Goodstein Sequence: Difference between revisions

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end for
end for
printf(1,"\n")
printf(1,"\n")
integer m64 = machine_bits()=64, maxi = iff(m64?16:15), alim = iff(m64?13:12)
printf(1,"The Nth term of Goodstein(N) sequence counting from 0, for values of N from 0 through 16:\n")
printf(1,"The Nth term of Goodstein(N) sequence counting from 0, for values of N from 0 through %d:\n",maxi)
bool m64 = machine_bits()=64
for i=0 to iff(m64?16:15) do
for i=0 to maxi do
string ia = iff(i>=iff(m64?13:12)?" (inaccurate)":""),
string ia = iff(i>=alim?" (inaccurate)":""),
gs = shorten(sprintf("%d",goodstein(i,i+1)[$]))
gs = shorten(sprintf("%d",goodstein(i,i+1)[$]))
printf(1,"Term %d of Goodstein(%d): %s%s\n",{i,i,gs,ia})
printf(1,"Term %d of Goodstein(%d): %s%s\n",{i,i,gs,ia})

Revision as of 11:05, 13 February 2024

Goodstein Sequence is a draft programming task. It is not yet considered ready to be promoted as a complete task, for reasons that should be found in its talk page.
Background

Goodstein sequences are sequences defined for a given counting number n by applying increasing bases to a representation of n after n has been used to construct a hereditary representation of that number, originally in base 2.

Start by defining the hereditary base-b representation of a number n. Write n as a sum of powers of b, staring with b = 2. For example, with n = 29, write 31 = 16 + 8 + 4 + 1. Now we write each exponent as a sum of powers of n, so as 2^4 + 2^3 + 2^1 + 2^0.

Continue by re-writing all of the current term's exponents that are still > b as a sum of terms that are <= b, using a sum of powers of b: so, n = 16 + 8 + 4 + 1 = 2^4 + 2^3 + 2 + 1 = 2^(2^2) + 2^(2 + 1) + 2 + 1.

If we consider this representation as a representation of a calculation with b = 2, we have the hereditary representation b^(b^b) + b^(b + 1) + b + 1.

Other integers and bases are done similarly. Note that an exponential term can be repeated up to (b - 1) times, so that, for example, if b = 5, 513 = b^3 + b^3 + b^3 + b^3 + b + b + 3 = 4 * b^3 + 2 * b + 3.

The Goodstein sequence for n, G(n) is then defined as follows:

The first term, considered the zeroeth term or G(n)(0), is always 0. The second term G(n)(1) is always n. For further terms, the m-th term G(n)(m) is defined by the following procedure: 

   1. Write G(n)(m - 1) as a hereditary representation with base (m - 1).
   2. Calculate the results of using the hereditary representation found in step 1 using base m rather than (m - 1)
   3. Subtract 1 from the result calculated in step 2.


Task
  • Create a function to calculate the Goodstein sequence for a given integer.
  • Use this to show the first 10 values of Goodstein(n) for the numbers from 0 through 7.
  • Find the nth term (counting from 0) of Goodstein(n) for n from 0 through 10.


Stretch task
  • Find the nth term (counting from 0) of Goodstein(n) for n = 11 through 16.



See also


Julia

""" Given nonnegative integer n and base b, return hereditary representation consisting of
    tuples (j, k) such that the sum of all (j * base^(evaluate(k)) = n.
"""
function decompose(n, b)
    if n < b
        return n
    end
    decomp = Vector{Union{typeof(n), Vector}}[]
    e = typeof(n)(0)
    while n != 0
        n, r = divrem(n, b)
        if r > 0
            push!(decomp, [r, decompose(e, b)])
        end
        e += 1
    end
    return decomp
end

""" Evaluate hereditary representation d under base b """
evaluate(d, b) = d isa Integer ? d : sum(j * b ^ evaluate(k, b) for (j, k) in d)

""" Return a vector of up to limitlength values of the Goodstein sequence for n """
function goodstein(n, limitlength = 10)
    seq = typeof(n)[]
    b = typeof(n)(2)
    while length(seq) < limitlength
        push!(seq, n)
        n == 0 && break
        d = decompose(n, b)
        b += 1
        n = evaluate(d, b) - 1
    end
    return seq
end

