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# Calendar for Sequence of the Day in September

Template:Sequence of the Day for September 1
 0.862240125868...

This is the binary number obtained by concatenating (after the "binary point") the binary representations of the positive integers in order: 1, 10, 11, 100, 101, 110, 111, 1000, and so on... as (0.1101110010111011110001...)2 which is then converted to base 10.

Template:Sequence of the Day for September 2

A014222: ${\displaystyle \scriptstyle a(0)\,=\,0\,}$, else ${\displaystyle \scriptstyle a(n)\,=\,3^{a(n-1)}\,=\,3\uparrow \uparrow \scriptstyle (n-1)\,}$ where ${\displaystyle \uparrow }$ is Knuth's (1976) arrow notation

 { 0, 1, 3, 27, 7625597484987 }

where 7625597484987 is the largest writeable exponent in Graham's number.

On this date 66 years ago WWII came to an abrupt end, and it was technology that played a crucial role in determining its outcome. Major developments in technology were made in weaponry, logistical support, medicine, industry, communications, and intelligence. To some extent these developments were the result of a quest for powerful computers, giving rise to Colossus, Mark 2, Zuse Z4, and ENIAC.

Today, integer sequences of tetrations (like this one) are nearly impossible to extend very far without powerful modern computers!

Template:Sequence of the Day for September 3

A064413: EKG sequence or electrocardiogram sequence: ${\displaystyle \scriptstyle a(1)\,=\,1\,}$; ${\displaystyle \scriptstyle a(2)\,=\,2}$; for ${\displaystyle \scriptstyle n\,>\,2,\,a(n)\,}$ is the smallest positive integer not already used which shares a factor with ${\displaystyle \scriptstyle a(n-1)\,}$.

 { 1, 2, 4, 6, 3, 9, 12, 8, 10, 5, 15, 18, 14, 7, 21, 24, 16, 20, 22, 11, 33, ... }

When plotted as a connect-the-dots plot (see Weisstein, Eric W., EKG Sequence, from MathWorld—A Wolfram Web Resource. [http://mathworld.wolfram.com/EKGSequence.html]) the sequence looks somewhat like an electrocardiogram (abbreviated "EKG" in medical circles), so this sequence became known as the EKG sequence. Every positive integer appears exactly once: this is a permutation of the positive integers. At the same time, the primes appear in ascending order. This sequence was discovered by Jonathan Ayres in 2001.

Template:Sequence of the Day for September 4

A010887: ${\displaystyle \scriptstyle 1\,+\,(n{\bmod {8}})\,}$

 { 1, 2, 3, 4, 5, 6, 7, 8, 1, 2, 3, 4, 5, 6, 7, 8, 1, 2, 3, 4, 5, 6, 7, 8, 1, 2, 3, ... }

Musicians sometimes joke that the only mathematical knowledge they need is to count to 4. However, in the case of Anton Bruckner (born September 4, 1824), he could and did often count to 8. Many of Bruckner's manuscripts, from sketches to full scores, contain his "metrical numbers," which show he thought of his compositions as being built up of 8-measure phrases. In part because of these "metrical numbers," musicologists can reconstruct Bruckner's incomplete Ninth Symphony with greater certainty than is possible with the famous unfinished Tenths of Beethoven and Mahler.

Template:Sequence of the Day for September 5

A059396: Number of primes less than ${\displaystyle \scriptstyle {\sqrt {p_{n}}}\,}$.

 { 0, 0, 1, 1, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 4, ... }

This is the number of trial divisions by smaller primes to show that the ${\displaystyle \scriptstyle n\,}$th prime is indeed prime. For example, ${\displaystyle a(32)=5}$ since the 32nd prime is 131 which is not divisible by 2, 3, 5, 7 or 11 (and does not need to be tested against 13, 17, 19 etc. since ${\displaystyle 13^{2}=169>131}$).

