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Calendar for Sequence of the Day in August
Calendar for Sequence of the Day in July * Calendar for Sequence of the Day in September
Template:Sequence of the Day for August 1 A025547: Least common multiples of the first odd numbers.
Repeated terms occur when is divisible by smaller odd numbers and enough of the necessary prime factors have accumulated.

Template:Sequence of the Day for August 2 A007598: , squares of the Fibonacci numbers
In the Fibonacci multiplication table, this sequence is in a very predictable diagonal. But closer inspection of the table yields an interesting formula for this sequence: . But this is just scratching the tip of the iceberg.
 
Template:Sequence of the Day for August 3 A001248: Squares of prime numbers

Template:Sequence of the Day for August 4 A181832: The strong phitorial of n (i.e. product of the positive integers that are strongly coprime to ), .
Back on November 5, 2010, we had A001783, the phitorial of , for Sequence of the Day. This led to the discovery of another one of those sequences that make us wonder "Why wasn't this added to the OEIS sooner?" On pondering the phitorial, one would think the next step would be to ask if the phitorial of is divisible by the divisorial of (A007955). It's not. However, Peter Luschny, who later on added the sequence to the OEIS, noticed that the phitorial of is divisible by the divisorial of .

Template:Sequence of the Day for August 5 A033168: Longest arithmetic progression of primes with difference 210 and minimal initial term.
These prime numbers are each 210 away from the previous and the next prime in the sequence. Note that in this sequence the primes are not required to be consecutive, a sequence with that extra condition being referred as an arithmetic progression of consecutive primes.

Template:Sequence of the Day for August 6 A061909: Base 10 skinny numbers
When these numbers are squared using long multiplication, it is completely unnecessary to carry anything. For example, 13 13  39 13  169

Template:Sequence of the Day for August 7 A109794: a(2n) = A001906(n+1), a(2n+1) = A002878(n). Bisection of Fibonacci numbers interleaved with bisection of Lucas numbers.
This sequence, on the face of it, is just another linear recurrence relation, e.g. It appears (proof?) that A109794() = A189761(), where A189761 gives numbers for which the set of residues is minimal, i.e. A066853 > A066853. It is also conjectured that the members of the sequence, for , are just those numbers for which the Pisano period is minimal: that is, conjecturally, A001175() > A001175() for all iff is in this sequence.

Template:Sequence of the Day for August 8 A115369: Decimal expansion of first zero of the Bessel function
According to Eric Weisstein, this is also the first zero of rather than 0 as given by Abramowitz and Stegun in their landmark Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Mathematica also gives this constant rather than 0 as the first zero of this function. See Weisstein, Eric W., Bessel Function of the First Kind, from MathWorld—A Wolfram Web Resource. [http://mathworld.wolfram.com/BesselFunctionoftheFirstKind.html]

Template:Sequence of the Day for August 9 A079397: Smallest prime containing smaller primes as substrings of its base 10 representation
Since 2 is the smallest prime and it contains no smaller primes as substrings of its base 10 representation (or any other standard positional base, for that matter), it is first in this list, corresponding to . Next, 13 contains 3 as a substring, but we no longer consider 1 prime. The substrings can overlap, as is the case with 113, containing as it does 11, 13 and 3 as substrings. The smallest prime in this list to contain a 9 is 23719.
 
Template:Sequence of the Day for August 10 A069567: Smaller of two consecutive primes which are anagrams of each other (base 10).
The 293^{rd} prime is 1913. The 294^{th} prime is 1931. It just so happens that both of these are anagrams of their base 10 digits, namely: two 1s, a 3 and a 9. These pairs of primes are called Ormiston pairs or rearrangement prime pairs. (An Ormiston pair can be generalized to an Ormiston ktuple: consecutive primes which are anagrams.) The relation of the prime counting function is important; it is not enough for the two primes to be anagrams, it must also be the case that . Thus, 179 and 197 don't count as an Ormiston pair because and . See:

