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A214836 Number of formula representations of n using addition, multiplication, exponentiation and the constant 1. 5
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

LINKS

Alois P. Heinz, Table of n, a(n) for n = 1..1000

Edinah K. Ghang, 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(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.

MAPLE

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);

MATHEMATICA

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-Fran├žois Alcover, Apr 11 2017, translated from Maple *)

CROSSREFS

Cf. A025280, A214833, A214843, A217250, A217253.

Sequence in context: A046866 A291255 A294004 * A176813 A000988 A185308

Adjacent sequences:  A214833 A214834 A214835 * A214837 A214838 A214839

KEYWORD

nonn

AUTHOR

Alois P. Heinz, Mar 07 2013

STATUS

approved

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Last modified June 2 18:09 EDT 2020. Contains 334787 sequences. (Running on oeis4.)