A045765 -id:A045765 - cp's OEIS frontend

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.

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))).

A262896 If n is in A262892, a(n) = A259934(n), otherwise the largest term in A045765 from which A259934(n) can be reached by iterating A049820, without visiting any other (larger) term of A259934.

Original entry on oeis.org

8, 2, 79, 12, 18, 40, 30, 140, 42, 52, 54, 66, 68, 123, 98, 90, 94, 116, 106, 126, 164, 121, 369, 133, 156, 168, 180, 184, 280, 229, 190, 194, 210, 218, 252, 246, 236, 242, 272, 254, 312, 324, 300, 364, 298, 302, 372, 356, 334, 342, 346, 354, 439, 366, 374, 390, 672, 414, 410, 438, 426, 460, 442, 452, 470, 466, 564, 496, 494, 524, 627, 530, 546, 558, 562, 566, 574, 592, 859, 660, 606, 642, 708, 650

Offset: 0

Antti Karttunen, Oct 06 2015

nonn

a(n) is the largest leaf-node among the finite subtrees branching from node n in the infinite trunk (A259934) of the tree generated by edge-relation A049820(child) = parent, and A259934(n) itself if it is one of the nonbranching nodes (A262897).

Note that without (so far undetected) regularity in A262509, there is no a priori upper bound for the value of a(n), and for some n this might not even be finite, if it happens that contrary to its conjectured nature, A259934 is not the unique infinite component, but just the lexicographically earliest instance of multiple infinite branches of the tree. In that case we might consider this sequence to be well-defined only up to the least such node branching to multiple infinite components, or alternatively, we might mark the nonfinite values at those points with -1.

Antti Karttunen, Table of n, a(n) for n = 0..10000
Max Alekseyev & Antti Karttunen, Standalone C++-program for computing this sequence

Cf. A000005, A049820, A082284, A259934, A262509, A262522, A262679, A262686, A262890, A262892, A262897, A262904, A263081.

(define (A262896 n) (let ((t (A259934 n))) (let loop ((m t) (k (A262686 t))) (cond ((<= k t) m) ((= t (A049820 k)) (loop (max m (A262522 k)) (- k 1))) (else (loop m (- k 1)))))))

a(n) = max(A259934(n), Max_{k = A082284(A259934(n)) .. A262686(A259934(n))} [A049820(k) = A259934(n)] * A262522(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:
A262904(a(n)) = n. [A262904 works as a left inverse for this sequence.]
A259934(n) = A262679(a(n)).
For all n >= 1:
a(A262892(n)) = A259934(A262892(n)) = A262897(n).

A262898 Square array A(row,col) read by antidiagonals: A(1,col) = A045765(col); for row > 1, if A(row-1,col) = 0 then A(row,col) = 0, otherwise A(row,col) = A049820(A(row-1,col)).

Original entry on oeis.org

7, 8, 5, 13, 4, 3, 19, 11, 1, 1, 20, 17, 9, 0, 0, 24, 14, 15, 6, 0, 0, 25, 16, 10, 11, 2, 0, 0, 28, 22, 11, 6, 9, 0, 0, 0, 33, 22, 18, 9, 2, 6, 0, 0, 0, 36, 29, 18, 12, 6, 0, 2, 0, 0, 0, 37, 27, 27, 12, 6, 2, 0, 0, 0, 0, 0, 40, 35, 23, 23, 6, 2, 0, 0, 0, 0, 0, 0, 43, 32, 31, 21, 21, 2, 0, 0, 0, 0, 0, 0, 0, 49, 41, 26, 29, 17, 17, 0, 0, 0, 0, 0, 0, 0, 0, 50, 46, 39, 22, 27, 15, 15, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Antti Karttunen, Oct 06 2015

The array is read by downwards antidiagonals: A(1,1), A(1,2), A(2,1), A(1,3), A(2,2), A(3,1), etc.
Column n gives the trajectory of iterates of A049820, when starting from A045765(n), thus stepping through successive parent-nodes when starting from the n-th leaf in the tree generated by edge-relation A049820(child) = parent, until finally reaching the fixed point 0, which is the root of the whole tree.
A portion of the hanging tail of each column (upward from the first encountered zero) converges towards A259934, although not in monotone fashion.

