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# Calendar for Sequence of the Day in August

Template:Sequence of the Day for August 1

A025547: Least common multiples of the first ${\displaystyle \scriptstyle n\,}$ odd numbers.

 { 1, 3, 15, 105, 315, 3465, 45045, 45045, ... }

Repeated terms occur when ${\displaystyle \scriptstyle 2n-1\,}$ is divisible by smaller odd numbers and enough of the necessary prime factors have accumulated.

Template:Sequence of the Day for August 2

A007598: ${\displaystyle \scriptstyle {F_{n}}^{2}\,}$, squares of the Fibonacci numbers

 { 1, 1, 4, 9, 25, 64, 169, 441, ... }

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: ${\displaystyle \scriptstyle (F_{n})^{2}\,=\,F_{n-i}F_{n+i}+(F_{i})^{2}(-1)^{i+n}\,}$. But this is just scratching the tip of the iceberg.

 A143212: Row sums 1 1 1 1 2 2 2 4 8 3 3 6 9 21 5 5 10 15 25 60 8 8 16 24 40 64 160

Template:Sequence of the Day for August 3

A001248: Squares of prime numbers

{ 4, 9, 25, 49, 121, 169, 289, ... }

These are the only numbers with exactly three divisors (1, their square root, themselves). In doing the sieve of Eratosthenes, this sequence gives the first composite number that you'll cross off after you identify the ${\displaystyle \scriptstyle n\,}$th prime (that is, unless you like to cross off again the composite numbers that have already been crossed off for a smaller prime factor – e.g., for the third prime, 5, the first number to cross off is 25, because 10, 15 and 20 should have already been crossed off for 2, 3 and 2 respectively.

Template:Sequence of the Day for August 4

A181832: The strong phi-torial of n (i.e. product of the positive integers ${\displaystyle \scriptstyle k\,\leq \,n\,}$ that are strongly coprime to ${\displaystyle \scriptstyle n\,}$), ${\displaystyle \scriptstyle n\,\geq \,0\,}$.

 { 1, 1, 1, 1, 1, 3, 1, 20, 15, 35, 7, 36288, 35, ... }

Back on November 5, 2010, we had A001783, the phi-torial of ${\displaystyle \scriptstyle n\,}$, 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 phi-torial, one would think the next step would be to ask if the phi-torial of ${\displaystyle \scriptstyle n\,}$ is divisible by the divisorial of ${\displaystyle \scriptstyle n\,}$ (A007955). It's not. However, Peter Luschny, who later on added the sequence to the OEIS, noticed that the phi-torial of ${\displaystyle \scriptstyle n\,}$ is divisible by the divisorial of ${\displaystyle \scriptstyle n-1\,}$.

Template:Sequence of the Day for August 5

A033168: Longest arithmetic progression of primes with difference 210 and minimal initial term.

 { 199, 409, 619, 829, 1039, 1249, 1459, 1669, 1879, 2089 }

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

 { 0, 1, 2, 3, 10, 11, 12, 13, ... }

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.

 { 1, 1, 3, 4, 8, 11, 21, 29, 55, 76, 144, 199, 377, 521, 987, 1364, 2584, 3571, 6765, 9349, 17711, 24476, ... }

This sequence, on the face of it, is just another linear recurrence relation, e.g.

${\displaystyle a(n)=3~a(n-2)-a(n-4),\,n\geq 4;\;}$
${\displaystyle a(0)=1,\,a(1)=1,\,a(2)=3,\,a(3)=4.\,}$

It appears (proof?) that A109794(${\displaystyle \scriptstyle n\,}$) = A189761(${\displaystyle \scriptstyle n+2\,}$), ${\displaystyle \scriptstyle n\,\geq \,4,\,}$ where A189761 gives numbers ${\displaystyle \scriptstyle n\,}$ for which the set of residues ${\displaystyle \scriptstyle \{{\rm {Fibonacci}}(k){\bmod {n}},\,k\,=\,0,\,1,\,2,\,\ldots \}\,}$ is minimal, i.e. ${\displaystyle \scriptstyle n\,>\,m\,\implies \,\,}$ A066853${\displaystyle \scriptstyle (n)\,}$ > A066853${\displaystyle \scriptstyle (m)\,}$. It is also conjectured that the members of the sequence, for ${\displaystyle \scriptstyle n\,\geq \,4\,}$, are just those numbers for which the Pisano period is minimal: that is, conjecturally, A001175(${\displaystyle \scriptstyle n\,}$) > A001175(${\displaystyle \scriptstyle m\,}$) for all ${\displaystyle \scriptstyle n\,>\,m\,}$ iff ${\displaystyle \scriptstyle m\,}$ is in this sequence.

Template:Sequence of the Day for August 8

A115369: Decimal expansion of first zero of the Bessel function ${\displaystyle \scriptstyle J_{1}(z)\,}$

 3.8317059702075...

