A suggestion for factordb.

[sweety439]

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

A      n! - (n-1)! + (n-2)! - ... 1!      A005165
B      Bell numbers      A000110
C      Catalan numbers      A000108
D      Distinct partition numbers      A000009
E      Euler zigzag numbers      A000111
F      Fermat numbers      A000215
G      Fubini numbers      A000670
H      H(m,n) = n-th m-Fibonacci numbers      A000045 A000129 A006190 A001076 ...
I      I(m,n) = n-th m-step Fibonacci numbers      A000045 A000073 A000078 A001591 ...
J      Coefficients of modular function j as power series in q = e^(2 Pi i t)      A000521
K      1! + 2! + 3! + ... + n!      A007489
L      L(m,n) = n-th m-Lucas numbers      A000032 A002203 A006497 A014448 ...
M      Mersenne numbers      A000225
N      N(m,n) = n!_(m) = m-factorial of n
O      O(m,n) = m-th cyclotomic polynomial evaluated at n
P      Partition numbers      A000041
Q      Perrin numbers      A001608
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      A000594
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      A000032 A001644 A073817 A074048 ...
Z      Motzkin numbers      A001006
!      Factorial
#      Primorial
%      Modulo operator
\      Integer division
@      n@ = n-th prime
&      Narayana's cows sequence      A000930
$      Padovan sequence      A000931
:      :(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))

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)

[sweety439]

the sequence in factordb can add the problem in https://oeis.org/A195264 and https://oeis.org/A316941 (like the home prime problem https://oeis.org/A037274), also the sequences in http://www.aliquotes.com/autres_proc...iteratifs.html (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.

[Stargate38]

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.

[sweety439]

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 Riesel problem, or 2*x-1, to get the Sierpinski problem, etc.

[sweety439]

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. n=20), and I think the sequence in factordb can add the generalization of A000945 and A000946 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 A005265 and A005266

[sweety439]

Also suggest to add:

* "unitary Aliquot sequence" (x --> A063919(x), see A003062, A063991, and A097030)
* "Aliquot-like sequence based on Dedekind psi function" (x --> A306927(x), see A323327)
* "exponential Aliquot sequence" (x --> A126164(x), see A126165 and A054979)
* "infinitary Aliquot sequence" (x --> A126168(x), see A126169)
* "bi-unitary Aliquot sequence" (x --> A331970(x), see A292981)
* "totient Aliquot sequence" (x --> A092693(x), see A286233 for "totient amicable numbers" and see A343243 for "totient sociable numbers")

These sequences can also use the sense of http://www.aliquotes.com/autres_proc...iteratifs.html (x --> A001065(x)+k with fixed integer k) to generate, i.e. x --> A063919(x)+k, x --> A306927(x)+k, etc.)

Besides, I think that factordb can include all numbers <=25 digits in database without new ID (currently factordb only include all numbers <=18 digits in database without new ID), also, factordb only show the first 10 digits and the last 2 digits for numbers >50 digits, I think that factordb can show the first 10 digits and the last 10 digits for numbers >50 digits.

[sweety439]

Does factordb use Baillie–PSW primality test to test the PRPs? Also, it is known that if generalized Riemann hypothesis is true, then all numbers n which passed the Miller Rabin primality test to all bases b<(ln n)^4 are primes, so I suggest factordb can also run this test and add a status "GRHP", i.e. if generalized Riemann hypothesis is true, then this number is definitely prime.

[MattcAnderson]

Hi again,

I sent an email to the guy in charge of Factordb.com (Markus), asking him how he feels about batch submissions of prime integer factorizations of positive integers.

Markus Tervooren (markustervooren [a] web.de) did not reply to me, and I do not plan on emailing him again. He is probably a busy guy. I feel factordb.com is a valuable service. A few people want to look up prime integer factroizations, and do not have computers set up to do it themselves.

If anyone else wants to contact Markus, that is okay.

Regards,
Matt

[Dr Sardonicus]

Originally Posted by sweety439

<snip>
Also, it is known that if generalized Riemann hypothesis is true, then all numbers n which passed the Miller Rabin primality test to all bases b<(ln n)^4 are primes,
<snip>

There's a much better conditional bound assuming GRH. 2*(log(n))² has been known for thirty years. It was proven in

E. Bach, Explicit bounds for primality testing and related problems, Math. Comp. 55 (1990), 355–380

Using Bach's bound for n around 10¹⁰⁰⁰, it would require over 700000 Miller-Rabin tests. So for large numbers, it would be a lot of bucks for not much bang.

EDIT: The number 700000 is a gross understatement. It's roughly the number of prime bases less than Bach's bound. That would be fine for the simple Fermat test.

With Rabin-Miller, though, it's not enough. It is possible to have n "pass" the test with prime bases p and q, but not to the base p*q. This can happen as follows: Suppose m = (n-1)/2^k is even,

p^2m == q^2m == 1 (mod n), and

p^m == q^m == -1 (mod n).

Then (pq)^m == 1 (mod n).

Note, however, that [using the hypothesis that m is even] p^m/2 and q^m/2 are square roots of -1 (mod n). If they are neither equal nor equal and opposite, then n is composite (more than two square roots of -1 (mod n)), and p*q is a "witness."

[sweety439]

Nice news!!!

I found a way to add many numbers in a sequence to factordb by one click!!!

First, enter a non-complete factored number (say the Fermat number F12)

Second, copy the b-file in OEIS (for the sequence you want to add) and paste the full b-file to the "report factors" box.

Then all numbers in this b-file will be added!!!

I used this way to add all numbers in these sequences:

A000110 Bell numbers (I copied this a-file instead of the b-file, since this a-file has 3016 terms (a(0) to a(3015))

A000111 Euler zigzag numbers

A000670 Fubini numbers

A001006 Motzkin numbers

A000041 partition numbers (but all numbers in this sequence seems to be already in factordb ....)

A000009 distinct partition numbers

A003422 left factorial numbers (since the b-file has only 100 terms, thus I used PARI/GP to compute more terms and copied them to factordb)

A005165 alternating factorial numbers (since the b-file has only 100 terms, thus I used PARI/GP to compute more terms and copied them to factordb)

A000108 Catalan numbers (although I know that this numbers are trivially 100% factored, like the factorial numbers and the primorial numbers)

A000105 polyomino numbers

A000602 alkane numbers

A000073 tribonacci numbers

A000078 tetranacci numbers

A000129 Pell numbers

A006190 3-Fibonacci numbers

A002315 NSW numbers

A000931 Padovan numbers

A000930 Narayana numbers

A001608 Perrin numbers

A001008 1-Wolstenholme numbers

A007406 2-Wolstenholme numbers

A000521 elliptic modular invariant J(tau)

A000594 Ramanujan's tau numbers (the most exciting!!! The b-file has 16090 terms (thus of course, after I added them, I get "You have reached your hourly limit of 6000 IDs created.", but you must remove the "-" for the negative numbers in the b-file, since factordb does not support negative numbers and not add their absolute values)

[Batalov]

Originally Posted by sweety439

... and paste the full b-file to the "report factors" box.

Congratulations, Captain Obvious!
It only took you a year of complaints and 99+ messages with "helpful suggestions" to finally find out that this was an obvious function of FactorDB? That is impressive!