A117607 - OEIS

%I #13 Aug 18 2025 12:25:31

%S 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 6,3,4,5,5

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

%C 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 lim inf = 3, lim sup = infinity. What is the average behavior of this sequence? - _Charles R Greathouse IV_, Jun 15 2012

%H Ed Pegg, Jr., <a href="https://www.mathpuzzle.com/MAA/17-Integer%20Complexity/mathgames_04_12_04.html">Integer Complexity</a>

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Multifactorial.html">Multifactorial.</a>

%H <a href="/index/Com#complexity">Index to sequences related to the complexity of n</a>

%e a(1) = 1 because there is one 1 in "1".

%e a(2) = 2 because "1 + 1".

%e a(6) = 3 because "(1+1+1)!".

%e a(7) = 4 because "(1+1+1)!+1".

%e a(8) = 4 because "(1+1+1+1)!!" using double factorial.

%e a(12) = 3 because "((1+1+1)!)!!!!" using quadruple factorial.

%e a(15) = 5 because "(1+1+1+1+1)!!" using double factorial.

%e a(16) = 4 because "((1+1+1+1)!!)!!!!!!" using double factorial and sextuple factorial.

%e a(24) = 3 because "(((1+1+1)!)!!!!)!!!!!!!!!!" using quadruple factorial and decuple factorial.

%e a(36) = 3 because "(((1+1+1)!)!!!!)!!!!!!!!!" using quadruple factorial and nonuple factorial.

%Y n! = A000142. n!! = A006882. n!!! = A007661. n!!!! = A007662. n!!!!! = A085157. n!!!!!! = A085158. n!!!!!!! = A114799. n!!!!!!!! = A114800. n!!!!!!!!! = A114806.

%K nonn

%O 1,2

%A _Jonathan Vos Post_, Apr 06 2006