A049820 -id:A049820 - cp's OEIS frontend

A259934 Infinite sequence starting with a(0)=0 such that A049820(a(k)) = a(k-1) for all k>=1, where A049820(n) = n - (number of divisors of n).

0, 2, 6, 12, 18, 22, 30, 34, 42, 46, 54, 58, 62, 70, 78, 90, 94, 102, 106, 114, 118, 121, 125, 129, 144, 152, 162, 166, 174, 182, 190, 194, 210, 214, 222, 230, 236, 242, 250, 254, 270, 274, 282, 294, 298, 302, 310, 314, 330, 342, 346, 354, 358, 366, 374, 390, 394, 402, 410, 418, 426, 434, 442, 446, 462, 466, 474, 486, 494, 510, 522, 530, 546, 558, 562, 566, 574, 582, 590

Offset: 0

Max Alekseyev, Jul 09 2015

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)

Robert Israel, Table of n, a(n) for n = 0..64800
M. Alekseyev et al., Apparently unique infinite sequences related to the sum of divisors, Discussion in SeqFan mailing list, 2015.
Michael De Vlieger, Poster illustrating A259934 and A263267
V. S. Guba et al., Sequence and divisors, 2015. (in Russian)

Cf. A000005, A049820, A060990, A259935 (first differences).
Topmost row of A263255. Cf. also irregular tables A263267 & A263265 and array A262898.
Cf. A262693 (characteristic function).
Cf. A155043, A262694, A262904 (left inverses).
Cf. A262514 (squares present), A263276 (their positions), A263277.
Cf. A262517 (odd terms).
Cf. A262509, A262510, A262897 (other subsequences).
Cf. also A175304, A260257, A262680.
Cf. A261089, A261103, A262503, A262506, A262516, A263279, A263280, A263085, A263086, A263253, A263257, A263278.
Cf. also A262679, A262896 (see the C++ program there).
No common terms with A045765 or A262903.
Positions of zeros in A262522, A262695, A262696, A262697, A263254.
Various metrics concerning finite side-trees: A262888, A262889, A262890.
Cf. also A262891, A262892 and A262895 (cf. its graph).
Cf. A260084, A260124 (variants).
Cf. also A179016 (a similar "beanstalk trunk sequence" but with more tractable and regular behavior).

N:= 10^4: # to get "guaranteed unique" terms <= N
S:= Vector(N,datatype=integer[1]):
for n from N+1 to 2*N do
  k:= n - numtheory:-tau(n);
  if k <= N then S[k]:= S[k]+1; B[k]:= n; fi;
od:
for n from N to 3 by -1 do
  if S[n] >= 1 then
    k:= n - numtheory:-tau(n);
    S[k]:= S[k]+1; B[k]:= n;
  fi
od:
A[0]:= 0: A[1]:= 2:
for n from 2 do
  b:= B[A[n-1]];
  if b > N or S[b] > 1 then break fi;
  A[n]:= b;
od:
seq(A[i],i=0..n-1); # Robert Israel, Jul 09 2015

NN = 10^4; (* to get "guaranteed unique" terms <= NN *)
Clear[A, B, S]; S[]=0; For[n = NN+1, n <= 2*NN, n++, k = n-DivisorSigma[0, n]; If[k <= NN, S[k] = S[k]+1; B[k]=n]]; For[n=NN, n >= 3, n--, If[S[n] >= 1 , k = n-DivisorSigma[0, n]; S[k] = S[k]+1; B[k]=n]]; A[0]=0; A[1]=2; For[n=2, True, n++, b = B[A[n-1]]; If[b>NN || S[b]>1, Break[]]; A[n]=b]; Table[A[i], {i, 0, n-1}] (* _Jean-François Alcover, Jul 22 2015, after Robert Israel *)

From Antti Karttunen, Nov 27 2015: (Start)
Other identities and observations. For all n >= 0:
a(n) = A262679(A262896(n)).
A155043(a(n)) = A262694(a(n)) = A262904(a(n)) = n.
A261089(n) <= a(n) <= A262503(n). [A261103 and A262506 give the distances of a(n) to these bounds.]
(End)

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)))))

A263255 Square array A(r,c), where each row r lists all numbers that are r edges distant from the infinite trunk (A259934) of the tree defined by edge-relation A049820(child) = parent.