Template:Sequence of the Day for August 11
In the classic Star Trek episode "The Trouble with Tribbles," Spock estimates the growth of the Tribble population aboard the Enterprise thus: "1,771,561. That's assuming one tribble, multiplying with an average litter of 10, producing a new generation every 12 hours over a period of three days." Note that for = 0 to 4 you get numbers which are concatenated single digit (base 10) binomial coefficients, while for ≥ 5 some binomial coefficients are now more than single digit (base 10) and thus overlap, e.g. 1 5 10 10 5 1  161051 Setting and in explains why this is so. See also:

Template:Sequence of the Day for August 12 A048855: , the totient function of the factorial of .
These are the numbers of integers less than that are coprime to said factorial (I choose to start the listing here with ). There's also a recurrence relation to explain these, which was found by Enoch Haga and goes by the offset shown here: , then, if is prime, , otherwise . Also note how small these numbers are compared to the factorials. For example, 20! = 2432902008176640000, while is just 416084687585280000, less than a fifth of 20!

Template:Sequence of the Day for August 13 A027606: Natural logarithm base , i.e. Euler's number, in the duodecimal numeral system.
This familiar number is of course in decimal 2.7182818284590452354...

Template:Sequence of the Day for August 14 A035287: Number of ways to place a nonattacking white and black rook on chessboard.
As it happens, this sequence has a very simple formula: , the product of two consecutive square numbers, the square of oblong numbers or the square of the falling factorial . This is the number of ways of placing two objects on an grid so that they don't share a row or a column. Now, the number of ways of placing objects on an grid so that they don't share a row or a column is (e.g. for 3 objects.) And then for an grid, the number of ways of placing objects so that they don't share a coordinate is which generalizes to higher dimensions... (Cf. {{(x)_n}}) for the falling factorial function template)

Template:Sequence of the Day for August 15 A065421: Decimal expansion of the twin primes Brun's constant : as runs through the twin prime pairs where the first twin prime pair is the only one that is not of the form .
For some constants, we can give thousands or even millions of decimal places. And for some constants we can barely give a dozen places, if that. Today's Sequence of the Day is an example of the latter, since it converges extremely slowly. For the few places that we do know, we have at least three different people to thank: Robert G. Wilson v, Neil Sloane and Pascal Sebah. It seems (is that the case?) that the number of decimal places obtained is about the square of the natural logarithm of the upper bound of the range for which we consider the twin prime pairs. For example, the above 13 decimal places have been obtained by considering all twin prime pairs up to 10^{16}, where . Also note that those 13 decimal places where obtained by a clever extrapolation method (which assumes the truth of the twin prime conjecture), whereas using direct estimation we would have to go up to 10^{530} just to reach 1.9! (Sebah and Gourdon) See:

Template:Sequence of the Day for August 16 A005910: Truncated octahedral numbers:
You can obtain these by taking the octahedral number and cutting off the [square] pyramidal number from each of the six vertices, i.e. (This description is thanks to Conway and Guy in their landmark Book of Numbers.)
 
Template:Sequence of the Day for August 17
This is the reciprocal of the arithmeticgeometric mean of 1 and . It was on May 30, 1799 that Carl Friedrich Gauß discovered the integral for this number shown above. Its simple continued fraction is (see A053002).

Template:Sequence of the Day for August 18 A182809: Fibonacci numbers that are base 10 cyclops numbers.
It's not often that a basedependent integer sequence (Keyword: base) piques my interest. When I first saw it, my first thoughts were "Keywords: Base and hard and more? We'll see about that." I thought a quick and dirty Mathematica program would prove that more terms are easy to come by and the keywords hard and more are totally unwarranted. Well, I was wrong. Indeed, a little reflection and commonsense will show that the likelihood that Fibonacci numbers with more and more digits will have only one instance of the digit 0, and that that digit will be precisely in the middle, is increasingly improbable. And yet, commonsense is insufficient to prove with mathematical certainty that this sequence is finite and given in full. The most that we can say for now is that there are no more terms less than .

Template:Sequence of the Day for August 19 A065474: Decimal expansion of the FellerTornier constant , where is the ^{th} prime.
This number is the density of squarefree numbers such that their successors are also squarefree (see A007674).

