Suggest lower case letters for variables, upper case letters for these functions: (remove: "!!" = double factorial and "##" = product of first n primes)

[CODE]
A n! - (n-1)! + (n-2)! - ... 1! [URL="https://oeis.org/A005165"]A005165[/URL]
B Bell numbers [URL="https://oeis.org/A000110"]A000110[/URL]
C Catalan numbers [URL="https://oeis.org/A000108"]A000108[/URL]
D Distinct partition numbers [URL="https://oeis.org/A000009"]A000009[/URL]
E Euler zigzag numbers [URL="https://oeis.org/A000111"]A000111[/URL]
F Fermat numbers [URL="https://oeis.org/A000215"]A000215[/URL]
G Fubini numbers [URL="https://oeis.org/A000670"]A000670[/URL]
H H(m,n) = n-th m-Fibonacci numbers [URL="https://oeis.org/A000045"]A000045[/URL] [URL="https://oeis.org/A000129"]A000129[/URL] [URL="https://oeis.org/A006190"]A006190[/URL] [URL="https://oeis.org/A001076"]A001076[/URL] ...
I I(m,n) = n-th m-step Fibonacci numbers [URL="https://oeis.org/A000045"]A000045[/URL] [URL="https://oeis.org/A000073"]A000073[/URL] [URL="https://oeis.org/A000078"]A000078[/URL] [URL="https://oeis.org/A001591"]A001591[/URL] ...
J Coefficients of modular function j as power series in q = e^(2 Pi i t) [URL="https://oeis.org/A000521"]A000521[/URL]
K 1! + 2! + 3! + ... + n! [URL="https://oeis.org/A007489"]A007489[/URL]
L L(m,n) = n-th m-Lucas numbers [URL="https://oeis.org/A000032"]A000032[/URL] [URL="https://oeis.org/A002203"]A002203[/URL] [URL="https://oeis.org/A006497"]A006497[/URL] [URL="https://oeis.org/A014448"]A014448[/URL] ...
M Mersenne numbers [URL="https://oeis.org/A000225"]A000225[/URL]
N N(m,n) = n!_(m) = m-factorial of n
O O(m,n) = m-th cyclotomic polynomial evaluated at n
P Partition numbers [URL="https://oeis.org/A000041"]A000041[/URL]
Q Perrin numbers [URL="https://oeis.org/A001608"]A001608[/URL]
R R(m,n) = repunit in base m with length n
S S(m,n) = (Smarandache numbers) the concatenate the first n integers in base m
T Ramanujan's tau function [URL="https://oeis.org/A000594"]A000594[/URL]
U U(n,p,q) = Lucas sequence U_n(p,q)
V V(n,p,q) = Lucas sequence V_n(p,q)
W W(m,n) = (Wolstenholme numbers) numerator of 1 + 1/(2^m) + 1/(3^m) + ... + 1/(n^m)
X X(m,n) = (Smarandache-Wellin numbers) the concatenate the first n primes in base m
Y Y(m,n) = n-th m-step Lucas numbers [URL="https://oeis.org/A000032"]A000032[/URL] [URL="https://oeis.org/A001644"]A001644[/URL] [URL="https://oeis.org/A073817"]A073817[/URL] [URL="https://oeis.org/A074048"]A074048[/URL] ...
Z Motzkin numbers [URL="https://oeis.org/A001006"]A001006[/URL]
! [URL="https://en.wikipedia.org/wiki/Factorial"]Factorial[/URL]
# [URL="https://en.wikipedia.org/wiki/Primorial"]Primorial[/URL]
% [URL="https://en.wikipedia.org/wiki/Modulo_(mathematics)"]Modulo operator[/URL]
\ [URL="https://en.wikipedia.org/wiki/Division_(mathematics)#Of_integers"]Integer division[/URL]
@ n@ = n-th prime
& Narayana's cows sequence [URL="https://oeis.org/A000930"]A000930[/URL]
$ Padovan sequence [URL="https://oeis.org/A000931"]A000931[/URL]
: :(m,n,r,s,...) = LCM(m,n,r,s,...) (least common multiple)
; ;(m,n,r,s,...) = GCD(m,n,r,s,...) (greatest common divisor)
~ ~(abc(de)^(fg)hij,n) = string "abcdedede...dededehij" (with fg de's, this fg is written in base 10) in base n (n must be >=2 and <=36) (the letters in the m of ~(m,n) must be upper case, and the m does not support any functions include +-*/, and if there are any variable in m, the variable is read in base n (except when the variable is in fg, in this case, the variable is still read in base 10))
[/CODE]

