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

Author

Ctibor O. Zizka, Jan 19 2009

Keywords

• nonn
• look

Comments

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)

Links

• 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.

Crossrefs

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).

Programs

Haskell

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

Maple

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

Mathematica

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 *)

PARI

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

Python

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

Scheme

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

Formula

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)

Extensions

Extended by Emeric Deutsch, Jan 26 2009

Name edited by Antti Karttunen, Sep 23 2015