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A214843 Number of formula representations of n using addition, exponentiation and the constant 1. 4
1, 1, 2, 6, 16, 48, 152, 502, 1694, 5832, 20420, 72472, 260096, 942304, 3441584, 12658128, 46842920, 174289108, 651610504, 2446686568, 9222628592, 34886505168, 132387975040, 503857644160, 1922782984688, 7355686851696, 28203617340756, 108368274550664 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

LINKS

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

Shalosh B. Ekhad, Everything About Formulas Representing Integers Using Additions and Exponentiation for integers from 1 to 8000

Edinah K. Ghang and Doron Zeilberger, Zeroless Arithmetic: Representing Integers ONLY using ONE, arXiv:1303.0885 [math.CO], 2013.

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) = 6: 1111+++, 111+1++, 11+11++, 111++1+, 11+1+1+, 11+11+^.

a(5) = 16: 11111++++, 1111+1+++, 111+11+++, 1111++1++, 111+1+1++, 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+.

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(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[1] = 1; a[n_] := a[n] = Sum[a[i]*a[n-i], {i, 1, n-1}] + Sum[a[n^(1/p)] * a[p], {p, Divisors[GCD @@ FactorInteger[n][[All, 2]]] ~Complement~ {0, 1} } ];

Table[a[n], {n, 1, 40}] (* Jean-Fran├žois Alcover, Nov 06 2017, after Alois P. Heinz *)

CROSSREFS

Cf. A213924, A214833, A214836.

Sequence in context: A046721 A230929 A291189 * A272411 A151528 A132803

Adjacent sequences:  A214840 A214841 A214842 * A214844 A214845 A214846

KEYWORD

nonn

AUTHOR

Alois P. Heinz, Mar 08 2013

STATUS

approved

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Last modified July 30 04:36 EDT 2021. Contains 346348 sequences. (Running on oeis4.)