Define to be the complexity of , the smallest number of 1s needed to write using an arbitrary combination of addition and multiplication. John Selfridge showed that for all , leading this author and Zelinsky to define the defect of , , to be the difference . Meanwhile, in the study of addition chains, it is common to consider , the number of small steps of , defined as , an integer quantity, where is the length of the shortest addition chain for . So here we analogously define , the integer defect of , an integer version of analogous to . Note that is not the same as .
We show that has additional meaning in terms of the defect well-ordering we considered in 2015, in that indicates which powers of the quantity lies between when one restricts to with lying in a specified congruence class modulo . We also determine all numbers with , and use this to generalize a result of Rawsthorne (1989).
"Integer complexity: the integer defect." Mosc. J. Comb. Number Theory 8 (3) 193 - 217, 2019. https://doi.org/10.2140/moscow.2019.8.193