Template:Sequence of the Day for September 6

A066888: Number of primes between the ${\displaystyle \scriptstyle n\,}$th triangular number (exclusive) and the next (inclusive): ${\displaystyle \scriptstyle \pi (T_{n+1})\,-\,\pi (T_{n}),\,n\,\geq \,0.\,}$

 { 0, 2, 1, 1, 2, 2, 1, 2, 3, 2, ... }
 Triangle begins 1 (0 prime in (T_0, T_1]) 2 3 (2 primes in (T_1, T_2]) 4 5 6 (1 prime in (T_2, T_3]) 7 8 9 10 (1 prime in (T_3, T_4]) 11 12 13 14 15 (2 primes in (T_4, T_5]) 

It is believed that the initial zero of this sequence is the only zero in the sequence. That is to say, that there is always a prime in ${\displaystyle \scriptstyle (T_{n},\,T_{n+1}]\,}$ for ${\displaystyle \scriptstyle n\,>\,0\,}$. (Or there is always a positive noncomposite for ${\displaystyle \scriptstyle n\,\geq \,0\,}$, since 1 is a unit.) (With the exception of 1 and 3, all triangular numbers are composite, since ${\displaystyle \scriptstyle T_{n}\,=\,{\frac {n(n+1)}{2}}\,}$.)

Template:Sequence of the Day for September 7

A051802: Nonzero multiplicative digital root of ${\displaystyle \scriptstyle n,\,n\,\geq \,0.\,}$

 { 1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 2, 2, 4, 6, ... }

Here we ignore the zeros in the number when multiplying the digits. (Note that for zero we get the empty product, giving 1.)

Template:Sequence of the Day for September 8

A051250: Numbers whose reduced residue system consists of 1 and prime powers only.

 { 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 14, 18, 20, 24, 30, 42, 60, ...? }

Reinhard Zumkeller has conjectured that this sequence is finite and given in full. He has verified it up to ${\displaystyle 10^{7}}$. If true, that would mean that there is no need for the ellipses above.

Template:Sequence of the Day for September 9

A038136: Dihedral calculator primes

...

Written in a typical calculator font and turned upside down, you will read a number that is also a prime (maybe a different prime, as in the case of 2; or the same prime, as in the case of 11).

Template:Sequence of the Day for September 10
 { 70, 836, 4030, 5830, 7192, 7912, 9272, ... }

It seems that the vast majority of abundant numbers (A005101) can be expressed as a sum of some of their proper divisors (they are pseudoperfect, listed in A005835), often in more than one way. For example, with 12, we have 12 = 1 + 2 + 3 + 6 (ignoring 4) or 12 = 2 + 4 + 6 (ignoring 1 and 3). Essentially, we figure out what the abundancy of the number is, figure out which divisors add up to that abundancy, and then add up the other divisors. Well, 70 is, like 12, an abundant number with proper divisors adding up to 4 more than itself. The eight divisors of 70 are 1, 2, 5, 7, 10, 14, 35, 70 (A018270). 1 and 2 add up to 3, one short of 4, and any sum of the other divisors is way too much. Isn't that weird?

However, note that of the known weird numbers, none of them are odd. Robert Hearn has verified all weird numbers up to ${\displaystyle 10^{17}}$ are even. Also, a weird number can't be ${\displaystyle \scriptstyle {\mathcal {S}}\,}$-perfect (a Granville number).

Template:Sequence of the Day for September 11

A064150: Numbers divisible by the sum of their ternary digits.

{ ..., 4, 6, 8, 9, 10, 12, 15, ... }

Today marks the 11th anniversary of a tragic event. Many things came to a halt, others slowed down. The OEIS received new sequences that day and this one was one of them.

Template:Sequence of the Day for September 12
"{ T is The firsT, fourTh, elevenTh, sixTeenTh, TwenTy-fourTh, TwenTy-ninTh, ThirTy-Third, ... } letter in this sentence, not counting spaces or commas"—A. J. Aronson

giving the sequence of ordinal numbers (expressed as decimal Hindu-Arabic numerals instead of English literals!)

 { 1, 4, 11, 16, 24, 29, 33, ... }

In his paper "Seven Staggering Sequences," Neil Sloane named Aronson's quote above as "a classic self-referential assertion," since it "indicates exactly which of its terms are t's."

