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2013-09-21

Random 100 sequences from the OEIS---a survey.

Summary: there were 16 holonomic, 16 prime, 11 digital, 7 constants, 4 arbitrary, 28 number theoretic, 13 combinatorial, four group theoretical, and one physics sequences in a random sample of 100 sequences from the OEIS.

The field I'm feeling most natural is mathematics, and I think my most successful work is associated with the OEIS database of integer sequences which sparked all my papers so far. To get an impression on what type of OEIS entries there are, I decided to work on a random sample of one hundred of them, and try to classify them.

So, let's get a random sample. Welcome to a hundred random numbers between 1 and 229000:
? for(i=1,100,print1(random(229000),","))
• First, there are polynomials in n, linear recurrences with constant coefficients (or lin-recs as the editors call them frequently), and other holonomic sequences. This is basic stuff, although not completely uninteresting. Many at first really interesting sequences later turn out to be of this type: A004315, A005056, A009671, A012845, A013081, A029920, A070358, A107396, A109794, A132200, A133886, A135493, A140405, A175485, A193931, A213036
• Then, the sequences involving primes. In my personal opinion most such sequences are random (no formulae possible), and you can't say much about them in terms of conjectures, although they may not be unimportant to have in the database: A003631, A007996, A013637, A022465, A045467, A066520, A086762, A088592, A090725, A100669, A105998, A118812, A120853, A122413, A142247, A188754
• Sequences involving decimal, and other digits: A034967, A037914, A053974, A061958, A075009, A092995, A095827, A102120, A117860, A141063, A209859
• A certain amount of OEIS entries are decimal expansions of constants. The justification to include them is the benefit for inverse calculations, and as a point where to collect statements and references about the respective constant: A088543, A153205, A154167, A196505, A196758, A198565, A201848
• Some sequences are so arbitrary that, although they could be interesting, it would be better to look at a definition or formula with small constants first and generalize from that. If the submitter gives no reason for the importance of such an arbitrary sequence, it is most likely unimportant. I found the following that fit this description: A030835, A040566, A152339, A182771
Now, the rest is what many OEIS editors agree to be interesting.
The really interesting sequences can be divided according to the field of mathematics they arise in, so let me list them so grouped. From here I will give one-liner definitions and make them clickable.

Number theory
A002547 Numerator of {n-th harmonic number H(n) divided by (n+1)}.
A004618 Divisible only by primes congruent to 4 mod 5.
A033831 Number of d dividing n such that d>=3 and 1<=n/d<=d-2.
A049384 a(0)=1, a(n+1) = (n+1)^a(n).
A060553 Symmetric patterns in the cellular automaton that generates Pascal's triangle modulo 2.
A064031 Product of non-unitary divisors of n!.
Distinct lines through the origin in n-dimensional cube of side length n.
A088138 Generalized Gaussian Fibonacci integers.
A088303 Smallest integer value of n!/ ( 1!a!b!c!...) ...
A089552 Sum of legs of primitive Pythagorean triangles having legs that add up to a square, sorted on hypotenuse.
A094234 Period of terms in continued fraction expansion of 2^n*tanh(1).
A117658 Number of solutions to x^(k+1)=x^k mod n for some k>=1.
A120615 sum(k=0,n,floor(phi*floor(n/phi))) where phi=(1+sqrt(5))/2.
A139799 n>=2 such that there is an integer k>1 with k divides n and k divides (n/k)+1.
A140418 Position of cubes in the EKG sequence.
A141321 Special sum of divisors of n.
A152066 Coefficients of certain polynomials.
A160394 Numbers n = p*q*r (p, q, r prime) congruent to 0 mod p+q+r.
A172819 Number of n X 9 0..4 arrays with row sums 9 and column sums n.
A173931 Primitive numbers k such that m/k is in the Cantor set for some m.
A178272 Number of collinear point 7-tuples in an n X n .. X n 4-dimensional cubical grid.
A178535 Matrix inverse of A178534.
A185383 Denominator of the fraction |n^2/A049417(n)-A064380(n)|.
A189675 Composition of Catalan and Fibonacci numbers.
A200521 Numbers n such that omega(n)=4 but bigomega(n)>4.
A218335 Even numbers n such that the largest value in trajectory of n under the juggler map is greater than n.
A227128 The twisted Euler phi-function for the non-principal Dirichlet character mod 3.
A227434 Value of row n in Pascal's triangle mod 3 seen as ternary number.

Enumerative combinatorics
A028461 Number of perfect matchings in graph P_{3} X C_{4} X P_{n}.
A057545 Maximum cycle size in range...
A124419 Number of partitions of the set {1,2,...n} having no blocks that contain both odd and even entries.
A135493 Number of ways to toss a coin n times and not get a run of six.
A149516 Number of walks within N^3 (the first octant of Z^3) starting...
A183882 Number of arrangements of n+2 numbers in 0..7 with ...
A185334 Number of not necessarily connected 3-regular simple graphs on 2n vertices with girth at least 4.
A186764 Permutations of {1,2,...,n} having k increasing even cycles.
A207224 Number of nX4 0..2 arrays avoiding the patterns ...
A208545 Number of 7-bead necklaces of n colors allowing reversal, with no adjacent beads having the same color.
A211359 Noncrossing partitions up to rotation and reflection of an n-set that contain k singleton blocks.
A214130 Partitions of n into parts congruent to +-2, +-3 (mod 13).
A227189 (k+1)-th part of the unordered partition which has been encoded in the binary expansion of n.

Group theory
A019537 Number of special orbits for dihedral group of degree n.
A057743 Maximal order of element of alternating group A_{2n+1}.
A170263 Number of reduced words of length n in Coxeter group on 14 generators
A214464 Degrees of irreducible representations of Suzuki group Sz(32).

Mathematical physics
A008199 Coordination sequence T4 for Zeolite Code MTW.

So, now you have a pretty good overview of what kind of OEIS entries exist, and what OEIS editors think are interesting submissions. If such pearls as the above can be found in a random sample of 100, what treasures might lurk there in the whole thing? Look for yourself!