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# Calendar for Sequence of the Day in January

Template:Sequence of the Day for January 1

A194602: Partitions interpreted as binary numbers

 { 0, 1, 3, 5, 7, 11, 15, 21, ... }

The compositions of ${\displaystyle \scriptstyle n\,}$ correspond to ${\displaystyle \scriptstyle 2^{n-1}\,}$ binary numbers. As partitions of ${\displaystyle \scriptstyle n\,}$ can be seen as compositions with addends ordered by size, binary numbers can also be assigned to partitions. The finite sequence of such binary numbers for partitions of ${\displaystyle \scriptstyle n\,}$ (in lexicographical order) is the beginning of the sequence for partitions of ${\displaystyle \scriptstyle n+1\,}$, which leads to an infinite sequence.

Template:Sequence of the Day for January 2

A004676: Primes written in base 2.

 { 10, 11, 101, 111, 1011, 1101, 10001, ... }

Since 2 is the only even prime, then of course it is the only one that ends in zero in base 2. This highlights the oddity of 2 being the only even prime number.

The Fermat primes (only five are known!) are the only primes with binary weight equal to two: {11, 101, 10001, 10000001, 10000000000001, ...more??? }

The repunits (base 2) are the Mersenne primes.

Template:Sequence of the Day for January 3

A000265: Largest odd divisor of ${\displaystyle \scriptstyle n.\,}$

 { 1, 1, 3, 1, 5, 3, 7, 1, ... }

Obviously ${\displaystyle \scriptstyle a(n)\,=\,1\,}$ only when ${\displaystyle \scriptstyle n\,}$ is a power of 2 and ${\displaystyle \scriptstyle a(n)\,=\,n\,}$ only when ${\displaystyle \scriptstyle n\,}$ is an odd noncomposite number. Somewhat less obviously, this is the numerator of ${\displaystyle \scriptstyle {\frac {n}{2^{n-1}}}\,}$ and the denominator of ${\displaystyle \scriptstyle {\frac {2^{n-1}}{n}}.\,}$

Template:Sequence of the Day for January 4

A198683: Number of distinct values taken by i^i^...^i (with ${\displaystyle \scriptstyle n\,}$ i 's and ${\displaystyle \scriptstyle n-2\,}$ pairs of parentheses inserted in all possible nontrivial ways) where i is the imaginary unit and ^ denotes the principal value of the power function.

 { 1, 1, 2, 3, 7, 15, 34, 77, ... }

In the case of say, ${\displaystyle \scriptstyle n\,=\,4\,}$, we're considering the expressions i^(i^(i^i)), i^((i^i)^i) = ((i^i)^i)^i, (i^i)^(i^i) = (i^(i^i))^i, viz.

${\displaystyle i^{(i^{(i^{i})})},\,i^{({(i^{i})}^{i})}={({(i^{i})}^{i})}^{i},\,{(i^{i})}^{(i^{i})}={(i^{(i^{i})})}^{i},\,}$

and whether or not these expressions result in distinct values or not. As it turns out, the second and third expressions give the same value ${\displaystyle \scriptstyle i^{-i}\,=\,\exp({\frac {\pi }{2}})\,}$ (approximately 4.81047738...), while the fourth and fifth expressions both result in ${\displaystyle \scriptstyle (\exp({\frac {-\pi }{2}}))^{\exp({\frac {-\pi }{2}})},\,}$ (approximately 0.721418...). (By the way, only the first of these expressions results in a number with a nonzero imaginary part.)

Template:Sequence of the Day for January 5

A000081: Number of rooted trees with ${\displaystyle n}$ nodes (or connected functions with a fixed point).

{ 1, 1, 2, 4, 9, 20, 48, 115, ... }

My interest in this sequence today has to do not with trees or anything in graph theory, but with the following number-theoretical problem:

Given ${\displaystyle n}$ instances of some constant ${\displaystyle c}$ with exponentiation carets and parentheses inserted in every valid way, how many distinct values are produced by the resulting expressions? The answer depends on what ${\displaystyle c}$ is. With four instances of ${\displaystyle c={\frac {1}{2}}}$, there are four distinct values, but with four instances of ${\displaystyle c=i}$ (the imaginary unit), only three distinct values are produced.

But what if you don't know what ${\displaystyle c}$ is? Is there an upper bound for ${\displaystyle n}$ instances of ${\displaystyle c}$? This sequence is the answer.

Template:Sequence of the Day for January 6

A002581: Decimal expansion of ${\displaystyle \scriptstyle {\sqrt[{3}]{3}}\,}$.

 1.4422495703074...

Maybe not as famous as ${\displaystyle \scriptstyle {\sqrt[{2}]{2}}\ (\,=\,{\sqrt[{4}]{4}})\,}$, but quite interesting in its own right.

Template:Sequence of the Day for January 7

A217281: Prime numbers on the front cover of the paperback editions of Marcus du Sautoy's book The Music of the Primes.