Template:Sequence of the Day for August 20 A053169: is in this sequence if and only if is not in sequence in the OEIS database
The number 40 is definitely not a prime number, yet in the OEIS it is the Anumber that indexes the prime numbers (see A000040). Since early on, contributors to the OEIS have been interested in this kind of question. Since Anumbers are generally assigned by computer, and the computer has no directive to try to choose an Anumber that would be "appropriate" or "ironic" for a given sequence, it is almost a matter of luck whether a new sequence will get an Anumber that is in the sequence itself. Here's two puzzles for the day:
See OnLine Encyclopedia of Integer Sequences#Selfreferential sequences—Wikipedia.org.

Template:Sequence of the Day for August 21
Looking at these in the binary numeral system, we see that if we read them backwards, we obtain different prime numbers.

Template:Sequence of the Day for August 22 A105999: Vos Post's semiprime^{th} recurrence
Just as with Wilson's prime^{th} recurrence (see A007097), we start with 1 (even though it's neither prime nor semiprime) and for the following terms use the recurrence (the semiprime equivalent of Wilson's .) I wonder if in this sequence 4 is the only square of a prime? (Quickest way to check several terms is to look for , provided of course one has several terms).

Template:Sequence of the Day for August 23 A092447: Concatenate odd primes in decreasing order
Quite selfexplanatory. Observe that 3, 53 and 171311753 are odd prime "concatenation of odd primes in decreasing order," are there any more? Yes, there are more: A092448 Primes in A092447.
 
Template:Sequence of the Day for August 24 A143212: Row sums of the Fibonacci multiplication table
is divisible by since

Template:Sequence of the Day for August 25 A002487: Stern's diatomic series
Stern's diatomic series is defined by he recursion , with base cases , . It magically gives, by the ratio of any two successive entries, all positive rationals exactly once. It is essentially a breadth first traversal of the ratios of the nodes in the SternBrocot tree.

Template:Sequence of the Day for August 26 A064510: Numbers such that the sum of the first divisors of is equal to for some .
Obviously all perfect numbers are included in this sequence. But if you take the perfect numbers out, you have the ErdősNicolas numbers left.

Template:Sequence of the Day for August 27 A008884: Collatz trajectory starting at 27 (see Collatz problem)
With the familiar Collatz function if is even and if is odd, we still can't say for sure whether or not every starting value of the iterated function eventually reaches 1. Often we can't even look at a random number and be able to tell how long it takes to reach 1: some numbers reach 1 fairly quickly (numbers like 5 and 341), while others go on a long, wild ride. Such is the case with 27. Term visibility in the main OEIS entry caps this sequence at the 62nd iteration, which has gotten up to 1079 (having hit some even higher values earlier on). In the bfile we see that the sequence later on reaches 9232 at the 77th iteration, and finally hits 16 at the 107th. Kind of amazing, really, that it did not hit a higher power of two.

Template:Sequence of the Day for August 28
the fractional part of which has the simple continued fraction (A058292) The neat thing about this constant is that it is almost the integer 262537412640768744; indeed if an ordinary calculator could show just a few more decimal places than that it would eventually give up and just show this integer (the Mac OS X Calculator, for example, gives the answer as 262,537,412,640,000,000).

Template:Sequence of the Day for August 29 A113307: Trott's third constant: decimal expansion coincides with its nonsimple continued fraction read serially.
We verify that M. Trott has discovered two other such constants. (How many such constants are there for base 10?) Since the concept is basedependent, one might wonder what similarly defined constants are for, e.g., base 2 (binary) or base (phinary)!

Template:Sequence of the Day for August 30 A079397: Smallest prime "recalling" previous primes in its base 10 representation.
For example, 137, "recalls" four smaller primes, namely: 3, 7, 13, 37. The first prime, 2, "recalls" zero smaller primes (it does not "recall" any smaller primes).
 
Template:Sequence of the Day for August 31 A131620: Weight distribution of a certain binary linear code of length 56 defined by AES (or Rijndael) Sbox.
This is the code of length 56 defined by taking a slice of four bytes and three bytes of successive AES round keys.