			The top left corner of the array:
7, 8, 13, 19, 20, 24, 25, 28, 33, 36, 37, 40, 43, 49, 50, 52, 55, 56
5, 4, 11, 17, 14, 16, 22, 22, 29, 27, 35, 32, 41, 46, 44, 46, 51, 48
3, 1,  9, 15, 10, 11, 18, 18, 27, 23, 31, 26, 39, 42, 38, 42, 47, 38
1, 0,  6, 11,  6,  9, 12, 12, 23, 21, 29, 22, 35, 34, 34, 34, 45, 34
0, 0,  2,  9,  2,  6,  6,  6, 21, 17, 27, 18, 31, 30, 30, 30, 39, 30
0, 0,  0,  6,  0,  2,  2,  2, 17, 15, 23, 12, 29, 22, 22, 22, 35, 22
0, 0,  0,  2,  0,  0,  0,  0, 15, 11, 21,  6, 27, 18, 18, 18, 31, 18
0, 0,  0,  0,  0,  0,  0,  0, 11,  9, 17,  2, 23, 12, 12, 12, 29, 12
0, 0,  0,  0,  0,  0,  0,  0,  9,  6, 15,  0, 21,  6,  6,  6, 27,  6
0, 0,  0,  0,  0,  0,  0,  0,  6,  2, 11,  0, 17,  2,  2,  2, 23,  2
0, 0,  0,  0,  0,  0,  0,  0,  2,  0,  9,  0, 15,  0,  0,  0, 21,  0
0, 0,  0,  0,  0,  0,  0,  0,  0,  0,  6,  0, 11,  0,  0,  0, 17,  0
0, 0,  0,  0,  0,  0,  0,  0,  0,  0,  2,  0,  9,  0,  0,  0, 15,  0
0, 0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  6,  0,  0,  0, 11,  0
0, 0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  2,  0,  0,  0,  9,  0
0, 0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  6,  0
...

Antti Karttunen, Table of n, a(n) for n = 1..5050; the first 100 antidiagonals of the array

Transpose: A262899.
Cf. A045765 (row 1), A262902 (row 2).
Cf. A049820, A259934.
Cf. also A257264.

(define (A262898 n) (A262898bi (A002260 n) (A004736 n)))
(define (A262898bi row col) (if (= 1 row) (A045765 col) (if (zero? (A262898bi (- row 1) col)) 0 (A049820 (A262898bi (- row 1) col)))))

A(1,col) = A045765(col), and for row > 1, if A(row-1,col) = 0 then A(row,col) = 0, otherwise A(row,col) = A049820(A(row-1,col)).

A262902 a(n) = A049820(A045765(n)); parent-nodes of the leaves of the tree generated by edge-relation A049820(child) = parent.

Original entry on oeis.org

5, 4, 11, 17, 14, 16, 22, 22, 29, 27, 35, 32, 41, 46, 44, 46, 51, 48, 57, 57, 58, 65, 62, 70, 69, 77, 81, 80, 92, 91, 101, 96, 107, 102, 111, 110, 111, 119, 118, 114, 129, 120, 129, 130, 128, 128, 139, 141, 138, 147, 144, 155, 148, 161, 158, 165, 152, 162, 166, 169, 162, 176, 181, 191, 187, 199, 192, 201, 214, 215, 222, 216, 227, 224, 231, 239, 238, 238, 249, 234, 249, 247, 255, 255

Offset: 1

Antti Karttunen, Oct 06 2015

nonn

The sequence is computed for each leaf of the tree (A045765), ordered by their magnitude, and it contains duplicates.

Antti Karttunen, Table of n, a(n) for n = 1..10000

Row 2 of A262898.
Cf. A262901 (same sequence sorted into ascending order, with duplicates removed).
Cf. A049820, A045765.
Cf. also A257507.

(define (A262902 n) (A049820 (A045765 n)))

a(n) = A049820(A045765(n)).

A263093 Numbers whose squares are in A045765.

Original entry on oeis.org

5, 6, 7, 8, 10, 14, 16, 18, 20, 22, 26, 27, 28, 34, 35, 37, 46, 47, 50, 54, 56, 58, 59, 60, 62, 67, 73, 78, 82, 85, 89, 90, 94, 95, 98, 100, 103, 104, 106, 110, 114, 116, 118, 122, 124, 125, 126, 127, 128, 130, 135, 140, 141, 142, 148, 150, 155, 158, 161, 164, 170, 172, 174, 177, 178, 182, 184, 188, 190, 199, 202, 205, 207

Offset: 1

Antti Karttunen, Oct 11 2015

nonn

Numbers n such that there is no such k for which k - d(k) = n^2, where d(k) is the number of divisors of k (A000005).

Numbers n for which A060990(n^2) = A263087(n) = 0.

Antti Karttunen, Table of n, a(n) for n = 1..12443
A. Karttunen, Ratio a(n)/A263092(n) drawn with the help of OEIS Plot2-utility

Cf. A000005, A000196, A045765, A049820, A060990, A263098.
Complement: A263092.
Positions of zeros in A263087 and positions of ones in A263088.
Cf. A263095 (the squares of these numbers).