Template:Sequence of the Day for September 13

A070243: ${\displaystyle \scriptstyle {\rm {Card~}}\{k\,|\,\varphi (k)\,\leq \,n\},\,n\,\geq \,1.\,}$

 { 2, 5, 5, 9, 9, 13, 13, 18, 18, 20, 20, 26, 26, 26, 26, 32, 32, 36, 36, 41, 41, 43, 43, 53, ... }

This sequence counts the number of positive integers ${\displaystyle \scriptstyle k\,}$ such that ${\displaystyle \scriptstyle \varphi (k)\,\leq \,n\,}$, where ${\displaystyle \scriptstyle \varphi (k)\,}$ is Euler's totient function. It has natural density ${\displaystyle \scriptstyle {\frac {\zeta (2)\zeta (3)}{\zeta (6)}}\,\approx \,1.9435\,}$, so in some sense the values of ${\displaystyle \scriptstyle \varphi (k)\,}$ are on average not much less than ${\displaystyle \scriptstyle k\,}$.

Note that ${\displaystyle \scriptstyle a(n)\,-\,a(n-1)\,=\,{\rm {Card~}}\{k\,|\,\varphi (k)\,=\,n\},\,n\,\geq \,1,\,}$ (Cf. A014197) and thus ${\displaystyle \scriptstyle a(n)\,=\,\sum _{m=1}^{n}{\rm {A014197}}(m)\,}$.

Template:Sequence of the Day for September 14

A093341: Decimal expansion of "lemniscate case".

 1.8540746773013719...

In the preface for A Dictionary of Real Numbers, the Borwein brothers credit Carl Friedrich Gauß realizing "that the number 1.85407467... is (essentially) a rational value of an elliptic integral" as a watershed moment "in the development of nineteenth century analysis." The numerical lexicographers then go on to wonder how modern researchers, lacking what Gauß had ("phenomenal insight and a prodigious memory"), can also come to such an epiphany. That is the point of their book, and also that of the OEIS.

Template:Sequence of the Day for September 15

A006752: Decimal expansion of Catalan's constant ${\displaystyle \scriptstyle K\,:=\,\sum _{i=0}^{\infty }{\frac {(-1)^{i}}{(2i+1)^{2}}}.\,}$

 0.915965594177219...

We have known since antiquity that ${\displaystyle \scriptstyle {\sqrt {2}}}$ is irrational. The jury is still out on Catalan's constant.

Template:Sequence of the Day for September 16

A059742: Decimal expansion of ${\displaystyle \scriptstyle \pi +e\,}$.

 5.859874482...

We don't know whether ${\displaystyle \scriptstyle \pi +e\,}$ is irrational (let alone transcendental) or not. We know that both ${\displaystyle \scriptstyle e\,}$ and ${\displaystyle \scriptstyle \pi \,}$ are not only irrational, but also transcendental. Does it naturally follow that their sum is irrational (and if so, then transcendental)? Or would that be jumping to conclusions?

It is not known whether the following are irrational (let alone transcendental) or not:

It is known that the following is transcendental (thus irrational):

Template:Sequence of the Day for September 17

A156090: Alternating sum of the squares of the first ${\displaystyle \scriptstyle n\,}$ Fibonacci numbers with index divisible by 3.

 { 0, –4, 60, –1096, 19640, –352460, ... }

As is the case with many sequences derived from the Fibonacci numbers, this one can be expressed in quite a few different ways, e.g. as a sum

${\displaystyle a(n)=\sum _{k=1}^{n}(-1)^{k}\,F(3k)^{2},\,}$

where for ${\displaystyle \scriptstyle n\,=\,0\,}$ we get the empty sum, i.e. 0, or with a closed formula

${\displaystyle a(n)=(-1)^{n}\,{\frac {F(6n+3)}{10}}-{\frac {2n+1}{5}},\,}$

to name just two.

Template:Sequence of the Day for September 18

A020988: ${\displaystyle \scriptstyle {\frac {2}{3}}\,(4^{n}-1)\,=\,2\,\left({\frac {4^{n}-1}{4-1}}\right)\,=\,2\,\sum _{k=0}^{n-1}4^{k}\,=\,\sum _{k=0}^{n-1}2^{2k+1},\quad n\,\geq \,0.\,}$

 { 0, 2, 10, 42, 170, 682, 2730, 10922, ... }

These are even numbers such that their binary representations have every other bit set to zero. Thus these numbers are also the sum of distinct powers of 2 that are not powers of 4. In hexadecimal, these numbers are either all As or an initial 2 followed by all As. For example, 715827882 in binary is 101010101010101010101010101010 and in hexadecimal 2AAAAAAA.

