A263254 -id:A263254 - 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)

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

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.

A263257 a(n) = least number with distance n to the infinite trunk (A259934) of the tree defined by edge-relation A049820(child) = parent.

Original entry on oeis.org

0, 1, 3, 5, 7, 19, 23, 27, 29, 31, 35, 37, 41, 43, 51, 53, 57, 59, 61, 65, 67, 71, 73, 77, 79, 223, 231, 233, 237, 239, 241, 249, 251, 263, 267, 269, 271, 277, 285, 291, 293, 299, 303, 315, 317, 321, 327, 331, 335, 337, 341, 347, 349, 357, 359, 369, 857, 859, 1077, 1081, 1087, 1095, 1097, 1101

Offset: 0

Antti Karttunen, Nov 07 2015

nonn

a(n) = least number k for which A263254(k) = n.

Also positions of records in A263254, thus the sequence is strictly increasing.

Antti Karttunen, Table of n, a(n) for n = 0..1835

Cf. A049820, A259934, A263254, A263258.

First column of array A263255.

A263274 One-based column index to array A263255.

Original entry on oeis.org

1, 1, 2, 1, 2, 1, 3, 1, 2, 2, 3, 3, 4, 3, 4, 4, 5, 2, 5, 1, 6, 2, 6, 1, 3, 4, 5, 1, 6, 1, 7, 1, 5, 2, 8, 1, 2, 1, 7, 2, 7, 1, 9, 1, 6, 2, 10, 2, 7, 8, 8, 1, 9, 1, 11, 2, 9, 1, 12, 1, 10, 1, 13, 2, 3, 1, 10, 1, 11, 2, 14, 1, 4, 1, 12, 2, 13, 1, 15, 1, 14, 8, 15, 11, 3, 12, 9, 5, 10, 4, 16, 5, 13, 2

Offset: 0

Antti Karttunen, Nov 07 2015

nonn

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

Cf. A259934, A263254, A263255.

Cf. also A263275.

(define (A263274 n) (let ((rowf (rowfun-for-A263255 (A263254 n)))) (let loop ((i 1)) (if (= n (rowf i)) i (loop (+ 1 i))))))
;; rowfun-for-A263255 given in A263255.

Other identities. For all n >= 0:

a(A259934(n)) = n+1.

A263275 One-based row-index to array A263255.

Original entry on oeis.org

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

Offset: 0

Antti Karttunen, Nov 07 2015

nonn

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

One more than A263254.
Cf. A049820, A262693, A263255.
Cf. also A263274.

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

a(n) = 1 + A263254(n).

A263262 Row 2 of A263255.

Original entry on oeis.org

3, 4, 11, 14, 32, 44, 48, 81, 86, 88, 116, 130, 135, 175, 184, 186, 192, 200, 244, 246, 252, 262, 266, 284, 286, 296, 320, 326, 336, 365, 370, 376, 378, 380, 384, 404, 406, 428, 430, 460, 482, 490, 524, 529, 534, 608, 624, 642, 652, 654, 656, 682, 690, 764, 768, 772, 786, 794, 796, 802, 806, 810, 884, 888

Offset: 1

Antti Karttunen, Nov 07 2015

nonn

Numbers n for which A263254(n) = 2. Numbers that are two edges removed from the infinite trunk (A259934) of the tree defined by edge-relation A049820(child) = parent.

Antti Karttunen, Table of n, a(n) for n = 1..9096

Cf. A049820, A259934, A263254, A263255.

A263261 Row 1 of A263255.

Original entry on oeis.org

1, 9, 10, 25, 26, 28, 38, 49, 52, 66, 68, 74, 76, 80, 82, 110, 112, 122, 124, 126, 127, 131, 133, 156, 168, 169, 170, 172, 176, 178, 180, 188, 218, 226, 234, 238, 258, 278, 288, 290, 300, 304, 306, 316, 318, 322, 334, 361, 362, 368, 398, 414, 422, 448, 452, 470, 478, 496, 514, 518, 526, 586

Offset: 1

Antti Karttunen, Nov 07 2015

nonn

Numbers n for which A263254(n) = 1. Numbers that are one edge removed from the infinite trunk (A259934) of the tree defined by edge-relation A049820(child) = parent.

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

Cf. A049820, A259934, A263254, A263255.

Cf. also A262901.

A263263 Row 3 of A263255.

Original entry on oeis.org

5, 8, 13, 15, 16, 20, 40, 50, 56, 60, 83, 85, 92, 134, 137, 138, 198, 204, 216, 268, 272, 289, 292, 308, 312, 328, 332, 344, 348, 360, 367, 382, 392, 396, 412, 416, 436, 438, 498, 533, 538, 620, 632, 636, 640, 664, 666, 694, 700, 704, 720, 776, 792, 814, 816, 828, 922, 936, 940, 956, 965, 966, 969, 1032

Offset: 1

Antti Karttunen, Nov 07 2015

nonn

Numbers n for which A263254(n) = 3. Numbers that are three edges removed from the infinite trunk (A259934) of the tree defined by edge-relation A049820(child) = parent.

Antti Karttunen, Table of n, a(n) for n = 1..2047

Cf. A049820, A259934, A263254, A263255.

A263264 Row 4 of A263255.

Original entry on oeis.org

7, 17, 24, 72, 87, 98, 139, 141, 142, 150, 202, 208, 225, 228, 297, 338, 340, 371, 372, 375, 386, 388, 408, 420, 424, 432, 502, 506, 537, 539, 542, 550, 625, 644, 660, 680, 684, 698, 712, 732, 804, 818, 822, 824, 926, 948, 952, 967, 968, 971, 973, 976, 990, 1056, 1062, 1120, 1122

Offset: 1

Antti Karttunen, Nov 07 2015

nonn

Numbers n for which A263254(n) = 4. Numbers that are four edges removed from the infinite trunk (A259934) of the tree defined by edge-relation A049820(child) = parent.

Antti Karttunen, Table of n, a(n) for n = 1..1023

Cf. A049820, A259934, A263254, A263255.

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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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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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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A263257 a(n) = least number with distance n to the infinite trunk (A259934) of the tree defined by edge-relation A049820(child) = parent.

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A263274 One-based column index to array A263255.

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A263275 One-based row-index to array A263255.

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A263262 Row 2 of A263255.

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A263261 Row 1 of A263255.

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A263263 Row 3 of A263255.

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