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A117607 Integer complexity of n represented with {1,+,!} and parentheses, where ! can be concatenated for multifactorials.

%I A117607 #13 Aug 18 2025 12:25:31
%S A117607 1,2,3,4,5,3,4,4,5,6,7,3,4,4,5,4,5,5,6,6,4,5,6,3,4,4,5,4,5,5,6,4,5,5,
%T A117607 6,3,4,5,5
%N A117607 Integer complexity of n represented with {1,+,!} and parentheses, where ! can be concatenated for multifactorials.
%C A117607 Using the set of symbols {1, +, !} and parentheses, how many 1's does it take to represent n? "!!" is double factorial, "!!!" is triple factorial and so forth.
%C A117607 lim inf = 3, lim sup = infinity. What is the average behavior of this sequence? - _Charles R Greathouse IV_, Jun 15 2012
%H A117607 Ed Pegg, Jr., <a href="https://www.mathpuzzle.com/MAA/17-Integer%20Complexity/mathgames_04_12_04.html">Integer Complexity</a>
%H A117607 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Multifactorial.html">Multifactorial.</a>
%H A117607 <a href="/index/Com#complexity">Index to sequences related to the complexity of n</a>
%e A117607 a(1) = 1 because there is one 1 in "1".
%e A117607 a(2) = 2 because "1 + 1".
%e A117607 a(6) = 3 because "(1+1+1)!".
%e A117607 a(7) = 4 because "(1+1+1)!+1".
%e A117607 a(8) = 4 because "(1+1+1+1)!!" using double factorial.
%e A117607 a(12) = 3 because "((1+1+1)!)!!!!" using quadruple factorial.
%e A117607 a(15) = 5 because "(1+1+1+1+1)!!" using double factorial.
%e A117607 a(16) = 4 because "((1+1+1+1)!!)!!!!!!" using double factorial and sextuple factorial.
%e A117607 a(24) = 3 because "(((1+1+1)!)!!!!)!!!!!!!!!!" using quadruple factorial and decuple factorial.
%e A117607 a(36) = 3 because "(((1+1+1)!)!!!!)!!!!!!!!!" using quadruple factorial and nonuple factorial.
%Y A117607 n! = A000142. n!! = A006882. n!!! = A007661. n!!!! = A007662. n!!!!! = A085157. n!!!!!! = A085158. n!!!!!!! = A114799. n!!!!!!!! = A114800. n!!!!!!!!! = A114806.
%K A117607 nonn
%O A117607 1,2
%A A117607 _Jonathan Vos Post_, Apr 06 2006