Original entry on oeis.org

0, 2, 1, 6, 9, 3, 12, 10, 4, 5, 18, 25, 11, 8, 7, 22, 26, 14, 13, 17, 19, 30, 28, 32, 15, 24, 21, 23, 34, 38, 44, 16, 72, 84, 93, 27, 42, 49, 48, 20, 87, 89, 95, 97, 29, 46, 52, 81, 40, 98, 91, 96, 99, 36, 31, 54, 66, 86, 50, 139, 143, 100, 104, 101, 33, 35, 58, 68, 88, 56, 141, 145, 149, 108, 105, 103, 109, 37

Offset: 0

Antti Karttunen, Nov 07 2015

The array A(row>=0,col>=1) is read by downwards antidiagonals: A(0,1), A(0,2), A(1,1), A(0,3), A(1,2), A(2,1), A(0,4), A(1,3), A(2,2), A(3,1), etc.

			Top left corner of the array:
   0,   2,   6,  12,  18,  22,  30,  34,  42,  46,  54,  58,  62,  70
   1,   9,  10,  25,  26,  28,  38,  49,  52,  66,  68,  74,  76,  80
   3,   4,  11,  14,  32,  44,  48,  81,  86,  88, 116, 130, 135, 175
   5,   8,  13,  15,  16,  20,  40,  50,  56,  60,  83,  85,  92, 134
   7,  17,  24,  72,  87,  98, 139, 141, 142, 150, 202, 208, 225, 228
  19,  21,  84,  89,  91, 143, 145, 146, 147, 148, 206, 220, 227, 301
  23,  93,  95,  96, 100, 149, 153, 154, 160, 212, 229, 240, 305, 356
  27,  97,  99, 104, 108, 151, 158, 224, 248, 307, 309, 379, 381, 385
  29,  36, 101, 105, 120, 155, 164, 232, 260, 264, 311, 324, 383, 387
  31,  33, 103, 107, 128, 132, 157, 159, 276, 280, 313, 389, 391, 453
  35, 109, 111, 136, 140, 161, 165, 393, 395, 399, 461, 465, 532, 540
  37,  39, 113, 115, 117, 163, 167, 171, 397, 401, 403, 405, 463, 467
  41,  45, 119, 173, 407, 471, 473, 475, 568, 571, 572, 573, 575, 659
  43,  47, 123, 177, 409, 411, 477, 483, 484, 577, 578, 579, 580, 585
  51, 179, 413, 415, 479, 481, 495, 581, 583, 587, 589, 594, 671, 676
  53,  55, 181, 183, 417, 485, 591, 595, 602, 612, 673, 681, 877, 879
  57, 185, 187, 189, 419, 423, 487, 489, 593, 610, 683, 685, 693, 881
  59,  63,  64, 191, 195, 196, 421, 425, 427, 491, 493, 597, 614, 618
  61, 193, 197, 429, 435, 497, 599, 603, 622, 691, 705, 893, 895, 897
  65, 199, 201, 431, 499, 501, 601, 605, 626, 628, 695, 711, 899, 901
  ...

Transpose: A263256.
Row 0: A259934, Row 1: A263261, Row 2: A263262, Row 3: A263263, Row 4: A263264.
Column 0: A263257.
Cf. A263254 (row index, zero-based), A263275 (row index, one-based), A263274 (column index, one-based).
Cf. also array A262898.

