A262508 -id:A262508 - cp's OEIS frontend

A263081 a(n) = largest k for which A155043(k) < A262508(n); a(n) = A262509(n) + A262909(n).

124340, 124340, 124340, 124340, 124340, 124340, 124340, 124340, 124340, 124340, 124340, 124340, 124340, 124340, 124340, 124340, 124340, 124340, 124340, 124340, 124340, 124340, 124340, 124340, 124340, 124340, 124340, 124340, 124340, 124340, 124340, 124340, 124340, 124340, 124340, 124340, 124340, 124340, 24684000, 24684000, 24684000, 24684000, 24684000, 24684000, 24684000

Offset: 1

Antti Karttunen, Oct 09 2015

nonn

a(n) = largest k for which A155043(k) < A155043(A262509(n)).
If a(n) > A262509(n) then it must be a leaf (see comments in A262909 for why). Particularly, we have A045765(40722) = 124340, A045765(8191770) = 24684000.
Terms of sequence (together with the corresponding values in A262508) give particularly clean values for the boundaries that are used for example in the C++-program which computes A262896.

Antti Karttunen, Table of n, a(n) for n = 1..68

Cf. A045765, A155043, A262508, A262509, A262896, A262909, A263077, A263083.

(define (A263081 n) (+ (A262509 n) (A262909 n)))

a(n) = A263077(A262509(n)).

a(n) = A262509(n) + A262909(n).

A155043 a(0)=0; for n >= 1, a(n) = 1 + a(n-d(n)), where d(n) is the number of divisors of n (A000005).

Original entry on oeis.org

0, 1, 1, 2, 2, 3, 2, 4, 3, 3, 3, 4, 3, 5, 4, 5, 5, 6, 4, 7, 5, 7, 5, 8, 6, 6, 6, 9, 6, 10, 6, 11, 7, 11, 7, 12, 10, 13, 8, 13, 8, 14, 8, 15, 9, 14, 9, 15, 9, 10, 10, 16, 10, 17, 10, 17, 10, 18, 11, 19, 10, 20, 12, 19, 19, 21, 12, 22, 13, 22, 13, 23, 11, 24, 14, 23, 14, 25, 14, 26, 14, 15, 15

Offset: 0

Ctibor O. Zizka, Jan 19 2009

From Antti Karttunen, Sep 23 2015: (Start)
Number of steps needed to reach zero 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).
The original name was: a(n) = 1 + a(n-sigma_0(n)), a(0)=0, sigma_0(n) number of divisors of n.
(End)

Antti Karttunen, Table of n, a(n) for n = 0..124340
B. Balamohan, A. Kuznetsov and S. Tanny, On the behavior of a variant of Hofstadter's Q-sequence, J. Integer Sequences, Vol. 10 (2007), #07.7.1.
Antti Karttunen, Graph plotted with OEIS Plot script up to n=10000
John A. Pelesko, Generalizing the Conway-Hofstadter $10,000 Sequence, Journal of Integer Sequences, Vol. 7 (2004), Article 04.3.5.

Cf. A000005, A049820, A060990, A259934.
Sum of A262676 and A262677.
Cf. A261089 (positions of records, i.e., the first occurrence of n), A262503 (the last occurrence), A262505 (their difference), A263082.
Cf. A262518, A262519 (bisections, compare their scatter plots), A262521 (where the latter is less than the former).
Cf. A261085 (computed for primes), A261088 (for squares).
Cf. A262507 (number of times n occurs in total), A262508 (values occurring only once), A262509 (their indices).
Cf. A263265 (nonnegative integers arranged by the magnitude of a(n)).
Cf. also A263077, A263078, A263079, A263080.
Cf. also A261104, A262680, A262904, A263254, A263259, A263260, A263266, A263270.
Cf. also A004001, A005185.
Cf. A264893 (first differences), A264898 (where repeating values occur).