{ 19, 23, 29, 167, 193, 5, 103, 107, 2, 47, 139, 263, 113, 109, 59, 577, 3 }

For the original hardcover printing of this book, the primes appear in order on the front cover. But for both paperback printings to date, the front cover shows this list of primes in apparently no order.

Template:Sequence of the Day for January 8

A185086: Fouvry-Iwaniec primes: primes of the form ${\displaystyle \scriptstyle k^{2}+p^{2}\,}$ where ${\displaystyle \scriptstyle p\,}$ is a prime.

 { 5, 13, 29, 41, 53, 61, 73, ... }

These numbers are prime in ${\displaystyle \scriptstyle \mathbb {Z} \,}$ but composite in ${\displaystyle \scriptstyle \mathbb {Z} [i]\,}$. In the latter, they are the product of ${\displaystyle \scriptstyle k+pi\,}$ and ${\displaystyle \scriptstyle k-pi\,}$ (see Gaussian integers.)

Template:Sequence of the Day for January 9

A073277: Irregular primes with irregularity index two.

{ 157, 353, 379, 467, 547, 587, ... }

These prime numbers are just some of the prime numbers that were such a headache for Gabriel Lamé in trying to prove Fermat's last theorem in the 19th century.

Template:Sequence of the Day for January 10

A007468: Sum of next ${\displaystyle n}$ primes.

{ 2, 8, 31, 88, 199, 384, 659, ... }

Or: if we arrange the prime numbers into a triangle, with 2 at the top, 3 and 5 in the second row, 7, 11 and 13 in the third row, and so on and so forth, this sequence gives the row sums.

Template:Sequence of the Day for January 11

A130111: Rearrangement of natural numbers such that each five terms sum up to a perfect square.

{ 1, 2, 3, 4, 6, 5, 7, 8, 9, 20, 10, 11, 12, 13, 18, 14, 15, 16, 17, 19, 21, 22, 23, 24, 31, ... }

"Rearrangement" means that every natural number appears and no number can occur twice. To be precise, one should also specify explicitly that the smallest possible choice is to be made for each ${\displaystyle a(n)}$, ${\displaystyle n=1,2,3,\ldots }$ (i.e., the lexicographically smallest sequence), which is often tacitly understood in such cases.

Template:Sequence of the Day for January 12

A006967: Number of graceful permutations of length ${\displaystyle n}$.

{ 1, 2, 4, 4, 8, 24, 32, 40, 120, ... }

There are exactly ${\displaystyle n!}$ permutations. But of those, only a few are graceful for ${\displaystyle n>2}$.

Template:Sequence of the Day for January 13

A1234567: Sequence name

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

More details...

Template:Sequence of the Day for January 14

A173623: Decimal expansion of ${\displaystyle {\frac {\pi \log 2}{2}}}$.

1.08879304515...

Multiplied by –1, this is the value of ${\displaystyle \int _{0}^{\frac {\pi }{2}}\log \sin(x)dx}$.

Template:Sequence of the Day for January 15

A160106: Decimal representation of Bernays' number, ${\displaystyle \scriptstyle 67^{257^{729}}\,}$

 3259153847986 (...) 8498607119427

This is an integer, but it has so many digits that here we are only showing the most significant and the least significant.

Bernays' number is approximately ${\displaystyle \scriptstyle 10^{1.2669\ldots \cdot 10^{1757}},\,}$ the number of digits of Bernays' number is ${\displaystyle \scriptstyle 1.2669\ldots \cdot 10^{1757},\,}$ the number of digits of the number of digits of Bernays' number is 1758.

Template:Sequence of the Day for January 16

A033715: Number of integer solutions ${\displaystyle \scriptstyle (x,\,y)\,}$ to the equation ${\displaystyle \scriptstyle x^{2}+2y^{2}\,=\,n.\,}$

 { 1, 2, 2, 4, 2, 0, 4, 0, 2, 6, 0, 4, 4, ... }

This pertains to a Diophantine equation that was studied by Ramanujan.

Template:Sequence of the Day for January 17

A1234567: Sequence name

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

More details...

Template:Sequence of the Day for January 18

A085667: Decimal expansion of ${\displaystyle \scriptstyle e\uparrow \uparrow 4\,=\,e^{e^{e^{e}}}\,=\,2.331504399...\cdot 10^{1656520}\,}$

 2331504399007...

As late as December 2011, this was the irrational constant in the OEIS with the largest integer part, having 1656521 digits to the left of the decimal point.

Template:Sequence of the Day for January 19

A1234567: Sequence name

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

More details...

Template:Sequence of the Day for January 20

A170739 G.f.: (1+x)/(1-19*x).

 { 1, 20, 380, 7220, 137180, 2606420, 49521980, 940917620, ... }

As a last resort, for any day of first month one might always use G.f.: (1+x)/(1-((day_of_month - 1) *x)...

Template:Sequence of the Day for January 21

A007542: Successive integers produced by Conway's PRIMEGAME.