According to Eric Weisstein, this is also the first zero of ${\displaystyle \scriptstyle J_{0}^{\prime }(z)\,}$ 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 ${\displaystyle \scriptstyle n\,}$ smaller primes as substrings of its base 10 representation

 { 2, 13, 23, 113, 137, 1237, 1733, ... }

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 ${\displaystyle \scriptstyle n\,=\,0\,}$. 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).

 { 1913, 18379, 19013, 25013, 34613, 35617, 35879, ... }

The 293rd prime is 1913. The 294th 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 k-tuple: ${\displaystyle \scriptstyle k\,}$ 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 ${\displaystyle \scriptstyle \pi (Op_{1})\,=\,\pi (Op_{2})-1\,}$. Thus, 179 and 197 don't count as an Ormiston pair because ${\displaystyle \scriptstyle \pi (179)\,=\,41\,}$ and ${\displaystyle \scriptstyle \pi (197)\,=\,45\,}$.

See:

Template:Sequence of the Day for August 11

A001020: ${\displaystyle \scriptstyle 11^{n},\,n\,\geq \,0,\,}$

 { 1, 11, 121, 1331, 14641, 161051, 1771561, ... }

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 ${\displaystyle \scriptstyle n\,}$ = 0 to 4 you get numbers which are concatenated single digit (base 10) binomial coefficients, while for ${\displaystyle \scriptstyle n\,}$ ≥ 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 ${\displaystyle \scriptstyle x\,=\,10\,}$ and ${\displaystyle \scriptstyle y\,=\,1\,}$ in

${\displaystyle (x+y)^{n}=\sum _{k=0}^{n}{\binom {n}{k}}x^{k}~y^{n-k}\,}$

explains why this is so.

A096884 ${\displaystyle \scriptstyle 101^{n},\,n\,\geq \,0.\,}$
A097659 ${\displaystyle \scriptstyle 1001^{n},\,n\,\geq \,0.\,}$

Template:Sequence of the Day for August 12

A048855: ${\displaystyle \scriptstyle \phi (n!)\,}$, the totient function of the factorial of ${\displaystyle \scriptstyle n\,}$.

 { ... 1, 2, 8, 32, 192, 1152, 9216, ... }

These are the numbers of integers less than ${\displaystyle \scriptstyle n!\,}$ that are coprime to said factorial (I choose to start the listing here with ${\displaystyle \scriptstyle a_{2}\,}$). There's also a recurrence relation to explain these, which was found by Enoch Haga and goes by the offset shown here: ${\displaystyle \scriptstyle a_{1}\,=\,1\,}$, then, if ${\displaystyle \scriptstyle n\,}$ is prime, ${\displaystyle \scriptstyle a_{n}\,=\,(n-1)\,a_{n-1}\,}$, otherwise ${\displaystyle \scriptstyle a_{n}\,=\,n\,a_{n-1}\,}$.

Also note how small these numbers are compared to the factorials. For example, 20! = 2432902008176640000, while ${\displaystyle \phi (20!)}$ is just 416084687585280000, less than a fifth of 20!

Template:Sequence of the Day for August 13

A027606: Natural logarithm base ${\displaystyle \scriptstyle e\,}$, i.e. Euler's number, in the duodecimal numeral system.

 2.8752360698219BA71971...

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 non-attacking white and black rook on ${\displaystyle \scriptstyle n{\rm {\,X\,}}n\,}$ chessboard.

 { 4, 36, 144, 400, 900, 1764, 3136, ... }

As it happens, this sequence has a very simple formula: ${\displaystyle \scriptstyle n^{2}(n-1)^{2}\,=\,(n(n-1))^{2}\,=\,\{(n)_{2}\}^{2}\,}$, the product of two consecutive square numbers, the square of oblong numbers or the square of the falling factorial ${\displaystyle \scriptstyle (n)_{2}\,}$.

This is the number of ways of placing two objects on an ${\displaystyle \scriptstyle n{\rm {\,X\,}}n\,}$ grid so that they don't share a row or a column. Now, the number of ways of placing ${\displaystyle \scriptstyle k\,}$ objects on an ${\displaystyle \scriptstyle n{\rm {\,X\,}}n\,}$ grid so that they don't share a row or a column is ${\displaystyle \scriptstyle \{(n)_{k}\}^{2}\,}$ (e.g. ${\displaystyle \scriptstyle \{(n)_{3}\}^{2}\,=\,(n(n-1)(n-2))^{2}\,}$ for 3 objects.) And then for an ${\displaystyle \scriptstyle n{\rm {\,X\,}}n{\rm {\,X\,}}n\,}$ grid, the number of ways of placing ${\displaystyle \scriptstyle k\,}$ objects so that they don't share a coordinate is ${\displaystyle \scriptstyle \{(n)_{k}\}^{3}\,}$ 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 ${\displaystyle \scriptstyle B_{2}\,}$: ${\displaystyle \scriptstyle \sum {\big (}{\frac {1}{p}}+{\frac {1}{p+2}}{\big )}\,}$ as ${\displaystyle \scriptstyle (p,\,p+2)\,}$ runs through the twin prime pairs