Template:Sequence of the Day for September 19

A003188: Decimal equivalent of Gray code for ${\displaystyle n}$.

 {0, 1, 3, 2, 6, 7, 5, 4, 12, 13, 15, 14, 10, 11, 9, 8, 24, 25, 27, 26, 30, 31, 29, ... }

In binary, visits every node of an ${\displaystyle n}$-cube, by a change of exactly one coordinate (0 to 1 or 1 to 0).

a(n) = n XOR floor(n/2)
a(n) = a(floor(n/2)) + floor((n+1)/2) mod 2

Template:Sequence of the Day for September 20

A000975: ${\displaystyle a(2n)=2a(2n-1)}$, ${\displaystyle a(2n+1)=2a(2n)+1}$.

 { 1, 2, 5, 10, 21, 42, 85, 170, 341, ... }

These are also numbers with no consecutive equal bits in their binary representations. (That is: 1, 10, 101, 1010, 10101, etc.)

Template:Sequence of the Day for September 21

A115020: Count backwards from 100 in steps of 7.

 { 100, 93, 86, 79, 72, 65, 58, 51, 44, 37, 30, 23, 16, 9, 2 }

This sequence is sometimes used to gauge the concentration ability of a patient with Alzheimer's disease. The arithmetic is simple, but if short-term memory and concentration are compromised, a mistake is likely to occur at some point. Other decrements can be used for this purpose, with the alternative step preferably having no factor in common with 100.

Today is Alzheimer's Awareness Day.

Template:Sequence of the Day for September 22
 1.251647597790463...

Bertrand's postulate (asserted by Joseph Bertrand in 1845, now a theorem (since 1850) proved by Pafnuty Chebyshev) shows that for each ${\displaystyle n>3}$ there is at least one prime ${\displaystyle n. Likewise for ${\displaystyle n>1}$ there is at least one prime ${\displaystyle n. Therefore a constant ${\displaystyle b}$ exists where ${\displaystyle \scriptstyle \lfloor 2^{2^{\ldots 2^{b}}}\ldots \rfloor }$ is a prime for one or more power operations, cf. A051501. Finding more digits for ${\displaystyle b}$ requires finding better bounds for the prime counting function.

Template:Sequence of the Day for September 23
 { 945, 1575, 2205, 2835, 3465, 4095, 4725, ... }

Yes, the very first abundant number to be odd is 945, the 232nd abundant number! The first abundant number is even, namely, 12.

No, the odd abundant numbers are not all divisible by 3 or 5, e.g.

• The first odd abundant number not divisible by 5 is 81081, this being the 175th odd abundant number (and the ?th abundant number).
• The first odd abundant number not divisible by 3 is 5391411025, this being the ?th odd abundant number (and the ?th abundant number).
• The first odd abundant number not divisible by neither 3 nor 5 is 20169691981106018776756331, this being the ?th odd abundant number (and the ?th abundant number).

Template:Sequence of the Day for September 24

A077761: Decimal expansion of Mertens' constant (or prime reciprocal constant),

${\displaystyle \lim _{k\to \infty }\left\{\left(\sum _{i=1}^{k}{\frac {1}{p_{i}}}\right)-\log \log p_{k}\right\}=0.2614972128476427837554...,\,}$

where ${\displaystyle \scriptstyle p_{i}\,}$ is the ${\displaystyle \scriptstyle i\,}$-th prime number.

 {2, 6, 1, 4, 9, 7, 2, 1, 2, 8, 4, 7, 6, 4, 2, 7, 8, ...}

Today's SOTD is about the harmonic series of the primes (sum of the reciprocals of the primes), September 26's SOTD will be about the sum of alternating series of reciprocals of primes.