"""Get the Nth term of Goodstein(n) sequence counting from 0, see https://oeis.org/A266201"""
A266201(n) = last(goodstein(BigInt(n), n + 1))

println("Goodstein(n) sequence (first 10) for values of n from 0 through 7:")
for i in 1:7
    println("Goodstein of $i: $(goodstein(i))")
end
println("\nThe Nth term of Goodstein(N) sequence counting from 0, for values of N from 0 through 16:")
for i in big"1":16
    println("Term $i of Goodstein($i}): $(A266201(i))")
end
Output:
Goodstein(n) sequence (first 10) for values of n from 0 through 7:
Goodstein of 0: [0]
Goodstein of 1: [1, 0]
Goodstein of 2: [2, 2, 1, 0]
Goodstein of 3: [3, 3, 3, 2, 1, 0]
Goodstein of 4: [4, 26, 41, 60, 83, 109, 139, 173, 211, 253]
Goodstein of 5: [5, 27, 255, 467, 775, 1197, 1751, 2454, 3325, 4382]
Goodstein of 6: [6, 29, 257, 3125, 46655, 98039, 187243, 332147, 555551, 885775]
Goodstein of 7: [7, 30, 259, 3127, 46657, 823543, 16777215, 37665879, 77777775, 150051213]

The Nth term of Goodstein(N) sequence counting from 0, for values of N from 0 through 16:
Term 0 of Goodstein(0): 0
Term 1 of Goodstein(1): 0
Term 2 of Goodstein(2): 1
Term 3 of Goodstein(3): 2
Term 4 of Goodstein(4): 83
Term 5 of Goodstein(5): 1197
Term 6 of Goodstein(6): 187243
Term 7 of Goodstein(7): 37665879
Term 8 of Goodstein(8): 20000000211
Term 9 of Goodstein(9): 855935016215
Term 10 of Goodstein(10): 44580503598539
Term 11 of Goodstein(11): 2120126221988686
Term 12 of Goodstein(12): 155568095557812625
Term 13 of Goodstein(13): 6568408355712901455
Term 14 of Goodstein(14): 295147905179358418247
Term 15 of Goodstein(15): 14063084452070776884879
Term 16 of Goodstein(16): 2771517379996516970665566613559367879596937714713289695169887161862950129194382447127464877388711781205972046374648603545513430106433206876557475731408608398953667881600740852227698037876781766310900319669456854530159244376159780346700931210394158247781113134808720678004134212529413831368888355854503034587880113970541681685966414888841800498150131839091463034162026108960280455620621355407543489960326268155088833218122810217973039385643494213235664908254695964740257569988152978579630435471016976693529875691083071137361386386918409765002837648351746984484967203877495399596876291343126699827442908994036031608979805166915596436929638418152127561722561465793969723556331679336828840983098559789555364076924597258115780567651772009250336359472037679350612341393780002377587368649157608579801815531133644879180066181854487069796160774056572568941004114162614925

Phix

Modified version of the python code from A059934 - tbh, I did not expect to get anywhere near this far using native atoms, and always planned to write a gmp version, but now that just feels like too much effort for too little gain. 

function digits(atom n, b)
    -- least significant first, eg 123,10 -> {3,2,1} or 6,2 -> {0,1,1}
    sequence r = {remainder(n,b)}
    while n>=b do
        n = floor(n/b)
        r &= remainder(n,b)
    end while
    return r
end function

function bump(atom n, b)
    atom res = 0
    for i,d in digits(n,b) do
        if d then
            res += d*round(power(b+1,bump(i-1,b)))
        end if
    end for
    return res
end function

function goodstein(atom n, maxterms = 10)
    sequence res = {n}
    while length(res)<maxterms and res[$]!=0 do
        res &= bump(res[$],length(res)+1)-1
    end while
    return res
end function

printf(1,"Goodstein(n) sequence (first 10) for values of n from 0 through 7:\n")
for i=0 to 7 do
    printf(1,"Goodstein of %d: %v\n",{i,goodstein(i)})
end for
printf(1,"\n")
integer m64 = machine_bits()=64, maxi = iff(m64?16:15), alim = iff(m64?13:12)
printf(1,"The Nth term of Goodstein(N) sequence counting from 0, for values of N from 0 through %d:\n",maxi)
for i=0 to maxi do
    string ia = iff(i>=alim?" (inaccurate)":""),
           gs = shorten(sprintf("%d",goodstein(i,i+1)[$]))  
    printf(1,"Term %d of Goodstein(%d): %s%s\n",{i,i,gs,ia})
end for
Output:

(on 64-bit) 

Goodstein(n) sequence (first 10) for values of n from 0 through 7:
Goodstein of 0: {0}
Goodstein of 1: {1,0}
Goodstein of 2: {2,2,1,0}
Goodstein of 3: {3,3,3,2,1,0}
Goodstein of 4: {4,26,41,60,83,109,139,173,211,253}
Goodstein of 5: {5,27,255,467,775,1197,1751,2454,3325,4382}
Goodstein of 6: {6,29,257,3125,46655,98039,187243,332147,555551,885775}
Goodstein of 7: {7,30,259,3127,46657,823543,16777215,37665879,77777775,150051213}

