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 A217253 Number of minimal length formulas representing n only using addition, multiplication, exponentiation and the constant 1. 3
 1, 1, 2, 7, 18, 4, 8, 2, 2, 4, 12, 36, 72, 16, 72, 14, 28, 4, 8, 8, 48, 24, 48, 8, 18, 36, 4, 8, 24, 96, 328, 18, 36, 164, 472, 4, 8, 24, 80, 144, 288, 224, 560, 216, 72, 144, 432, 56, 8, 52, 232, 72, 144, 8, 16, 16, 32, 48, 96, 256, 512, 656, 32, 20, 40, 120 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 LINKS Alois P. Heinz, Table of n, a(n) for n = 1..10000 Edinah K. Ghang and Doron Zeilberger, Zeroless Arithmetic: Representing Integers ONLY using ONE, arXiv:1303.0885 [math.CO], 2013 Shalosh B. Ekhad, Everything About Formulas Representing Integers Using Additions, Multiplication and Exponentiation for integers from 1 to 8000 Wikipedia, Postfix notation Index to sequences related to the complexity of n EXAMPLE a(6) = 4: there are 58 formulas representing 6 only using addition, multiplication, exponentiation and the constant 1. The 4 formulas with minimal length 9 are: 11+111++*, 11+11+1+*, 111++11+*, 11+1+11+*. a(8) = 2: 11+111++^, 11+11+1+^. a(9) = 2: 111++11+^, 11+1+11+^. a(10) = 4: 1111++11+^+, 111+1+11+^+, 111++11+^1+, 11+1+11+^1+. All formulas are given in postfix (reverse Polish) notation but other notations would give the same results. MAPLE with(numtheory): b:= proc(n) option remember; local d, i, l, m, p, t; if n=1 then [1, 1] else l, m:= infinity, 0; for i to n-1 do t:= 1+b(i)[1]+b(n-i)[1]; if t=l then m:= m +b(i)[2]*b(n-i)[2] elif t b(n)[2]: seq(a(n), n=1..100); MATHEMATICA b[1] = {1, 1}; b[n_] := b[n] = Module[{d, i, l, m, p, t}, {l, m} = { Infinity, 0}; For[i=1, i <= n-1, i++, t = 1 + b[i][[1]] + b[n - i][[1]]; Which[t==l, m = m + b[i][[2]]*b[n-i][[2]], t

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Last modified December 4 05:36 EST 2023. Contains 367541 sequences. (Running on oeis4.)