A261085 -id:A261085 - cp's OEIS frontend

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).

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

A261089 a(n) = least k such that A155043(k) = n; positions of records in A155043.

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, 331, 335, 337, 341, 347, 349, 357, 359, 369, 515

Offset: 0

Antti Karttunen, Sep 23 2015

nonn

Note that there are even terms besides 0, and they all seem to be squares: a(915) = 7744 (= 88^2), a(41844) = 611524 (= 782^2), a(58264) = 872356 (= 934^2), a(66936) = 1020100 (= 1010^2), a(95309) = 1503076 (= 1226^2), a(105456) = 1653796 (= 1286^2), ...

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

Cf. A000005, A155043, A049820, A259934, A261103.
Cf. A262503 (the last occurrence of n in A155043).
Cf. A262505 (difference between the last and the first occurrence).
Cf. A262507 (the number of occurrences of n in A155043).
Cf also A261085, A261088.

import Data.List (elemIndex); import Data.Maybe (fromJust)
a261089 = fromJust . (`elemIndex` a155043_list)
-- Reinhard Zumkeller, Nov 27 2015

lim = 80; a[0] = 0; a[n_] := a[n] = 1 + a[n - DivisorSigma[0, n]]; t = Table[a@ n, {n, 0, 12 lim}]; Table[First@ Flatten@ Position[t, n] - 1, {n, 0, lim}] (* Michael De Vlieger, Sep 29 2015 *)

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)]);
A155043 = n -> if(!n,n,v155043[n]);
n=0; k=0; while(k <= 10000, if(A155043(n)==k, write("b261089.txt", k, " ", n); k++); n++;);

;; With Antti Karttunen's IntSeq-library, two variants.
(definec (A261089 n) (let loop ((k 0)) (if (= n (A155043 k)) k (loop (+ 1 k)))))
(define A261089 (RECORD-POS 0 0 A155043))

Other identities. For all n >= 0:

A155043(a(n)) = n.

A261088 Number of steps needed to reach zero when starting from k = n^2 and repeatedly applying the map that replaces k with k - d(k), where d(k) is the number of divisors of k (A000005).

Original entry on oeis.org

0, 1, 2, 3, 5, 6, 10, 10, 19, 15, 19, 21, 24, 28, 39, 33, 53, 44, 49, 53, 60, 61, 69, 72, 79, 82, 92, 93, 117, 108, 115, 115, 140, 121, 174, 146, 205, 155, 233, 217, 267, 192, 295, 209, 225, 222, 238, 249, 267, 270, 299, 290, 336, 313, 373, 328, 411, 347, 451, 380, 486, 400, 534, 422, 447, 441, 460, 460, 511, 479, 496, 504, 545, 529, 602, 553, 579, 577, 626, 612, 681, 632, 747, 665, 796, 695

Offset: 0

Antti Karttunen, Sep 23 2015

nonn

Antti Karttunen, Table of n, a(n) for n = 0..5792

Cf. A000005, A000290, A155043, A049820, A259934, A261085.

Cf. also A260732, A261222.

f[n_]:=Length[NestWhileList[#-DivisorSigma[0,#]&,n^2,#!= 0&]]-1;f/@Range[0,85] (* Ivan N. Ianakiev, Sep 25 2015 *)

allocatemem((2^31)+(2^30));
uplim = 2^25;
v155043 = vector(uplim);
v155043[1] = 1; v155043[2] = 1;
for(i=3, uplim, v155043[i] = 1 + v155043[i-numdiv(i)]; if(!(i%65536),print1(i,", ")););
A155043 = n -> if(!n,n,v155043[n]);
A261088 = n -> A155043(n^2);
for(n=0, 5792, write("b261088.txt", n, " ", A261088(n)));

(define (A261088 n) (A155043 (A000290 n)))

a(n) = A155043(A000290(n)) = A155043(n^2).

A261087 Primes p for which A155043(p) < A155043(prevprime(p)), where A155043 gives the number of steps needed to reach zero when repeatedly applying the map that replaces k with k - A000005(k).

Original entry on oeis.org

83, 127, 227, 281, 307, 367, 443, 541, 631, 677, 853, 863, 967, 1091, 1217, 1427, 1451, 1487, 1523, 1667, 1787, 1861, 1997, 2027, 2153, 2207, 2297, 2339, 2411, 2477, 2543, 2693, 2711, 2837, 2909, 2963, 3089, 3251, 3313, 3323, 3467, 3533, 3593, 3677, 3719, 3797, 3863, 3917, 3989, 4007, 4019, 4091, 4259, 4447, 4493, 4643, 4657, 4783, 4787, 4799, 4877, 4937, 5087, 5119, 5441

Offset: 1

Antti Karttunen, Sep 23 2015

nonn

			A155043(83) = 16 although A155043(79) = 26, thus 83 is included in this sequence.

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

Cf. A000005, A000040, A155043, A261085, A261086.

(define (A261087 n) (A000040 (A261086 n)))

a(n) = A000040(A261086(n)).

A330877 Number of steps needed to reach zero or a cycle when starting from k = n and repeatedly applying the map that replaces k by k - d(k) if k is even, by k + d(k) if k is odd, where d(k) is the number of divisors of k (A000005).

Original entry on oeis.org

0, 2, 1, 7, 3, 6, 2, 5, 4, 4, 3, 12, 3, 11, 4, 10, 13, 10, 4, 9, 5, 8, 5, 8, 14, 7, 6, 32, 6, 32, 6, 31, 7, 30, 7, 29, 33, 29, 8, 28, 8, 28, 8, 27, 9, 26, 9, 12, 9, 11, 10, 25, 10, 25, 10, 24, 10, 23, 11, 23, 10, 22, 12, 21, 24, 21, 12, 21, 13

Offset: 0

Ctibor O. Zizka, Apr 29 2020

nonn

First cycle we see for n = 83. The length of the cycle is 38 steps. To reach a cycle means the time to first step into the loop.

			n = 1, mapping steps are 1 + 1 = 2, 2 - 2 = 0, a(1) = 2;
n = 2, mapping steps are 2 - 2 = 0, a(2) = 1;
n = 3, mapping steps are 3 + 2 = 5, 5 + 2 = 7, 7 + 2 = 9, 9 + 3 = 12, 12 - 6 = 6, 6 - 4 = 2, 2 - 2 = 0, a(3) = 7;
n = 4, mapping steps are 4 - 3 = 1, 1 + 1 = 2, 2 - 2 = 0, a(4) = 3;
n = 5, mapping steps are 5 + 2 = 7, 7 + 2 = 9, 9 + 3 = 12, 12 - 6 = 6, 6 - 4 = 2, 2 - 2 = 0, a(5) = 6.

Cf. A000005, A155043, A261085, A261088.

Cf. A049820, A062249.

cp's OEIS Frontend

A261086 Numbers n > 1 for which A261085(n) < A261085(n-1).

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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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A261089 a(n) = least k such that A155043(k) = n; positions of records in A155043.

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A261088 Number of steps needed to reach zero when starting from k = n^2 and repeatedly applying the map that replaces k with k - d(k), where d(k) is the number of divisors of k (A000005).

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Mathematica

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A261087 Primes p for which A155043(p) < A155043(prevprime(p)), where A155043 gives the number of steps needed to reach zero when repeatedly applying the map that replaces k with k - A000005(k).

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A330877 Number of steps needed to reach zero or a cycle when starting from k = n and repeatedly applying the map that replaces k by k - d(k) if k is even, by k + d(k) if k is odd, where d(k) is the number of divisors of k (A000005).

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