import Data.List (genericIndex)
a155043 n = genericIndex a155043_list n
a155043_list = 0 : map ((+ 1) . a155043) a049820_list
-- Reinhard Zumkeller, Nov 27 2015

with(numtheory): a := proc (n) if n = 0 then 0 else 1+a(n-tau(n)) end if end proc: seq(a(n), n = 0 .. 90); # Emeric Deutsch, Jan 26 2009

a[0] = 0; a[n_] := a[n] = 1 + a[n - DivisorSigma[0, n]]; Table[a@n, {n, 0, 82}] (* Michael De Vlieger, Sep 24 2015 *)

uplim = 110880; \\ = A002182(30).
v155043 = vector(uplim);
v155043[1] = 1; v155043[2] = 1;
for(i=3, uplim, v155043[i] = 1 + v155043[i-numdiv(i)]);
A155043 = n -> if(!n,n,v155043[n]);
for(n=0, uplim, write("b155043.txt", n, " ", A155043(n)));
\\ Antti Karttunen, Sep 23 2015

from sympy import divisor_count as d
def a(n): return 0 if n==0 else 1 + a(n - d(n))
print([a(n) for n in range(101)]) # Indranil Ghosh, Jun 03 2017

(definec (A155043 n) (if (zero? n) n (+ 1 (A155043 (A049820 n)))))
;; Antti Karttunen, Sep 23 2015

From Antti Karttunen, Sep 23 2015 & Nov 26 2015: (Start)
a(0) = 0; for n >= 1, a(n) = 1 + a(A049820(n)).
a(n) = A262676(n) + A262677(n). - Oct 03 2015.
Other identities. For all n >= 0:
a(A259934(n)) = a(A261089(n)) = a(A262503(n)) = n. [The sequence works as a left inverse for sequences A259934, A261089 and A262503.]
a(n) = A262904(n) + A263254(n).
a(n) = A263270(A263266(n)).
A263265(a(n), A263259(n)) = n.
(End)

Extended by Emeric Deutsch, Jan 26 2009

Name edited by Antti Karttunen, Sep 23 2015

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)))))

A262509 Numbers n such that there is no other number k for which A155043(k) = A155043(n).

Original entry on oeis.org

0, 119143, 119147, 119163, 119225, 119227, 119921, 119923, 120081, 120095, 120097, 120101, 120281, 120293, 120349, 120399, 120707, 120747, 120891, 120895, 120903, 120917, 120919, 121443, 121551, 121823, 122079, 122261, 122263, 122273, 122277, 122813, 122961, 123205, 123213, 123223, 123237, 123257, 123765, 24660543, 24660549, 24662311, 24662329, 24663759, 24664997, 24665023, 24665351

Offset: 0

Antti Karttunen, Sep 25 2015

nonn

Starting offset is zero, because a(0) = 0 is a special case in this sequence.
Numbers where A155043 takes a unique value. (Those values are given by A262508.)
Numbers n such that there does not exist any other number h from which one could reach zero in exactly the same number of steps as from n by repeatedly applying the map where k is replaced by k - A000005(k) = A049820(k). Thus in the tree where zero is the root and parent-child relation is given by A049820(child) = parent, all the numbers > n+t (where t is a small value depending on n) have n as their common ancestor. As it is guaranteed that there is at least one infinite path in such a tree, any n in this sequence can be neither a leaf nor any other vertex in a finite side-tree, as then at least one node in the infinite part would have the same distance to the root, thus it must be that n itself is in the infinite part and thus has an infinite number of descendant vertices. Also, for the same reason, the tree cannot branch to two infinite parts from any ancestor of n (which are nodes nearer to the root of the tree, zero).
From the above it follows that if this sequence is infinite, then A259934 is guaranteed to be the only infinite sequence which starts with a(0)=0 and satisfies the condition A049820(a(k)) = a(k-1) for all k>=1, where A049820(n) = n-d(n) and d(n) is the number of divisors of n (A000005).  This is a sufficient condition for the uniqueness of A259934, although not necessary. See e.g. A179016 which is the unique infinite solution to a similar problem, even though A086876 contains no ones after its two initial terms.
Is it possible for any even terms to occur after zero? If not, then apart from zero, this would be a subsequence of A262517.