\\ Compute A263093 and A263095 at the same time:
A060990(n) = { my(k = n + 1440, s=0); while(k > n, if(((k-numdiv(k)) == n),s++); k--;); s}; \\ Hard limit 1440 is good for at least up to A002182(67) = 1102701600 as A002183(67) = 1440.
n = 1; k = 0; while((n^2)<1102701600, if((0 == A060990(n*n)), k++; write("b263093.txt", k, " ", n); write("b263095.txt", k, " ", (n*n)); ); n++; if(!(n%8192),print1(n,",k=", k, ", ")); );

;; With Antti Karttunen's IntSeq-library.
(define A263093 (MATCHING-POS 1 1 (lambda (n) (zero? (A060990 (* n n))))))
(define A263093 (ZERO-POS 1 0 A263087))

a(n) = A000196(A263095(n)).

A263095 Squares in A045765; numbers n^2 such that there is no such k for which k - d(k) = n^2, where d(k) is the number of divisors of k (A000005).

Original entry on oeis.org

25, 36, 49, 64, 100, 196, 256, 324, 400, 484, 676, 729, 784, 1156, 1225, 1369, 2116, 2209, 2500, 2916, 3136, 3364, 3481, 3600, 3844, 4489, 5329, 6084, 6724, 7225, 7921, 8100, 8836, 9025, 9604, 10000, 10609, 10816, 11236, 12100, 12996, 13456, 13924, 14884, 15376, 15625, 15876, 16129, 16384, 16900, 18225, 19600, 19881, 20164, 21904, 22500, 24025, 24964, 25921, 26896

Offset: 1

Antti Karttunen, Oct 10 2015

nonn

Some of the terms are shared with A262687, but none with A262514.

Antti Karttunen, Table of n, a(n) for n = 1..12443

Cf. A000005, A049820, A010052, A060990, A262514, A262687, A263098.
Cf. A263093 (gives the square roots).
Intersection of A000290 and A045765.
Cf. also A263091.

lim = 40000; Take[Select[Complement[Range@ lim, Sort@ DeleteDuplicates@ Table[n - DivisorSigma[0, n], {n, lim}]], IntegerQ@ Sqrt@ # &], 60] (* Michael De Vlieger, Oct 13 2015 *)

PARI
```
\\ See code in A263093.
```

;; With Antti Karttunen's IntSeq-library.
(define A263095 (MATCHING-POS 1 1 (lambda (n) (and (= 1 (A010052 n)) (zero? (A060990 n))))))

A049820 a(n) = n - d(n), where d(n) is the number of divisors of n (A000005).

Original entry on oeis.org

0, 0, 1, 1, 3, 2, 5, 4, 6, 6, 9, 6, 11, 10, 11, 11, 15, 12, 17, 14, 17, 18, 21, 16, 22, 22, 23, 22, 27, 22, 29, 26, 29, 30, 31, 27, 35, 34, 35, 32, 39, 34, 41, 38, 39, 42, 45, 38, 46, 44, 47, 46, 51, 46, 51, 48, 53, 54, 57, 48, 59, 58, 57, 57, 61, 58, 65

Offset: 1

Clark Kimberling

a(n) is the number of non-divisors of n in 1..n. - Jaroslav Krizek, Nov 14 2009
Also equal to the number of partitions p of n such that max(p)-min(p) = 1. The number of partitions of n with max(p)-min(p) <= 1 is n; there is one with k parts for each 1 <= k <= n. max(p)-min(p) = 0 iff k divides n, leaving n-d(n) with a difference of 1. It is easiest to see this by looking at fixed k with increasing n: for k=3, starting with n=3 the partitions are [1,1,1], [2,1,1], [2,2,1], [2,2,2], [3,2,2], etc. - Giovanni Resta, Feb 06 2006 and Franklin T. Adams-Watters, Jan 30 2011
Number of positive numbers in n-th row of array T given by A049816.
Number of proper non-divisors of n. - Omar E. Pol, May 25 2010
a(n+2) is the sum of the n-th antidiagonal of A225145. - Richard R. Forberg, May 02 2013
For n > 2, number of nonzero terms in n-th row of triangle A051778. - Reinhard Zumkeller, Dec 03 2014
Number of partitions of n of the form [j,j,...,j,i] (j > i). Example: a(7)=5 because we have [6,1], [5,2], [4,3], [3,3,1], and [2,2,2,1]. - Emeric Deutsch, Sep 22 2016

			a(7) = 5; the 5 non-divisors of 7 in 1..7 are 2, 3, 4, 5, and 6.
The 5 partitions of 7 with max(p) - min(p) = 1 are [4,3], [3,2,2], [2,2,2,1], [2,2,1,1,1] and [2,1,1,1,1,1]. - _Emeric Deutsch_, Mar 01 2006

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
G. E. Andrews, M. Beck, N. Robbins, Partitions with fixed differences between largest and smallest parts, arXiv preprint arXiv:1406.3374 [math.NT], 2014-2015.