The Nth term of Goodstein(N) sequence counting from 0, for values of N from 0 through 16:
Term 0 of Goodstein(0): 0
Term 1 of Goodstein(1): 0
Term 2 of Goodstein(2): 1
Term 3 of Goodstein(3): 2
Term 4 of Goodstein(4): 83
Term 5 of Goodstein(5): 1197
Term 6 of Goodstein(6): 187243
Term 7 of Goodstein(7): 37665879
Term 8 of Goodstein(8): 20000000211
Term 9 of Goodstein(9): 855935016215
Term 10 of Goodstein(10): 44580503598539
Term 11 of Goodstein(11): 2120126221988686
Term 12 of Goodstein(12): 155568095557812625
Term 13 of Goodstein(13): 6568408355712901452 (inaccurate)
Term 14 of Goodstein(14): 295147905179358418240 (inaccurate)
Term 15 of Goodstein(15): 14063084452070776847260 (inaccurate)
Term 16 of Goodstein(16): 27715173799965170860...62604488626682848248 (862 digits) (inaccurate)

Python

Translation of: Julia
def decompose(n, b):
    if n < b:
        return n
    decomp = []
    e = 0
    while n != 0:
        n, r = divmod(n, b)
        if r > 0:
            decomp.append([r, decompose(e, b)])
        e += 1

    return decomp


def evaluate(d, b):
    if type(d) is int:
        return d
    return sum(j * b ** evaluate(k, b) for j, k in d)


def goodstein(n, maxlen=10):
    seq = []
    b = 2
    while len(seq) < maxlen:
        seq.append(n)
        if n == 0:
            break
        d = decompose(n, b)
        b += 1
        n = evaluate(d, b) - 1

    return seq


def A266201(n):
    """Get the Nth term of Goodstein(n) sequence counting from 0, see https://oeis.org/A266201"""
    return goodstein(n, n + 1)[-1]


if __name__ == "__main__":

    print("Goodstein(n) sequence (first 10) for values of n from 0 through 7:")
    for i in range(8):
        print(f"Goodstein of {i}: {goodstein(i)}")

    print(
        "\nThe Nth term of Goodstein(N) sequence counting from 0, for values of N from 0 through 16:"
    )
    for i in range(17):
        print(f"Term {i} of Goodstein({i}): {A266201(i)}")
Output:
Goodstein(n) sequence (first 10) for values of n from 0 through 7:
Goodstein of 0: [0]
Goodstein of 1: [1, 0]
Goodstein of 2: [2, 2, 1, 0]
Goodstein of 3: [3, 3, 3, 2, 1, 0]
Goodstein of 4: [4, 26, 41, 60, 83, 109, 139, 173, 211, 253]
Goodstein of 5: [5, 27, 255, 467, 775, 1197, 1751, 2454, 3325, 4382]
Goodstein of 6: [6, 29, 257, 3125, 46655, 98039, 187243, 332147, 555551, 885775]
Goodstein of 7: [7, 30, 259, 3127, 46657, 823543, 16777215, 37665879, 77777775, 150051213]

The Nth term of Goodstein(N) sequence counting from 0, for values of N from 0 through 16:
Term 0 of Goodstein(0): 0
Term 1 of Goodstein(1): 0
Term 2 of Goodstein(2): 1
Term 3 of Goodstein(3): 2
Term 4 of Goodstein(4): 83
Term 5 of Goodstein(5): 1197
Term 6 of Goodstein(6): 187243
Term 7 of Goodstein(7): 37665879
Term 8 of Goodstein(8): 20000000211
Term 9 of Goodstein(9): 855935016215
Term 10 of Goodstein(10): 44580503598539
Term 11 of Goodstein(11): 2120126221988686
Term 12 of Goodstein(12): 155568095557812625
Term 13 of Goodstein(13): 6568408355712901455
Term 14 of Goodstein(14): 295147905179358418247
Term 15 of Goodstein(15): 14063084452070776884879
Term 16 of Goodstein(16): 2771517379996516970665566613559367879596937714713289695169887161862950129194382447127464877388711781205972046374648603545513430106433206876557475731408608398953667881600740852227698037876781766310900319669456854530159244376159780346700931210394158247781113134808720678004134212529413831368888355854503034587880113970541681685966414888841800498150131839091463034162026108960280455620621355407543489960326268155088833218122810217973039385643494213235664908254695964740257569988152978579630435471016976693529875691083071137361386386918409765002837648351746984484967203877495399596876291343126699827442908994036031608979805166915596436929638418152127561722561465793969723556331679336828840983098559789555364076924597258115780567651772009250336359472037679350612341393780002377587368649157608579801815531133644879180066181854487069796160774056572568941004114162614925