Antti Karttunen, Table of n, a(n) for n = 0..68

Cf. A000005, A049820, A070319, A155043, A262508.
Subsequence of A259934, A261089, A262503 and A262513.
Cf. A262510 (gives the parent nodes for the terms a(1) onward), A262514, A262516, A262517.
Cf. also A086876, A179016.

\\ Compute A262508 and A262509 at the same time:
allocatemem((2^31)+(2^30));
\\ The limits used are quite ad hoc. Beware of the horizon-effect if you change these.
\\ As a post-check, test that A262509(n) = A259934(A262508(n)) for all the terms produced by this program.
uplim1 = 43243200 + 672; \\ = A002182(54) + A002183(54).
uplim2 = 36756720; \\ = A002182(53).
uplim3 = 10810800; \\
v155043 = vector(uplim1);
v262503 = vector(uplim3);
v262507 = vector(uplim3);
v155043[1] = 1; v155043[2] = 1;
for(i=3, uplim1, v155043[i] = 1 + v155043[i-numdiv(i)]);
A155043 = n -> if(!n,n,v155043[n]);
for(i=1, uplim1, v262503[v155043[i]] = i; v262507[v155043[i]]++; if(!(i%1048576),print1(i,", ")));
A262503 = n -> if(!n,n,v262503[n]);
A262507 = n -> if(!n,1,v262507[n]);
k=0; for(n=0, uplim3, if((1==A262507(n)) && (A262503(n) <= uplim2), write("b262508.txt", k, " ", n); write("b262509.txt", k, " ", A262503(n)); k++));

(define (A262509 n) (A261089 (A262508 n)))

a(n) = A261089(A262508(n)) = A262503(A262508(n)) = A259934(A262508(n)).

A262507 a(n) = number of times n occurs in A155043.

Original entry on oeis.org

1, 2, 3, 5, 4, 5, 6, 4, 4, 4, 8, 4, 4, 5, 8, 7, 7, 7, 7, 8, 5, 6, 6, 8, 10, 7, 8, 7, 7, 5, 5, 6, 6, 8, 6, 7, 7, 7, 4, 5, 5, 6, 6, 8, 7, 5, 5, 6, 7, 11, 5, 4, 5, 8, 12, 7, 9, 5, 8, 8, 9, 10, 14, 11, 12, 11, 9, 11, 13, 12, 12, 11, 11, 11, 12, 12, 10, 9, 9, 9, 8, 6, 10, 9, 10, 8, 7, 7, 8, 11, 10, 10, 12, 9, 7, 6, 5, 5, 5, 5, 4, 7, 8, 6, 7, 9, 7, 5, 11, 13, 13, 8, 10, 12, 13, 10, 12, 16, 9, 8, 12

Offset: 0

Antti Karttunen, Sep 25 2015

nonn

Records are: 1, 2, 3, 5, 6, 8, 10, 11, 12, 14, 16, 17, 19, 21, 22, 24, 25, 26, 27, 31, 35, 39, 44, ... and they occur at positions: 0, 1, 2, 3, 6, 10, 24, 49, 54, 62, 117, 236, 445, 484, 892, 893, 1022, 1784, 1911, 1912, 1913, 20600, 50822, ...

a(n) gives the length of each row of irregular table A263265.

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

Cf. A000005, A060990, A155043, A262502, A262505, A263265, A263270, A263279, A263280.
Cf. A261089, A262503.
Cf. A262508 (positions of ones).
Cf. A263260 (partial sums).

allocatemem(123456789);
uplim = 2162160; \\ = A002182(41).
v155043 = vector(uplim);
v155043[1] = 1; v155043[2] = 1;
for(i=3, uplim, v155043[i] = 1 + v155043[i-numdiv(i)]);
uplim2 = 110880; \\ = A002182(30).
v262507 = vector(uplim2);
for(i=1, uplim, if(v155043[i] <= uplim2, v262507[v155043[i]]++));
A262507 = n -> if(!n,1,v262507[n]);
for(n=0, uplim2, write("b262507.txt", n, " ", A262507(n)));

(define (A262507 n) (add (lambda (k) (if (= (A155043 k) n) 1 0)) n (A262502 (+ 2 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)))))))

a(n) = Sum_{k=n..A262502(2+n)} [A155043(k) == n]. (Here [...] denotes the Iverson bracket, resulting 1 when A155043(k) is n and 0 otherwise.)
Other identities. For all n >= 0:
a(n) = A263279(n) + A263280(n).

