1,2

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

See A393294 for the maybe more self-consistent approach where the starting value consists of 128-bits 0 rather than the empty string, and in subsequent iterations, the 128 bits = 16 bytes of the (binary) digest are fed into the MD5 function. As the above comment says, here the 128-bit return value is converted to a message of 32 bytes = 256 bits, processed in the next iteration. - M. F. Hasler, Mar 08 2026

Ben Whitmore, Table of n, a(n) for n = 1..38 (first 35 terms from Sean A. Irvine)

Wikipedia, MD5

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.

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 *)

#!/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

Cf. A393294 (different approach: initial value = 128 bits 0).

Sequence in context: A077349 A368827 A257020 * A164564 A211329 A104298

Adjacent sequences: A234846 A234847 A234848 * A234850 A234851 A234852

nonn,fini,nice

Sean A. Irvine, Dec 31 2013

approved