A213923 - OEIS

%I #29 Oct 05 2021 15:32:53

%S 1,3,5,7,9,9,11,11,11,13,15,13,15,15,15,15,17,15,17,17,17,19,21,17,19,

%T 19,17,19,21,19,21,19,21,21,21,19,21,21,21,21,23,21,23,23,21,23,25,21,

%U 23,23,23,23,25,21,23,23,23,25,27,23,25,25,23,23,25,25,27,25,27,25,27,23,25,25,25,25,27,25

%N Minimal lengths of formulas representing n only using addition, multiplication and the constant 1.

%H Alois P. Heinz, <a href="/A213923/b213923.txt">Table of n, a(n) for n = 1..10000</a>

%H Shalosh B. Ekhad, <a href="http://www.math.rutgers.edu/~zeilberg/tokhniot/oArithFormulas1">Everything About Formulas Representing Integers Using Additions and Multiplication for integers from 1 to 8000</a>.

%H Edinah K. Ghang, Doron Zeilberger, <a href="http://arxiv.org/abs/1303.0885">Zeroless Arithmetic: Representing Integers ONLY using ONE</a>, arXiv:1303.0885 [math.CO], 2013

%F a(n) = 2*A005245(n)-1.

%e a(3) = 5 because for n = 3, the minimum is length = 5, formula = "11+1+" or "111++".

%p with(numtheory):

%p a:= proc(n) option remember;

%p        1+ `if`(n=1, 0, min(seq(a(i)+a(n-i), i=1..n/2),

%p        seq(a(d)+a(n/d), d=divisors(n) minus {1, n})))

%p     end:

%p seq(a(n), n=1..100);  # _Alois P. Heinz_, Mar 07 2013

%t a[n_] := a[n] = 1 + If[n == 1, 0, Min[Join[Table[a[i] + a[n-i], {i, 1, n/2}], Table[a[d] + a[n/d], {d, Divisors[n] ~Complement~ {1, n}}]]]]; Table[a[n], {n, 1, 100}] (* _Jean-François Alcover_, Feb 01 2017, after _Alois P. Heinz_ *)

%Y Cf. A005245, A214833.

%K nonn

%O 1,2

%A _Jonathan Vos Post_, Mar 06 2013