Goodstein Sequence: Difference between revisions
m →{{header|Phix}}: cut out the middleman, fix a trailing ".0" which had crept in |
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for i,d in digits(n,b) do |
for i,d in digits(n,b) do |
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if d then |
if d then |
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res += d*power(b+1,bump(i-1,b)) |
res += d*round(power(b+1,bump(i-1,b))) |
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end if |
end if |
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end for |
end for |
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return res |
return res |
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end function |
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function A059934(atom n, k) |
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for i=1 to k do |
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n = bump(n, i+1)-1 |
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end for |
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return n |
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end function |
end function |
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sequence res = {n} |
sequence res = {n} |
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while length(res)<maxterms and res[$]!=0 do |
while length(res)<maxterms and res[$]!=0 do |
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res &= |
res &= bump(res[$],length(res)+1)-1 |
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end while |
end while |
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return res |
return res |
Revision as of 10:54, 13 February 2024
- 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(abs(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[$],length(res)+1)-1
end while
return res
end function
printf(1,"Goodstein(n) sequence (first 10) for values of n from 0 through 7:\n")
for i=0 to 7 do
printf(1,"Goodstein of %d: %v\n",{i,goodstein(i)})
end for
printf(1,"\n")
printf(1,"The Nth term of Goodstein(N) sequence counting from 0, for values of N from 0 through 16:\n")
bool m64 = machine_bits()=64
for i=0 to iff(m64?16:15) do
string ia = iff(i>=iff(m64?13:12)?" (inaccurate)":""),
gs = shorten(sprintf("%d",goodstein(i,i+1)[$]))
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
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
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