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A378758 Number of 1's required to build n using +, -, and ^.

%I A378758 #12 Dec 21 2024 00:51:30
%S A378758 1,2,3,4,5,6,6,5,5,6,7,8,9,8,7,6,7,8,9,10,11,10,9,8,7,7,6,7,8,9,8,7,8,
%T A378758 9,9,8,9,10,11,12,12,13,12,13,12,11,10,9,8,9,10,11,12,12,12,12,13,13,
%U A378758 12,11,10,9,8,7,8,9,10,11,12,13,13,12,12,13
%N A378758 Number of 1's required to build n using +, -, and ^.
%C A378758 All intermediate steps in building the number should be integers as well, for consistency with related sequences.
%C A378758 A348262(n) >= a(n) >= A091334(n) for all n, as the available operators in A348262 are a subset of the available operators here, and the available operators here are a subset of the available operators in A091334.
%H A378758 Jake Bird, <a href="/A378758/b378758.txt">Table of n, a(n) for n = 1..300</a>
%e A378758 a(22) = 10 because 22 = (1+1+1+1+1)^(1+1)-(1+1+1), which has 10 occurrences of the symbol "1", and there is no way of making 22 with fewer using these rules.
%e A378758 Note that A348262(22) = 12 because 22 = (1+1)^(1+1)^(1+1)+(1+1)^(1+1)+1+1; subtraction allows for two fewer occurrences of the symbol "1" to be used here. Similarly, A091334(22) = 9 because 22 = ((1+1+1)^(1+1)+1+1)*(1+1); multiplication allows for one fewer occurrence of the symbol "1" to be used there. 22 is the least n such that A348262(n) > a(n) > A091334(n).
%Y A378758 Cf. A000027 {1,+}, {1,+,-}
%Y A378758 Cf. A005245 {1,+,*}
%Y A378758 Cf. A348262 {1,+,^}
%Y A378758 Cf. A091333 {1,+,-,*}
%Y A378758 Cf. A025280 {1,+,*,^}
%Y A378758 Cf. A378759 {1,+,/,^}
%Y A378758 Cf. A091334 {1,+,-,*,^}
%Y A378758 Cf. A348089 {1,+,-,*,/,^}
%K A378758 nonn
%O A378758 1,2
%A A378758 _Jake Bird_, Dec 06 2024