{ 2, 15, 825, 725, 1925, 2275, 425, ... }

After the initial 2, each number that has an integer for a binary logarithm is also a prime number; that is to say powers of 2 with composite exponents (A073718) don't show up in this sequence.

Template:Sequence of the Day for January 22

A1234567: Sequence name

{ 1, 22, 2012, 3, 4, 7, 6, 5, ... }

More details...

Template:Sequence of the Day for January 23

A020806: Decimal expansion of ${\displaystyle \scriptstyle {\frac {1}{7}}\,}$.

 0.14285714285714...

7 is a "full reptend prime" ("long period prime" or "maximal period prime") in base 10 since ${\displaystyle \scriptstyle {\frac {1}{7}}\,}$ has period length 6.

Since ${\displaystyle \scriptstyle {\frac {10^{6}-1}{7}}\,=\,{\frac {999999}{7}}\,=\,142857\,}$ is a cyclic number, we have

${\displaystyle \scriptstyle n\,}$ ${\displaystyle \scriptstyle 142857\times n\,}$ ${\displaystyle \scriptstyle {\frac {n}{7}}\,}$
1 142857 × 1 = 142857 0.142857142857...
2 142857 × 2 = 285714 0.285714285714...
3 142857 × 3 = 428571 0.428571428571...
4 142857 × 4 = 571428 0.571428571428...
5 142857 × 5 = 714285 0.714285714285...
6 142857 × 6 = 857142 0.857142857142...

Template:Sequence of the Day for January 24

A001021: ${\displaystyle 12^{n}}$

{ 1, 12, 144, 1728, 20736, 248832, 2985984, ... }

These numbers are the basis of the duodecimal numeral system. Some of us might be old enough to remember in school having to learn multiplication tables up to 12 rather than 10, and perhaps also having to learn at least the small powers of 12. Starting with 12, this sequence appears in the film Vollmond (1998, dir. Fredi Murer), when the children write it on the sidewalk at night.

Template:Sequence of the Day for January 25

A1234567: Sequence name

{ 1, 25, 2012, 3, 4, 7, 6, 5, ... }

More details...

Template:Sequence of the Day for January 26

A021030: Decimal expansion of ${\displaystyle {\frac {1}{26}}}$.

0.03846153846...

A tool code breakers sometimes use is the index of coincidence, ${\displaystyle I_{c}}$. According to cryptanalysis expert Christopher Swenson, "the theoretically perfect ${\displaystyle I_{c}}$ is if all characters occur the exact same number of times so that none was more likely than any other to be repeated." For cyphertext encrypted from English text (using an alphabet of 26 letters) of infinite length, this means there exists the infinite limit ${\displaystyle {\frac {n-1}{26n-1}}}$ which by L'Hôpital's rule works out to ${\displaystyle {\frac {1}{26}}}$.

Template:Sequence of the Day for January 27

A078437: Decimal expansion of the sum of the alternating series of reciprocals of the primes.

0.26960635...

Is this number, ${\displaystyle \sum _{i=1}^{\infty }(-1)^{i+1}{\frac {1}{p_{i}}}={\frac {1}{2}}-{\frac {1}{3}}+{\frac {1}{5}}-{\frac {1}{7}}}$ ${\displaystyle +{\frac {1}{11}}-{\frac {1}{13}}+{\frac {1}{17}}-{\frac {1}{19}}+\ldots }$ irrational? Transcendental even?

Template:Sequence of the Day for January 28

A1234567: Sequence name

{ 1, 28, 2012, 3, 4, 7, 6, 5, ... }

More details...

Template:Sequence of the Day for January 29

A082813: Prime numbers of the form ${\displaystyle \sum _{i=1}^{\omega (m)}{p_{i}}^{m}}$ for some composite ${\displaystyle m}$.

{ 539633, 3(288) + 1188, ... }

Already the second term has 92 decimal digits, and it is thus more concise to express it as the sum of ${\displaystyle m}$th powers of ${\displaystyle m}$'s (not necessarily distinct) prime factors. There's a third term that's known, and potentially a few more that are not known (the sequence could be infinite, but we can only speculate to that). But only for the first term is the specified expression longer than the number written out: 212 + 212 + 312.

Although getting this sequence is computationally intensive, it is at least possible to quickly cull out many ${\displaystyle m}$ that would simply not work, such as ${\displaystyle m}$ of the form ${\displaystyle 2^{3}3^{\alpha }}$ (because then the resulting number, though quite large, would be divisible by 3).

Template:Sequence of the Day for January 30

A1234567: Sequence name

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

More details...

Template:Sequence of the Day for January 31

A051006: The prime constant

0.4146825098511116...

The formula is ${\displaystyle \sum _{k=1}^{\infty }{\frac {\chi _{p}(k)}{2^{k}}}}$, where ${\displaystyle \chi _{p}(n)}$ is the characteristic function of the primes. Essentially, this is an encoding of the prime numbers into a single real number in the unit interval.