OFFSET
1,2
COMMENTS
The MD5 message digest algorithm converts a string into a 128-bit number represented as a 32-digit lowercase hexadecimal string. Consider iterating this function starting with the empty string, MD5("") = d41d8cd98f00b204e9800998ecf8427e, MD5^{(2)}("") = MD5("d41d8cd98f00b204e9800998ecf8427e") = 74be16979710d4c4e7c6647856088456.
The sequence gives the iteration numbers where a record value is achieved.
Because each iteration produces a 128-bit number, the sequence must be finite, although it is unknown what the maximum value attained by the above iteration is.
From Ben Whitmore, Apr 03 2018: (Start)
It appears likely that a(39) will be much larger than a(38). This is because the first 46 bits of MD5^{a(38)}("")=fffffffffffcfade870a33bf9bba701c are all 1's, while a(38) < 2^40 << 2^46.
a(39) > 2^40. (End)
Each iteration consists of converting the previous 16-byte MD5 hash result into a 32 byte lowercase hexadecimal string, and then taking the MD5 hash of that. This sequence only includes values that reach a new maximum. - Delbert L. Johnson, Mar 12 2023
LINKS
Ben Whitmore, Table of n, a(n) for n = 1..38 (first 35 terms from Sean A. Irvine)
Wikipedia, MD5
EXAMPLE
a(2)=15 because MD5^{(15)}("")=d4f6f4e928303d5361d83531beb5260f is larger than MD^{(n)}("") for all n<15.
a(28)=149333691 because MD5^{(149333691)}("")=fffffff5fedfdd0975cc5bbd15245b22 is larger than all preceding values.
MATHEMATICA
f[w_] := FromDigits[If[# < 57, # - 48, # - 87] & /@ Flatten[ToCharacterCode /@ Characters@ w], 16]; With[{s = f /@ Rest@ NestList[Hash[#, "MD5", "HexString"] &, "", 2^16]}, Map[FirstPosition[s, #][[1]] &, Union@FoldList[Max, s]]] (* Michael De Vlieger, Apr 07 2018, Version 11.3 *)
PROG
#!/bin/bash
function md5() {
echo -n "$1" | md5sum - | gawk '{print $1}'
}
it=0
while :; do
it=$[$it+1]
prev=$(md5 "$prev")
if [ "$prev" == "$best" ]; then
echo "Cycle detected"
exit
elif [ "$prev" \> "$best" ]; then
best=$prev
echo $it $prev
fi
done
(Python)
from hashlib import md5
def afind(limit):
record = hash = ""
for k in range(1, limit+1):
hash = md5(hash.encode('utf-8')).hexdigest()
if hash > record:
print(k, end=", ")
record = hash
afind(10**7) # Michael S. Branicky, Jul 02 2022
(C#)
public Enumerable<BigInteger> A234849()
{
yield return BigInteger.One;
var record = Convert.ToHexString(
MD5.HashData(Array.Empty<byte>())).ToLower();
var current = record;
for(var count = new BigInteger(2); ; count++)
{
current = Convert.ToHexString(
MD5.HashData(Encoding.UTF8.GetBytes(current))).ToLower();
var compareResult = string.Compare(current, record);
if (compareResult >= 0)
{
if(compareResult == 0)
{
yield break; // cycle detected
}
yield return count;
record = current;
}
}
} // Delbert L. Johnson, Mar 12 2023
CROSSREFS
KEYWORD
nonn,fini,nice
AUTHOR
Sean A. Irvine, Dec 31 2013
STATUS
approved