${\displaystyle {\bigg (}{\frac {1}{3}}+{\frac {1}{5}}{\bigg )}+{\bigg (}{\frac {1}{5}}+{\frac {1}{7}}{\bigg )}+{\bigg (}{\frac {1}{11}}+{\frac {1}{13}}{\bigg )}+\ldots \,}$

where the first twin prime pair is the only one that is not of the form ${\displaystyle \scriptstyle (6k-1,6k+1)\,}$.

 1.902160583104...

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 1016, where ${\displaystyle \scriptstyle (\log(10^{16}))^{2}\,=\,13.0077\ldots \,}$. 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 10530 just to reach 1.9! (Sebah and Gourdon)

See:

Template:Sequence of the Day for August 16

A005910: Truncated octahedral numbers: ${\displaystyle \scriptstyle 16n^{3}-33n^{2}+24n-6,\,n\,\geq \,1.\,}$

 { 1, 38, 201, 586, 1289, 2406, 4033, ... }

You can obtain these by taking the octahedral number ${\displaystyle \scriptstyle {\rm {Oct}}_{3n-2}\,=\,{\frac {k(2k^{2}+1)}{3}}{\big |}_{3n-2}\,}$ and cutting off the [square] pyramidal number ${\displaystyle \scriptstyle {\rm {Pyr}}_{n-1}^{(4)}\,=\,{\frac {k(k+1)(2k+1)}{6}}{\big |}_{n-1}\,}$ from each of the six vertices, i.e.

${\displaystyle 16n^{3}-33n^{2}+24n-6=\,}$
${\displaystyle {\rm {Oct}}_{3n-2}-6\cdot {\rm {Pyr}}_{n-1}^{(4)},\quad n\,\geq \,1.\,}$

(This description is thanks to Conway and Guy in their landmark Book of Numbers.)

Template:Sequence of the Day for August 17

A014549: Gauß's constant ${\displaystyle \scriptstyle {\frac {2}{\pi }}\int _{0}^{1}{\frac {1}{\sqrt {1-x^{4}}}}dx\,}$

 0.834626841674...

This is the reciprocal of the arithmetic-geometric mean of 1 and ${\displaystyle \scriptstyle {\sqrt {2}}\,}$. It was on May 30, 1799 that Carl Friedrich Gauß discovered the integral for this number shown above.

Its simple continued fraction is ${\displaystyle \scriptstyle {\frac {1}{{1}+{\frac {1}{{5}+{\frac {1}{{21}+{\frac {1}{\ddots }}}}}}}}\,}$ (see A053002).

Template:Sequence of the Day for August 18

A182809: Fibonacci numbers that are base 10 cyclops numbers.

 {0, 75025, 6557470319842, 14472334024676221, 99194853094755497, ... }

It's not often that a base-dependent 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 ${\displaystyle \scriptstyle F_{10^{7}}\,}$.

Template:Sequence of the Day for August 19

A065474: Decimal expansion of the Feller-Tornier constant ${\displaystyle \scriptstyle \prod _{i=1}^{\infty }\left(1-{\frac {2}{{p_{i}}^{2}}}\right)\,}$, where ${\displaystyle \scriptstyle p_{i}\,}$ is the ${\displaystyle \scriptstyle i\,}$th prime.

 0.322634098939...

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: ${\displaystyle \scriptstyle n\,}$ is in this sequence if and only if ${\displaystyle \scriptstyle n\,}$ is not in sequence ${\displaystyle \scriptstyle An\,}$ in the OEIS database

 { 4, 7, 9, 11, 12, 13, 15, ... }

The number 40 is definitely not a prime number, yet in the OEIS it is the A-number that indexes the prime numbers (see A000040). Since early on, contributors to the OEIS have been interested in this kind of question. Since A-numbers are generally assigned by computer, and the computer has no directive to try to choose an A-number that would be "appropriate" or "ironic" for a given sequence, it is almost a matter of luck whether a new sequence will get an A-number that is in the sequence itself.

Here's two puzzles for the day:

1. Should the number 51369 be in A053169? (Welcome to Russel's paradox!)
2. Should the number 0 be in A053169? (Daniel Forgues had suggested using A000000 for the empty sequence, and Neil Sloane was amenable to the idea, but since that sequence has no terms, the lookup programs would not be able to handle it.)