A262510 Parent nodes of nonzero terms of A262509: a(n) = A049820(A262509(n)).

Original entry on oeis.org

119139, 119143, 119147, 119213, 119225, 119919, 119921, 120073, 120091, 120095, 120097, 120277, 120291, 120347, 120391, 120703, 120739, 120883, 120891, 120895, 120915, 120917, 121435, 121543, 121819, 122075, 122257, 122261, 122271, 122273, 122809, 122953, 123197, 123205, 123219, 123231, 123251, 123749, 24660527, 24660543, 24662309, 24662321, 24663755, 24664989, 24665019, 24665347, 24665929, 24665977, 24669139, 24669833

Offset: 1

Antti Karttunen, Sep 25 2015

nonn

These numbers are one step nearer (than those of A262509) to the root (zero) of the tree where the parent-child relation is given by A049820(child) = parent. Like the terms of A262509, they are also vertices in the infinite trunk of that tree. Cf. A259934.

Antti Karttunen, Table of n, a(n) for n = 1..68

Cf. A049820, A262508, A262509.
Subsequence of A259934 and A262511.
Also a subsequence of A262517 (provided all terms are odd).

(define (A262510 n) (A049820 (A262509 n)))

a(n) = A049820(A262509(n)).

a(n) = A259934(A262508(n)-1).

A262505 Distance between the positions of the first and the last occurrence of n in A155043: a(n) = A262503(n) - A261089(n).

Original entry on oeis.org

0, 1, 3, 7, 11, 9, 13, 15, 19, 21, 31, 41, 49, 59, 67, 77, 81, 87, 55, 57, 65, 59, 63, 67, 77, 79, 89, 37, 27, 33, 37, 47, 55, 65, 67, 71, 85, 99, 65, 67, 77, 89, 93, 103, 95, 105, 109, 117, 121, 131, 135, 115, 117, 119, 133, 133, 153, 133, 143, 141, 147, 159, 175, 181, 201, 205, 223, 239, 245, 249, 259, 275, 269, 271, 273, 271, 289, 303, 77

Offset: 0

Antti Karttunen, Sep 25 2015

nonn

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

Cf. A155043, A259934, A261089, A261103, A262503, A262506.
Cf. A262508 (positions of zeros).
Cf. A262507 (the number of occurrences of n in A155043).

(define (A262505 n) (- (A262503 n) (A261089 n)))

a(n) = A262503(n) - A261089(n).

a(n) = A261103(n) + A262506(n).

A262892 Indices of nonbranching nodes in the infinite trunk (A259934) of the tree generated by edge-relation A049820(child) = parent.

Original entry on oeis.org

1, 3, 4, 6, 8, 10, 15, 16, 18, 21, 30, 31, 32, 36, 37, 39, 44, 45, 49, 50, 51, 53, 54, 55, 58, 60, 62, 65, 68, 71, 72, 73, 74, 75, 76, 80, 83, 84, 90, 91, 93, 96, 109, 112, 117, 122, 123, 124, 126, 127, 131, 134, 135, 137, 141, 142, 144, 145, 147, 149, 152, 154, 155, 161, 162, 165, 166, 170, 178, 187, 189, 190, 191, 193, 195, 199, 201, 203, 205, 211, 212, 213, 219, 223, 225

Offset: 1

Antti Karttunen, Oct 04 2015

nonn

Positions of zeros in A262890. Positions of ones in A262891.

