Heavily referenced sequences

[franktaw at netscape.net]

I took a look at the most referenced sequences (from in the database) that do not have the "core" keyword.  These are candidates to be considered core sequences.  Here are the top 25, with my votes:
 
1. A002275 (666 references) Repunits: (10^n - 1)/9
There are a lot of sequences like "Numbers n such that <k1>*R_n + <k2> is prime, where R_n = 11...1 is the repunit (A002275) of length n." or "Numbers n such that <k1>*10^n + <k2>*R_n + <k3> is prime...".  On balance, I would vote no.
 
2. A001358 (540 references) Products of two primes.
Again, there may be an excess of references to these, but primes are more mathematical than repdigits.  My vote is yes in this case.
 
3. A002110 (447 references) Primorial numbers (1st definition): product of first n primes.  Sometimes written p#.
These are pretty significant.  Yes.
 
4. A001221 (362 references) Number of distinct primes dividing n (also called omega(n)).
Yes.
 
5. A001222 (361 references) Number of prime divisors of n (counted with multiplicity).
Interesting that these should wind up next to each other.  Yes.
 
6. A007318 (337 references) Pascal's triangle read by rows: C(n,k)  binomial(n,k) = n!/(k!*(n-k)!), 0<=k<=n.
Definitely yes.  It makes no sense for this not to be a core sequence.
 
7. A007088 (293 references) Numbers written in base 2.
I would vote no on this one.  Should it have the base keyword?
 
8. A006530 (292 references) Largest prime dividing n (with a(1) = 1).
Probably not.
 
9. A014486 (271 references) Totally balanced sequences of 2n binary digits, their decimal representation.  Binary expansion of each term contains n 0's and n 1's, and reading from left to right (the most significant to the least significant bit), the number of 0's never exceeds the number of 1's.
On balance, I would vote against this one.
 
10. A001045 (267 references) Jacobsthal sequences: a(n) = a(n-1) + 2a(n-2).
Very closely related to powers of 2.  Still, I'd give it a yes.  The volume of comments and formulas is enormous.
 
11. A000533 (239 references) 10^n + 1, n>=1.
More base related stuff.  No.
 
12. A000796(239 references) Decimal expansion of Pi.
I'm inclined to think that this and A001113 (Decimal expansion of e) should be core sequences.
 
13. A005117 (231 references) Square-free numbers.
Yes.
 
14. A000032 (228 references) Lucas numbers (beginning at 2): L(n) = L(n-1) + L(n-2).
A000204 Lucas numbers (beginning at 1) is a core sequence.  If only one of the two is, I would favor this one.  Both would not be a miscarriage of justice.
 
15. A049310 (215 references) Triangle of coefficients of Chebyshev's S(n,x):=U(n,x/2) polynomials (exponents in increasing order).
I think so.
 
16. A000521 (213 references) Coefficients of modular function j as power series in q=e^(2Pi i t).
This is referenced from sequences for McKay-Thompson series for the Monster.  There is too little general interest here to justify it being a core sequence.
 
17. A001223 (206 references) Differences between consecutive primes.
This is too derivative.  No.
 
18. A007241 (200 references) McKay-Thompson series of class 2A for Monster.
19. A007267 (199 references) McKay-Thompson series of class 2A for Monster.
20. A045478 (199 references) McKay-Thompson series of class 2A for Monster.
21. A007240 (198 references) McKay-Thompson series of class 1A for Monster; another version of j-function.
22. A014708 (198 references) Coefficients of the modular function J = j - 744.
No.  See 16 above.
 
23. A000043 (185 references) Primes p such that 2^p - 1 is prime. 2^p - 1 is then called a Mersenne prime.
Yes.
 
24. A014137 (175 references) Partial sums of Catalan numbers (A000108).
A bit too derivative.
 
25. A020639 (175 references) Lpf(n): least prime dividing n (a(1)=1).
No.
 
 
 
Franklin T. Adams-Watters
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