Template:Sequence of the Day for August 21

A080790: Binary emirps

 { 11, 13, 23, 29, 37, 41, 43, ... }

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 semiprimeth recurrence

 { 1, 4, 10, 26, 77, 235, 779, ... }

Just as with Wilson's primeth recurrence (see A007097), we start with 1 (even though it's neither prime nor semiprime) and for the following terms use the recurrence ${\displaystyle \scriptstyle a(n)\,=\,sp_{a(n-1)}\,}$ (the semiprime equivalent of Wilson's ${\displaystyle \scriptstyle a(n)\,=\,p_{a(n-1)}\,}$.) I wonder if in this sequence 4 is the only square of a prime? (Quickest way to check several terms is to look for ${\displaystyle \scriptstyle \mu (a(n))\,=\,0\,}$, provided of course one has several terms).

Template:Sequence of the Day for August 23

A092447: Concatenate odd primes in decreasing order

 { 3, 53, 753, 11753, 1311753, 171311753, 19171311753, ... }

Quite self-explanatory. 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.

3 (1 digit)
53 (2 digits)
171311753 (9 digits)
8983797371676159534743413731292319171311753 (43 digits)
a(5) starts with odd prime 383 (198 digits) (cf. A100003)
a(6) starts with odd prime 8831 (4202 digits) (cf. A100003)
a(7) starts with (odd prime > 4400-th prime) (more than 20000 digits) (cf. A100003)
I'm hinting that this sequence is infinite... but I have no clue how one could come up with a proof! — DF

Template:Sequence of the Day for August 24

A143212: Row sums of the Fibonacci multiplication table

 { 1, 2, 8, 21, 60, 160, 429, ... }

${\displaystyle \scriptstyle a(n)\,}$ is divisible by ${\displaystyle \scriptstyle F_{n}\,}$ since

${\displaystyle a(n)=F_{n}\sum _{i=1}^{n}F_{i}=F_{n}\,(F_{n+2}-1).\,}$

Template:Sequence of the Day for August 25

A002487: Stern's diatomic series

{0, 1, 1, 2, 1, 3, 2, 3, 1, 4, 3, 5, 2, 5, 3, 4, 1, 5, 4, 7, 3, 8, 5, 7, 2, 7, 5, 8, 3, 7, ... }

Stern's diatomic series is defined by he recursion ${\displaystyle a(2n)=a(n)}$, ${\displaystyle a(2n+1)=a(n)+a(n+1)}$ with base cases ${\displaystyle a(0)=0}$, ${\displaystyle a(1)=1}$.

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 Stern-Brocot tree.

Template:Sequence of the Day for August 26

A064510: Numbers ${\displaystyle n}$ such that the sum of the first ${\displaystyle k}$ divisors of ${\displaystyle n}$ is equal to ${\displaystyle n}$ for some ${\displaystyle k}$.

{ 1, 6, 24, 28, 496, 2016, 8128, 8190, ... }

Obviously all perfect numbers are included in this sequence. But if you take the perfect numbers out, you have the Erdős-Nicolas numbers left.

Template:Sequence of the Day for August 27

A008884: Collatz trajectory starting at 27 (see Collatz problem)

{ 27, 82, 41, 124, 62, 31, 94, ... }

With the familiar Collatz function ${\displaystyle \scriptstyle f(n)\,=\,{\frac {n}{2}}\,}$ if ${\displaystyle \scriptstyle n\,}$ is even and ${\displaystyle \scriptstyle f(n)\,=\,3n+1\,}$ if ${\displaystyle \scriptstyle n\,}$ 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 b-file 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

A060295: Ramanujan's constant ${\displaystyle \scriptstyle e^{\pi {\sqrt {163}}}\,}$

 262537412640768743.999999999999250072597...

the fractional part of which has the simple continued fraction (A058292)

${\displaystyle {\cfrac {1}{{1}+{\cfrac {1}{{1333462407511}+{\cfrac {1}{{1}+\,\ddots }}}}}}\,}$

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 non-simple continued fraction read serially.

 0.48267728...

We verify that

${\displaystyle 0+{\cfrac {4}{{8}+{\cfrac {2}{{6}+{\cfrac {7}{{7}+\ddots }}}}}}=0.482677\ldots \,}$

M. Trott has discovered two other such constants. (How many such constants are there for base 10?)

Since the concept is base-dependent, one might wonder what similarly defined constants are for, e.g., base 2 (binary) or base ${\displaystyle \scriptstyle \phi \,}$ (phinary)!

Template:Sequence of the Day for August 30

A079397: Smallest prime "recalling" ${\displaystyle \scriptstyle n,\,n\,\geq \,0,\,}$ previous primes in its base 10 representation.

 { 2, 13, 23, 113, 137, 1237, 1733, ... }

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) S-box.

{ 1, 0, 8, 27, 55, 240, 701, ... }

This is the code of length 56 defined by taking a slice of four bytes and three bytes of successive AES round keys.