Cf. A000005.
One less than A062968, two less than A059292.
Cf. A161664 (partial sums).
Cf. A060990 (number of solutions to a(x) = n).
Cf. A045765 (numbers not occurring in this sequence).
Cf. A236561 (same sequence sorted into ascending order), A236562 (with also duplicates removed), A236565, A262901 and A262903.
Cf. A262511 (numbers that occur only once).
Cf. A055927 (positions of repeated terms).
Cf. A245388 (positions of squares).
Cf. A155043 (number of steps needed to reach zero when iterating a(n)), A262680 (number of nonzero squares encountered).
Cf. A259934 (an infinite trunk of the tree defined by edge-relation a(child) = parent, conjectured to be unique).
Cf. tables and arrays A047916, A051731, A051778, A173540, A173541.
Cf. also arrays A225145, A262898, A263255 and tables A263265, A263267.
Other related sequences: A006218, A006590, A051953, A070824, A094181, A062249, A067391, A076627, A128508, A131187, A134156, A140826, A161886, A177235, A177236, A227874, A228453, A230653, A230654, A231167, A245197, A253473.

List([1..80],n->n-Tau(n)); # Muniru A Asiru, Sep 28 2018

a049820 n = n - a000005 n  -- Reinhard Zumkeller, Feb 06 2012

A049820 := n->n-numtheory[tau](n):
seq(A049820(n), n=1..100);

Table[n - DivisorSigma[0, n], {n, 100}] (* Wesley Ivan Hurt, Nov 19 2014 *)
Array[(# - DivisorSigma[0, #])&, 70] (* Vincenzo Librandi, Dec 29 2015 *)

PARI
```
a(n)=n-numdiv(n)
```

(define (A049820 n) (- n (A000005 n))) ;; Antti Karttunen, Nov 27 2015

a(n) = Sum_{k=1..n} ceiling(n/k)-floor(n/k). - Benoit Cloitre, May 11 2003
G.f.: Sum_{k>0} x^(2*k+1)/(1-x^k)/(1-x^(k+1)). - Emeric Deutsch, Mar 01 2006
a(n) = A006590(n) - A006218(n) = A161886(n) - A000005(n) - A006218(n) + 1 for n >= 1. - Jaroslav Krizek, Nov 14 2009
a(n) = Sum_{k=1..n} A000007(A051731(n,k)). - Reinhard Zumkeller, Mar 09 2010
a(n) = A076627(n) / A000005(n). - Reinhard Zumkeller, Feb 06 2012
For n >= 2, a(n) = A094181(n) / A051953(n). - Antti Karttunen, Nov 27 2015
a(n) = Sum_{k=1..n} ((n mod k) + (-n mod k))/k. - Wesley Ivan Hurt, Dec 28 2015
G.f.: Sum_{j>=2} (x^(j+1)*(1-x^(j-1))/(1-x^j))/(1-x). - Emeric Deutsch, Sep 22 2016
Dirichlet g.f.: zeta(s-1)- zeta(s)^2. - Ilya Gutkovskiy, Apr 12 2017
a(n) = Sum_{i=1..n-1} sign(i mod n-i). - Wesley Ivan Hurt, Sep 27 2018

Edited by Franklin T. Adams-Watters, Jan 30 2012

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).

Original entry on oeis.org

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)

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)))))

cp's OEIS Frontend

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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A262896 If n is in A262892, a(n) = A259934(n), otherwise the largest term in A045765 from which A259934(n) can be reached by iterating A049820, without visiting any other (larger) term of A259934.

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A262898 Square array A(row,col) read by antidiagonals: A(1,col) = A045765(col); for row > 1, if A(row-1,col) = 0 then A(row,col) = 0, otherwise A(row,col) = A049820(A(row-1,col)).

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A262902 a(n) = A049820(A045765(n)); parent-nodes of the leaves of the tree generated by edge-relation A049820(child) = parent.

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A263093 Numbers whose squares are in A045765.

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A263095 Squares in A045765; numbers n^2 such that there is no such k for which k - d(k) = n^2, where d(k) is the number of divisors of k (A000005).

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A049820 a(n) = n - d(n), where d(n) is the number of divisors of n (A000005).

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