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A214836
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Number of formula representations of n using addition, multiplication, exponentiation and the constant 1.
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5
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1, 1, 2, 7, 18, 58, 180, 613, 2076, 7270, 25752, 92918, 338432, 1246092, 4624536, 17290646, 65047436, 246079536, 935484928, 3571960668, 13692523960, 52675401248, 203299385584, 786949008100, 3054440440486, 11884949139900, 46351113658232, 181153317512536
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OFFSET
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1,3
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LINKS
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EXAMPLE
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a(1) = 1: 1.
a(2) = 1: 11+.
a(3) = 2: 111++, 11+1+.
a(4) = 7: 1111+++, 111+1++, 11+11++, 111++1+, 11+1+1+, 11+11+*, 11+11+^.
a(5) = 18: 11111++++, 1111+1+++, 111+11+++, 1111++1++, 111+1+1++, 111+11+*+, 111+11+^+, 11+111+++, 11+11+1++, 111++11++, 11+1+11++, 1111+++1+, 111+1++1+, 11+11++1+, 111++1+1+, 11+1+1+1+, 11+11+*1+, 11+11+^1+.
All formulas are given in postfix (reverse Polish) notation but other notations would give the same results.
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MAPLE
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with(numtheory):
a:= proc(n) option remember; `if`(n=1, 1,
add(a(i)*a(n-i), i=1..n-1)+
add(a(d)*a(n/d), d=divisors(n) minus {1, n})+
add(a(root(n, p))*a(p), p=divisors(igcd(seq(i[2],
i=ifactors(n)[2]))) minus {0, 1}))
end:
seq(a(n), n=1..40);
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MATHEMATICA
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a[n_] := a[n] = If[n==1, 1, Sum[a[i]*a[n-i], {i, 1, n-1}] + Sum[a[d]*a[n/d], {d, Divisors[n] ~Complement~ {1, n}}] + Sum[a[n^(1/p)] * a[p], {p, Divisors[GCD @@ Table[i[[2]], {i, FactorInteger[n]}]] ~Complement~ {0, 1}}]]; Array[a, 40] (* Jean-François Alcover, Apr 11 2017, translated from Maple *)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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