# Extend the sequences of numbers and names

Find the next entry in each of this pair of related sequences:

1, 4, 6, 8, 9, ?

Bobby, Joe, Johnny, Carl, Ted, ?

Note: There's no way to solve this without making use of both sequences - it's unsolvable with only the numbers.

Hint:

The next entries in the sequences after the ones I'm looking for are 26 and Wade respectively.

• Is it safe to assume that no worldly knowledge is needed, such as familiarity with actual persons with those names?
– humn
Oct 8, 2017 at 6:14
• @humn That is not safe to assume. Oct 8, 2017 at 15:01

1, 4, 6, 8, 9, ?

... 14, 26, 27, 34, 45   =   uniform numbers retired by baseball’s Boston Red Sox

Bobby, Joe, Johnny, Carl, Ted, ?

... Jim, Wade, Carlton, David, Pedro   =   first names of the players

The hint made it clear.   You see, ever since...

...Babe Ruth pitched for Boston (who didn’t bother to retire his uniform #), I’ve been a fan of Boston’s lefty batters.   The “Wade” (Boggs) hint rang a bell for “Ted” (Williams) and “Carl” (Yastremski, who was on the team with two others here when I once got to a game at Fenway Park).   Here’s the list:

Bobby Doerr, uniform #1
Joe Cronin, uniform #4
Johnny Pesky, uniform #6
Carl Yastremski, uniform #8
Ted Williams, uniform #9
Jim Rice, uniform #14
Wade (from hint) Boggs, uniform #26
Carlton Fisk, uniform #27
David Ortiz, uniform #34
Pedro Martinez, uniform #45

• Uh, oh, baseball.:) (After the first partial answer was out, I hoped that the numbers would be non-primes and the names those of prime ministers of some country or other ...) Oct 10, 2017 at 5:25
• Like minds..., @MOehm, before the hint I looked up prime ministers too! (And those who weren't elected PM.)
– humn
Oct 10, 2017 at 5:27
• nice, i got wade boggs but know nothing about baseball and couldn't link the other numbers to players Oct 11, 2017 at 2:31

The next term of the first sequence is 10.

Why? Because the first sequence is the sequence of non-prime integers (OEIS A018252), and this is how we get the first 5 terms:$$1\space\space\color{red}2\space\space\color{red}3\space\space4\space\space\color{red}5\space\space6\space\space\color{red}7\space\space8\space\space9$$The terms in red are excluded. The sequence is (0-indexed)$$\alpha_n=f(n+1,0)\\f(x,y)=((x=0\rightarrow y)\land(x>0\rightarrow f(x-((\Omega(y)\ne1\rightarrow1)\land(\Omega(y)=1\rightarrow0)),y+1)))\\\Omega(x)=\text{Count of prime factors of }x\text{ including duplicates}$$Since $10$ is the next non-prime integer, it's the next term of the sequence.

• It could just as well be 12: non-prime numbers of the form \$2^a 3^b\$ Oct 8, 2017 at 12:42
• @PeterTaylor Can you post that as an answer please? Oct 8, 2017 at 13:07

The next number in the list can be

22

Based on this list

These can be non-prime numbers:

https://oeis.org/search?q=1%2c%204%2c%206%2c%208%2c%209%2c