A188163 Smallest m such that A004001(m) = n.

1, 3, 5, 6, 9, 10, 11, 13, 17, 18, 19, 20, 22, 23, 25, 28, 33, 34, 35, 36, 37, 39, 40, 41, 43, 44, 46, 49, 50, 52, 55, 59, 65, 66, 67, 68, 69, 70, 72, 73, 74, 75, 77, 78, 79, 81, 82, 84, 87, 88, 89, 91, 92, 94, 97, 98, 100, 103, 107, 108, 110, 113, 117, 122

Offset: 1

Author

Reinhard Zumkeller, Jun 03 2011

Keywords

• nonn

Comments

How is this related to A088359? - R. J. Mathar, Jan 09 2013

It is not hard to show that a(n) exists for all n, and in particular a(n) < 2^n. - Charles R Greathouse IV, Jan 13 2013

From Antti Karttunen, Jan 10 & 18 2016: (Start)

Positions of records in A004001. After 1 the positions where A004001 increases (by necessity by one).

An answer to the question of R. J. Mathar above: This sequence is equal to A088359 with prepended 1. This follows because at each of its unique values (terms of A088359), A004001 must grow, but it can grow nowhere else. See Kubo and Vakil paper and especially the illustrations of Q and R-trees on pages 229-230 (pages 5 & 6 in PDF) and also in sequence A265332.

Obviously A004001 can obtain unique values only at points which form a subset (A266399) of this sequence.

(End)

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
• T. Kubo and R. Vakil, On Conway's recursive sequence, Discr. Math. 152 (1996), 225-252.
• User "Panurge", Frankl's conjecture and Oeis sequence A188163, Mathoverflow.net, Mar 29 2016.
• Eric Weisstein's World of Mathematics, Hofstadter-Conway $10,000 Sequence.
• Wikipedia, Hofstadter sequence
• Index entries for Hofstadter-type sequences

Crossrefs

Equal to A088359 with prepended 1.

Column 1 of A265901, Row 1 of A265903.

Cf. A051135 (first differences).

Cf. A087686 (complement, apart from the initial 1).

Cf. A004001 (also the least monotonic left inverse of this sequence).

Cf. A093879, A265332.

Cf. A266399 (a subsequence).

Programs

Haskell

import Data.List (elemIndex)
import Data.Maybe (fromJust)
a188163 n = succ $ fromJust $ elemIndex n a004001_list

Magma

h:=[n le 2 select 1 else Self(Self(n-1)) + Self(n - Self(n-1)): n in [1..500]]; // h=A004001
A188163:= function(n)
   for j in [1..2*n+1] do
       if h[j] eq n then return j; end if;
   end for;
end function;
[A188163(n): n in [1..100]]; // G. C. Greubel, May 20 2024

Maple

A188163 := proc(n)
    for a from 1 do
        if A004001(a) = n then
            return a;
        end if;
    end do:
end proc: # R. J. Mathar, May 15 2013

Mathematica

h[1] = 1; h[2] = 1; h[n_] := h[n] = h[h[n-1]] + h[n - h[n-1]];
a[n_] := For[m = 1, True, m++, If[h[m] == n, Return[m]]];
Array[a, 64] (* Jean-François Alcover, Jan 27 2018 *)

SageMath

@CachedFunction
def h(n): return 1 if (n<3) else h(h(n-1)) + h(n - h(n-1)) # h=A004001
def A188163(n):
    for j in range(1,2*n+2):
        if h(j)==n: return j
[A188163(n) for n in range(1,101)] # G. C. Greubel, May 20 2024

Scheme

(define A188163 (RECORD-POS 1 1 A004001))
;; Code for A004001 given in that entry. Uses also my IntSeq-library. - Antti Karttunen, Jan 18 2016

Formula

Other identities. For all n >= 1:

A004001(a(n)) = n and A004001(m) < n for m < a(n).

A051135(n) = a(n+1) - a(n).