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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.
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
:) (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 ...)
$\endgroup$
Partial answer.
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.
