as its result 1 only when A049820(k) = n, and 0 otherwise).

A262522 a(n)=0 if n is in A259934, otherwise the largest term in A045765 from which one can reach n by iterating A049820 zero or more times.

Original entry on oeis.org

0, 8, 0, 7, 8, 7, 0, 7, 8, 79, 20, 79, 0, 13, 20, 79, 24, 79, 0, 19, 20, 79, 0, 79, 24, 25, 40, 79, 28, 79, 0, 79, 40, 33, 0, 79, 36, 37, 140, 79, 40, 43, 0, 43, 50, 79, 0, 79, 140, 49, 50, 79, 52, 79, 0, 55, 56, 79, 0, 79, 140, 79, 0, 63, 64, 79, 66, 67, 68, 79, 0, 79, 140, 79, 74, 75, 123, 79, 0, 79, 88, 123, 98, 123, 140, 85, 98, 123, 88, 103, 0, 123, 98, 103, 0, 123

Offset: 0

Antti Karttunen, Oct 04 2015

nonn

If n is itself in A045765, we iterate 0 times, and thus 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 of 8, see the example in A262695).
For n=9, its transitive closure is {9, 11, 13, 15, 16, 17, 19, 21, 23, 24, 27, 29, 31, 33, 35, 36, 37, 39, 41, 43, 45, 47, 51, 53, 55, 57, 59, 61, 63, 64, 65, 67, 69, 71, 73, 75, 77, 79}. The largest term is 79, thus a(9) = 79.

Antti Karttunen, Table of n, a(n) for n = 0..17724

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

Cf. A262679, A262695, A262696, A262697.

If A262693(n) = 1 [when n is in A259934],
  then a(n) = 0,
otherwise, if A060990(n) = 0 [when n is one of the leaves, A045765],
  then a(n) = n,
otherwise:
  a(n) = Max_{k = A082284(n) .. A262686(n)} [A049820(k) = n] * a(k).
(In the last clause [ ] stands for Iverson bracket, giving as its result 1 only when A049820(k) = n, and 0 otherwise).
Other identities. For all n >= 1:
a(A262511(n)) = a(A262512(n)) = a(A082284(A262511(n))).

A262890 a(n) = total number of nodes in the finite subtrees branching from node n in the infinite trunk (A259934) of the tree generated by edge-relation A049820(child) = parent.

Original entry on oeis.org

6, 0, 41, 0, 0, 5, 0, 16, 0, 2, 0, 1, 1, 26, 4, 0, 0, 3, 0, 1, 13, 0, 105, 2, 1, 1, 2, 5, 18, 7, 0, 0, 0, 1, 3, 3, 0, 0, 5, 0, 4, 13, 2, 7, 0, 0, 7, 6, 1, 0, 0, 0, 53, 0, 0, 0, 90, 1, 0, 5, 0, 2, 0, 1, 1, 0, 12, 1, 0, 3, 61, 0, 0, 0, 0, 0, 0, 2, 117, 7, 0, 2, 10, 0, 0, 1, 23, 1, 1, 1, 0, 0, 1, 0, 5, 1, 0, 3, 2, 2, 568, 1, 1, 1, 4, 1, 5, 9, 3, 0, 22, 1, 0, 9, 2, 1, 7, 0, 2, 10, 1, 1, 0

Offset: 0

Antti Karttunen, Oct 04 2015

nonn

Antti Karttunen, Table of n, a(n) for n = 0..8107

Cf. A000005, A049820, A082284, A259934, A262686, A262697, A262888, A262889.
Cf. A262892 (positions of zeros).
Cf. A262893 (partial sums).
Cf. also A255330.

(define (A262890 n) (let ((t (A259934 n))) (let loop ((s 0) (k (A262686 t))) (cond ((<= k t) s) ((= t (A049820 k)) (loop (+ s (A262697 k)) (- k 1))) (else (loop s (- k 1)))))))

a(n) = Sum_{k = A082284(A259934(n)) .. A262686(A259934(n))} [A049820(k) = A259934(n)] * A262697(k).
(Here [ ] stands for Iverson bracket, giving as its result 1 only when A049820(k) = A259934(n), and 0 otherwise.)
Other identities. For all n >= 0:
a(n) = A262888(n) + A262889(n).

cp's OEIS Frontend

A265751 Square array A(row,col): A(row,0) = row and for col >= 1, if A082284(row) is 0, then A(row,col) = 0, otherwise A(row,col) = A(A082284(row),col-1).

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A266110 If A082284(n) = 0, a(n) = 0, otherwise a(n) = 1 + 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).

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A266111 If A082284(n) = 0, a(n) = 1, otherwise a(n) = 1 + 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).

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A060990 Number of solutions to x - d(x) = n, where d(n) is the number of divisors of n (A000005).

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A263267 Breadth-first traversal of the tree defined by the edge-relation A049820(child) = parent.

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A262686 a(n) = largest number k such that k - d(k) = n, or 0 if no such number exists, where d(n) = the number of divisors of n (A000005).

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A262697 a(n)=0 if n is in A259934, otherwise number of nodes (including leaves and the node n itself) in that finite subtree whose root is n and edge-relation is defined by A049820(child) = parent.

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A262695 a(n)=0 if n is in A259934, otherwise 1 + number of steps to reach the farthest leaf in that finite branch of the tree defined by edge-relation A049820(child) = parent.

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A262522 a(n)=0 if n is in A259934, otherwise the largest term in A045765 from which one can reach n by iterating A049820 zero or more times.

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