An Interesting Sequence

Here's an interesting question that was posed by PUZZLE A DAY on 30th October 2024: What number comes next in this sequence and why? 1, 7, 11, 27, ...

Mathematically, there doesn't seem to be any pattern and I needed the clue provided to proceed any further. Here is the clue:

Clue: Say the number aloud.

Immediately we know that the sequence is about how the number sounds and not about anything to do with the numbers per se. When we sound the numbers out, we have:

• one ... single syllable
• seven ... two syllables
• eleven ... three syllables
• twenty seven ... four syllables

Clearly we are looking for the next number that has five syllables and that numbers happens to be 77 or seventy seven. So we now have 1, 7, 11, 27, 77, ...

That's where the puzzle ends but the question then arises as to the next number in the sequence. This will take us into the hundreds and we need to decide whether to sound out the "and" or not. For example, 101 can be read as "one hundred and one" or "one hundred one". Let's go with the latter and search for the next number. 

As far as I can determine this must be 107 or one hundred seven that has five syllables. After that it would be 111 or one hundred eleven with six syllables. So now we have: 1, 7, 11, 27, 77, 107, 111. 

After this the next number with seven syllables is 127 or one hundred twenty seven. For eight syllables, it would 177 or one hundred seventy seven. Let's stop there and then list what we have so far is 1, 7, 11, 27, 77, 107, 111, 127, 177, ...

The puzzle's focus on the sound of the number rather than the number itself reminds of aban, eban, iban, oban and uban numbers. See my blog post Iban Numbers. It's also reminscent of the Look and Say Sequence. Finally, the Mathematics section of PUZZLE A DAY can be found here.