Template:Sequence of the Day for September 25

A002064: Cullen numbers ${\displaystyle C_{n}=n2^{n}+1}$

 { 1, 3, 9, 25, 65, 161, 385, 897, ... }

Grau proved that that there is no Cullen number with the Lehmer property. Hence, if ${\displaystyle \phi (C_{n})|(C_{n}-1)}$, then ${\displaystyle C_{n}}$ is prime. A composite integer ${\displaystyle m}$ is called a Lehmer number if ${\displaystyle \phi (m)|(m-1)}$, where ${\displaystyle \phi (m)}$ is the totient function.

Template:Sequence of the Day for September 26

A078437: Decimal expansion of sum of alternating series of reciprocals of primes ${\displaystyle \scriptstyle \sum _{i=1}^{\infty }(-1)^{i-1}{\frac {1}{p_{i}}}.\,}$

 0.26960635197167...

Here is another reminder of the infinitude of prime numbers. We may think we know a lot of prime numbers, but when it comes to computing this alternating series, this is the most precise we can get. The next digit is 4 or 5, we can't say for sure just yet.

Today's SOTD is about the sum of alternating series of reciprocals of primes, September 24's SOTD was about the harmonic series of the primes (sum of the reciprocals of the primes).

Template:Sequence of the Day for September 27

A192038: The Fibonacci "logarithm" for 4.

 4.5491125565...

What is the ten millionth Fibonacci number? Without the Binet formula, answering that question would require computing ten million Fibonacci numbers. Thanks to the Binet formula, we can just plug in ${\displaystyle x=10^{7}}$ with just a few operations instead of ten million or so additions.

The thought then occurs: Must we plug in an integer? What happens if we put in an irrational number? Defining ${\displaystyle f(x)={\frac {\phi ^{x}-\phi ^{-x\cos(\pi x)}}{\sqrt {5}}}}$, where ${\displaystyle \phi ={\frac {1+{\sqrt {5}}}{2}}}$ is the golden ratio, (a slight variation of the Binet formula, Clark Kimberling tells us) does not readily indicate any impediments to plugging in an irrational number, and so, for example, we have ${\displaystyle f(2{\sqrt {5}})\approx 3.84257859447}$.

The next question then is: Can we find a value of ${\displaystyle x}$ that will give us an integer that is not actually a Fibonacci number, like, say, 4? Indeed we can. Setting ${\displaystyle x\approx 4.5491125565}$ we obtain ${\displaystyle f(x)=4}$. That's one solution, there are others, which are all negative numbers.

Template:Sequence of the Day for September 28

A024670: Numbers that are sums of two distinct positive cubes, ${\displaystyle n=x^{3}+y^{3}.}$

 { 9, 28, 35, 65, 72, 91, 126, ... }

And yes, 1729 is in this sequence, as it exceeds the minimum requirements for inclusion since it is the smallest integer which is the sum of two [distinct] cubes in two different ways. Also, 1729 is the 52nd term of the sequence. (Which means that if you find a term of the sequence once a week...)

As M. F. Hasler observed, this sequence contains no primes because ${\displaystyle x^{3}+y^{3}}$ has the polynomial factorization ${\displaystyle (x^{2}-xy+y^{2})(x+y)}$, thus the sum of two [distinct] positive cubes yields a composite number.

Fermat's last theorem implies that this sequence contains no cubes. (Although it trivially yields an infinity of quasi-cubes ${\displaystyle n^{3}+1}$, since 1 is a cube.)

Template:Sequence of the Day for September 29

A070027: Prime numbers whose initial, all intermediate and final iterated sums of [decimal] digits are primes.

 { ... 11, 23, 29, 41, 43, 47, 61, 83, 101, 113, ... }

The more interesting members of this sequence are those for which there is more than one step to get to the digital root. For example, with 29, we have 2 + 9 = 11 and then 1 + 1 = 2, whereas with 41 it just goes 4 + 1 = 5.

Template:Sequence of the Day for September 30

A987654: Sequence name

 9.87654321...

Paragraph or two of info.

By the way, ${\displaystyle \scriptstyle {\sqrt {9.87654321}}}$ = 3.142696805293... pretty close to ${\displaystyle \pi }$!