# The Very Model of a Number Sequence

Number sequence questions are normally not received well here but I thought this one might warrant a bit more attention as it has the following interesting properties:

1. It is a finite sequence.
2. Each element of the sequence appears exactly once.
3. It does not appear on OEIS but is closely related to another sequence that does.
4. Many would say it is too short.

Here are the first few entries:

5, 51, 47, 48, 37, 97, 7, 6, 56, 91, ?

Can you tell me what comes next?

Hint 1

Most people would say the sequence needs an additional 16 elements.

Hint 2

The sequence was created in 1959

Hint 3

There is a very subtle hint in the title

• Of course I can! Of course I can't! May 7, 2020 at 16:06
• Does hint 1 include the ?, or is it in addition to it?
– CG.
May 21, 2020 at 8:51
• @CG. The entire sequence is much longer than presented here, I have only listed the first 10 entries and want you to name the 11th. May 21, 2020 at 8:54
• Also, I'm fairly certain hint 1 is a straight out lie. Most people don't know much about number sequences. :-P
– CG.
May 21, 2020 at 8:55
• @Kalaivanan Are your referring to A167409? If so, then I will say no. May 21, 2020 at 9:03

The next element in the sequence is:

102

Why? Because each item in this sequence corresponds to:

The order of Periodic Table elements in the song 'The Elements' by Tom Lehrer (1959), in which Lehrer sings the names of 102 elements of the Periodic Table to the tune of "I Am the Very Model of a Modern Major-General" from Gilbert and Sullivan's 1879 comic opera The Pirates of Penzance. (Bolded text in the title of this puzzle, note...)

By considering the elements in Periodic Table order, the nth term of the sequence corresponds to the ordered position in the song lyrics of the nth chemical element, like so:

5 (5th position in the song) = Hydrogen (element 1 in the Periodic Table)
51 = Helium (2)
47 = Lithium (3)
48 = Beryllium (4)
37 = Boron (5)
97 = Carbon (6)
7 = Nitrogen (7)
6 = Oxygen (8)
56 = Fluorine (9)
91 = Neon (10)

The next member of the sequence should therefore correspond to the position of the 11th element of the Periodic Table - Sodium - in the song. Sodium is the last one mentioned - in position 102.

Of course, at the latest count there are actually 118 elements in the Periodic Table, which is why many may consider the sequence needs an additional 16 elements, as per Hint 1...

If you're interested in calculating further terms of the sequence...

I recommend you watch the linked video! But also here are the lyrics to Lehrer's song, with the positions of each term of the sequence relevant to this question emphasised in bold, and every tenth element numbered for ease of counting:

There's antimony, arsenic, aluminum, selenium,
And hydrogen[5] and oxygen[6] and nitrogen[7] and rhenium,
And nickel, neodymium[10], neptunium, germanium,
And iron, americium, ruthenium, uranium,
And lanthanum and osmium and astatine and radium,
And gold and protactinium and indium and gallium,
And iodine and thorium[30] and thulium and thallium.

There's yttrium, ytterbium, actinium, rubidium,
And strontium and silicon and silver and samarium,
And bismuth, bromine, lithium[47], beryllium[48], and barium.

There's holmium[50] and helium[51] and hafnium and erbium,
And phosphorus and francium and fluorine[56] and terbium,
And manganese and mercury, molybdenum[60], magnesium,
Dysprosium and scandium and cerium and caesium.
And lead, praseodymium and platinum, plutonium,
And tantalum, technetium, titanium, tellurium,
And cadmium and calcium and chromium[80] and curium.

There's sulfur, californium and fermium, berkelium,
And also mendelevium, einsteinium, nobelium,
And argon, krypton[90], neon[91], radon, xenon, zinc and rhodium,
And chlorine, carbon[97], cobalt, copper, tungsten[100], tin and sodium[102].

These are the only ones of which the news has come to Hahvard,
And there may be many others but they haven't been discahvered!

• Yep, that's it. Nice work! May 21, 2020 at 13:12
• Thanks @hexomino - when it finally clicked it was that fabled proper 'aha' moment! Fun idea for a sequence :)
– Stiv
May 21, 2020 at 14:02