A236562 Numbers n such that A049820(x) = n has a solution.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 9, 10, 11, 12, 14, 15, 16, 17, 18, 21, 22, 23, 26, 27, 29, 30, 31, 32, 34, 35, 38, 39, 41, 42, 44, 45, 46, 47, 48, 51, 53, 54, 57, 58, 59, 60, 61, 62, 65, 69, 70, 71, 72, 73, 76, 77, 78, 80, 81, 82, 83, 84, 86, 87, 89, 90, 91, 92, 93, 94

Offset: 1

Jaroslav Krizek, Feb 09 2014

nonn

Complement of A045765.

			10 is in sequence because A049820(14) = 14 - A000005(14) = 14 - 4 = 10.

Jaroslav Krizek, Table of n, a(n) for n = 1..10000

Cf. A049820, A236561, A236565.

Take[Sort@ DeleteDuplicates@ Table[n - DivisorSigma[0, n], {n, 1200}], 67] (* Michael De Vlieger, Oct 13 2015 *)

A060990(a(n)) > 0.

A262680 Number of squares encountered before zero is reached when iterating A049820 starting from n: a(0) = 0 and for n >= 1, a(n) = A010052(n) + a(A049820(n)).

Original entry on oeis.org

0, 1, 0, 1, 2, 1, 0, 1, 2, 1, 0, 1, 0, 1, 0, 1, 2, 1, 0, 1, 0, 1, 0, 1, 2, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 2, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 2, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 2, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1

Offset: 0

Antti Karttunen, Oct 03 2015

nonn

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.

			For n=1, we subtract 1 - A000005(1) = 0, thus we reach zero in one step, and the starting value 1 is a square, thus a(1) = 1. Also, the parity changes once, from odd to even as we go from 1 to 0.
For n=24, when we start repeatedly subtracting the number of divisors (A000005), we obtain the following numbers: 24 - A000005(24) = 24 - 8 = 16, 16 - A000005(16) = 16 - 5 = 11, 11 - 2 = 9, 9 - 3 = 6, 6 - 4 = 2, 2 - 2 = 0. Of these numbers, 16 and 9 are squares larger than zero, thus a(24)=2. Also, we see that the parity changes twice: from even to odd at 16 and then back from odd to even at 9.

Antti Karttunen, Table of n, a(n) for n = 0..10201

Bisections: A262681, A262682.
Cf. A262687 (positions of records).
Cf. A000005, A000290, A010052, A049820, A155043, A259934, A261088, A262509, A262514, A262676, A262677.

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).

A263254 If A262693(n) = 1, then a(n) = 0, otherwise a(n) = 1 + a(A049820(n)).

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, Nov 07 2015

nonn

Distance of node n from the infinite trunk (A259934) of the tree defined by edge-relation A049820(child) = parent.

Zero-based row index to array A263255.

Antti Karttunen, Table of n, a(n) for n = 0..10002

One less than A263275.
Cf. A263257 (positions of records, where each n first occurs).
Cf. A000005, A049820, A155043, A259934, A262693, A262695, A262904, A263255, A263274.

If A262693(n) = 1 [when n is in A259934], then a(n) = 0, otherwise a(n) = 1 + a(A049820(n)).
a(n) = A155043(n) - A262904(n).
a(n) = A263275(n) - 1.

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

A259934 Infinite sequence starting with a(0)=0 such that A049820(a(k)) = a(k-1) for all k>=1, where A049820(n) = n - (number of divisors of n).

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

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A263255 Square array A(r,c), where each row r lists all numbers that are r edges distant from the infinite trunk (A259934) of the tree defined by edge-relation A049820(child) = parent.

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A236562 Numbers n such that A049820(x) = n has a solution.

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A262680 Number of squares encountered before zero is reached when iterating A049820 starting from n: a(0) = 0 and for n >= 1, a(n) = A010052(n) + a(A049820(n)).

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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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A263254 If A262693(n) = 1, then a(n) = 0, otherwise a(n) = 1 + a(A049820(n)).

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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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