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%I #31 Apr 04 2024 09:59:01
%S 0,1,2,2,3,3,4,3,4,4,5,4,5,5,5,4,5,5,6,5,6,6,7,5,6,6,6,6,7,6,7,5,6,6,
%T 7,6,7,7,7,6,7,7,8,7,7,8,9,6,7,7,7,7,8,7,8,7,8,8,9,7,8,8,8,6,7,7,8,7,
%U 8,8,9,7,8,8,8,8,8,8,9,7,8,8,9,8,8,9,9,8,9,8,9,9,9,10,9,7,8,8,8,8,9,8,9,8,9
%N Length of shortest sequence b with b(0) = 1, b(i+1) = b(i)+d where d|b(i) and b(k) = n.
%C This is similar to the shortest addition chain for n. Both the binary method and the divisor method for finding an addition chain will find a sequence of this type. The smallest few n where there is an addition chain shorter than this sequence are 23,43,46,47,59. The first few n where this sequence is smaller than the shortest addition chain are 143,267,275,286,407. The smallest few n such that a(n) = a(2n) are 86,213,285,342,383.
%C The greedy inverse (index of first occurrences of n = 0, 1, 2, 3...) starts 1, 2, 3, 5, 7, 11, 19, 23, 43, 47, 94, 167, 283, 359, 718, 719, 1439, 2447, 4079, 7559,.. , different from A105017. - _R. J. Mathar_, Mar 03 2022
%H David W. Wilson, <a href="/A117497/b117497.txt">Table of n, a(n) for n = 1..10000</a>
%H John M. Campbell, <a href="https://arxiv.org/abs/2403.20073">A binary version of the Mahler-Popken complexity function</a>, arXiv:2403.20073 [math.NT], 2024. See pp. 5-6.
%H <a href="/index/Com#complexity">Index to sequences related to the complexity of n</a>
%F a(1)=0, a(n) = 1 + min_{d|n, d<n} a(n-d).
%F a(2^n) = n. - _R. J. Mathar_, Mar 03 2022
%e The sequence 1,2,4,8,16,32,64,128,132,143 gets 143 in 9 steps, so a(143) = 9.
%p A117497 := proc(n)
%p option remember ;
%p local prev,d,a ;
%p if n <= 2 then
%p n-1 ;
%p else
%p a := n ;
%p for prev from n-1 to 1 by -1 do
%p for d in numtheory[divisors](prev) do
%p if d+prev = n then
%p a := min(a,procname(prev)+1) ;
%p end if;
%p end do:
%p end do:
%p a ;
%p end if;
%p end proc:
%p seq(A117497(n),n=1..100) ; # _R. J. Mathar_, Mar 02 2022
%t a[n_] := a[n] = If[n == 1, 0, With[{m = Log2[n]}, If[IntegerQ[m], m,
%t 1 + Min[a[n-#]& /@ Most[Divisors[n]]]]]];
%t Table[a[n], {n, 1, 105}] (* _Jean-François Alcover_, Aug 05 2022 *)
%o (Python)
%o from functools import lru_cache
%o from sympy import divisors
%o @lru_cache(maxsize=None)
%o def A117497(n): return 0 if n == 1 else 1 + min(A117497(n-d) for d in divisors(n,generator=True) if d < n) # _Chai Wah Wu_, Mar 03 2022
%Y Cf. A003313, A117498, A352079.
%K nonn,changed
%O 1,3
%A _Franklin T. Adams-Watters_, Mar 22 2006
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