Also let factordb can support expressions with negative numbers or noninteger rational numbers as middle result, but the final result is nonnegative integer (e.g. "3-5+7" and "5/10*12"), and then we can type expressions such as (142/999)*(10^(3*n+1)-10)+1, also we can type "3.4*5" to get the result 17, if so, then I also suggest use "|n" for absolute value of n and "?n" for numerator of n and "_n" for floor(n) and "`n" for ceiling(n) and 'n for gaussian_rounding(n) (also: "-n" for addition inverse of n and "/n" for multiplicative inverse of n), if the final result is negative number or noninteger rational number, then factordb return "Error: Negative" or "Error: Not divisible" (Note: For expression m^n, n can be negative number, but n cannot be noninteger number, since the result of m^n will be transcendental number)

the [URL="http://factordb.com/sequences.php"]sequence[/URL] in factordb can add the problem in [URL="https://oeis.org/A195264"]https://oeis.org/A195264[/URL] and [URL="https://oeis.org/A316941"]https://oeis.org/A316941[/URL] (like the home prime problem [URL="https://oeis.org/A037274"]https://oeis.org/A037274[/URL]), also the sequences in [URL="http://www.aliquotes.com/autres_processus_iteratifs.html"]http://www.aliquotes.com/autres_processus_iteratifs.html[/URL] (n --> sigma(n)-n+k for fixed constant k>=-1) (the original Aliquot sequence is n --> sigma(n)-n, i.e. k=0) (the k=-1 case can be called "quasi-Aliquot sequence", like "quasiperfect number" is n=sigma(n)-n-1 and "quasi-amicable pair" is betrothed pair, such as 48 and 75; also the k=1 case can be called "almost Aliquot sequence", like "almost perfect number" is n=sigma(n)-n+1)

Also, the home prime problem and inverse home prime problem can include all bases 2<=b<=36

Also I think that the sequence "Greatest prime factor ^2+1" is silly, as the sequences never terminate at a prime unless the start number is power of 2, also "Greatest prime factor ^2-1", "Greatest prime factor ^3+1", etc.

I use those "Greatest Prime Factor" sequences (mostly GPF^3+1 and GPF^3-1, but I've also run the q^2-1 sequences of all factored RSA numbers) to see if they ever end in a cycle, so we can't get rid of those.

Silvester/Euclid sequence in factordb is repeatedly x^2-x+1 until reaching a prime, I think the polynomial "x^2-x+1" can be generalized to any irreducible polynomial with positive leading coefficient (can be added to factordb, users may enter the polynomial and the start number), e.g. 2*x+1, to get the [URL="https://oeis.org/A050412"]Riesel problem[/URL], or 2*x-1, to get the Sierpinski problem, etc.

Silvester/Euclid sequence (and the sequences which I suggest in the previous post) should not stop when the number is not fully factored (e.g. [URL="http://factordb.com/sequences.php?se=28&aq=20&action=last20&fr=0&to=100"]n=20[/URL]), and I think the sequence in factordb can add the generalization of [URL="https://oeis.org/A000945"]A000945[/URL] and [URL="https://oeis.org/A000946"]A000946[/URL] starting with given number n, e.g. A000945 starting with 12 is 12, 13, 157, 7, 5, 857221, 734826985621, 67, 36178036819339404422591341, 619, 196501174273319, ..., and A000946 starting with 12 is 12, 13, 157, 3499, 85697509, 1025850392947, 4054085938915939, 229936221387049265462828227782402791669, ..., A000945 should stop only when the number has no known prime factor, but A000946 should stop when the number is not fully factored.

Also [URL="https://oeis.org/A005265"]A005265[/URL] and [URL="https://oeis.org/A005266"]A005266[/URL]