Sequence b(n) = A262508(n)-1 = 9235, 9236, 9237, 9246, 9247, 9329, 9330, 9352, 9355, ..., is a subsequence.

Antti Karttunen, Table of n, a(n) for n = 1..4085

Cf. A049820, A259934, A262890, A262891, A262896.
Cf. A262897 (nonbranching nodes themselves).
Cf. A262508.

Other identities. For all n >= 1:

A262897(n) = A259934(a(n)) = A262896(a(n)).

A262909 a(n) = greatest k such that A155043(k+A262509(n)) < A155043(A262509(n)).

Original entry on oeis.org

5197, 5193, 5177, 5115, 5113, 4419, 4417, 4259, 4245, 4243, 4239, 4059, 4047, 3991, 3941, 3633, 3593, 3449, 3445, 3437, 3423, 3421, 2897, 2789, 2517, 2261, 2079, 2077, 2067, 2063, 1527, 1379, 1135, 1127, 1117, 1103, 1083, 575, 23457, 23451, 21689, 21671, 20241, 19003, 18977, 18649, 18063, 18019, 14853, 14159, 13659, 12707, 11681, 10993, 10991, 10297, 10281, 9151, 9149, 9145, 9111, 8897, 8535, 8147, 6835, 6813, 5539, 5537

Offset: 1

Antti Karttunen, Oct 09 2015

nonn

a(n) = largest k such that A155043(k+A262509(n)) < A262508(n).
There might occur also negative terms, but no zeros.
For all terms a(n) > 0, a(n)+A262509(n) = A263081(n) is by necessity one of the leaves (A045765) in the tree generated by edge-relation A049820(child) = parent. See also comments in A262908.

Cf. A000005, A049820, A045765, A262508, A262509, A262908, A263078, A263081.

a(n) = A263078(A262509(n)).
a(n) = A263081(n) - A262509(n).
Other identities. For all n >= 1:
a(n) >= A262908(n).

A264988 The left edge of A263267.

Original entry on oeis.org

0, 1, 3, 5, 7, 13, 17, 19, 23, 27, 29, 31, 35, 37, 41, 43, 51, 53, 57, 59, 61, 65, 67, 71, 73, 77, 79, 143, 149, 151, 155, 157, 161, 163, 173, 177, 179, 181, 185, 191, 193, 199, 203, 209, 211, 215, 219, 223, 231, 233, 237, 239, 241, 249, 251, 263, 267, 269, 271, 277, 285, 291, 293, 299, 303, 315, 317, 321, 327, 333, 335, 337, 341, 347, 349, 357, 359, 369, 517, 531, 535, 523, 527

Offset: 0

Antti Karttunen, Nov 29 2015

nonn

The first point where the sequence is nonmonotonic is the dip from a(80) = 535 to a(81) = 523.

Antti Karttunen, Table of n, a(n) for n = 0..10001

Cf. A263260, A262508, A262509.
The left edge of irregular table A263267.
Cf. A263269 (the other edge).
Differs from A261089 for the first time at n=69, where a(69) = 333, while A261089(69) = 331.

(define (A264988 n) (if (zero? n) n (A263267 (A263260 (- n 1)))))

a(0) = 0; for n >= 1, a(n) = A263267(A263260(n-1)).
Other identities. For all n >= 0:
A155043(a(n)) = n.
a(A262508(n)) = A262509(n) = A263269(A262508(n)). [In case A262508 and A262509 are infinite sequences.]

cp's OEIS Frontend

A263081 a(n) = largest k for which A155043(k) < A262508(n); a(n) = A262509(n) + A262909(n).

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A155043 a(0)=0; for n >= 1, a(n) = 1 + a(n-d(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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A262509 Numbers n such that there is no other number k for which A155043(k) = A155043(n).

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A262507 a(n) = number of times n occurs in A155043.

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A262510 Parent nodes of nonzero terms of A262509: a(n) = A049820(A262509(n)).

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A262505 Distance between the positions of the first and the last occurrence of n in A155043: a(n) = A262503(n) - A261089(n).

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