Wren

Translation of: Julia
Library: Wren-big
Library: Wren-fmt
import "./big" for BigInt
import "./fmt" for Fmt

// Given non-negative integer n and base b, return hereditary representation
// consisting of tuples (j, k) so sum of all (j * b^(evaluate(k, b)) = n.
var decompose // recursive
decompose = Fn.new { |n, b|
    if (n < b) return n
    var decomp = []
    var e = BigInt.zero
    while (n != 0) {
        var t = n.divMod(b)
        n = t[0]
        var r = t[1]
        if (r > 0) decomp.add([r, decompose.call(e, b)])
        e = e.inc
    }
    return decomp
}

// Evaluate hereditary representation d under base b.
var evaluate // recursive
evaluate = Fn.new { |d, b|
    if (d is BigInt) return d
    var sum = BigInt.zero
    for (a in d) {
        var j = a[0]
        var k = a[1]
        sum = sum + j * b.pow(evaluate.call(k, b))
    }
    return sum
}

// Return a vector of up to limitlength values of the Goodstein sequence for n.
var goodstein = Fn.new { |n, limitLength|
    var seq = []
    var b = BigInt.two
    while (seq.count < limitLength) {
        seq.add(n)
        if (n == 0) break
        var d = decompose.call(n, b)
        b = b.inc
        n = evaluate.call(d, b) - 1
    }
    return seq
}

// Get the nth term of the Goodstein(n) sequence counting from 0
var a266201 = Fn.new { |n| goodstein.call(n, (n + 1).toSmall)[-1] }

System.print("Goodstein(n) sequence (first 10) for values of n in [0, 7]:")
for (i in BigInt.zero..7) System.print("Goodstein of %(i): %(goodstein.call(i, 10))")

System.print("\nThe nth term of Goodstein(n) sequence counting from 0, for values of n in [0, 16]:")
for (i in BigInt.zero..16) {
    Fmt.print("Term $2i of Goodstein($2i): $i", i, i, a266201.call(i, 10))
}
Output:
Goodstein(n) sequence (first 10) for values of n in [0, 7]:
Goodstein of 0: [0]
Goodstein of 1: [1, 0]
Goodstein of 2: [2, 2, 1, 0]
Goodstein of 3: [3, 3, 3, 2, 1, 0]
Goodstein of 4: [4, 26, 41, 60, 83, 109, 139, 173, 211, 253]
Goodstein of 5: [5, 27, 255, 467, 775, 1197, 1751, 2454, 3325, 4382]
Goodstein of 6: [6, 29, 257, 3125, 46655, 98039, 187243, 332147, 555551, 885775]
Goodstein of 7: [7, 30, 259, 3127, 46657, 823543, 16777215, 37665879, 77777775, 150051213]

The nth term of Goodstein(n) sequence counting from 0, for values of n in [0, 16]:
Term  0 of Goodstein( 0): 0
Term  1 of Goodstein( 1): 0
Term  2 of Goodstein( 2): 1
Term  3 of Goodstein( 3): 2
Term  4 of Goodstein( 4): 83
Term  5 of Goodstein( 5): 1197
Term  6 of Goodstein( 6): 187243
Term  7 of Goodstein( 7): 37665879
Term  8 of Goodstein( 8): 20000000211
Term  9 of Goodstein( 9): 855935016215
Term 10 of Goodstein(10): 44580503598539
Term 11 of Goodstein(11): 2120126221988686
Term 12 of Goodstein(12): 155568095557812625
Term 13 of Goodstein(13): 6568408355712901455
Term 14 of Goodstein(14): 295147905179358418247
Term 15 of Goodstein(15): 14063084452070776884879
Term 16 of Goodstein(16): 2771517379996516970665566613559367879596937714713289695169887161862950129194382447127464877388711781205972046374648603545513430106433206876557475731408608398953667881600740852227698037876781766310900319669456854530159244376159780346700931210394158247781113134808720678004134212529413831368888355854503034587880113970541681685966414888841800498150131839091463034162026108960280455620621355407543489960326268155088833218122810217973039385643494213235664908254695964740257569988152978579630435471016976693529875691083071137361386386918409765002837648351746984484967203877495399596876291343126699827442908994036031608979805166915596436929638418152127561722561465793969723556331679336828840983098559789555364076924597258115780567651772009250336359472037679350612341393780002377587368649157608579801815531133644879180066181854487069796160774056572568941004114162614925
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