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# Calendar for Sequence of the Day in February

Template:Sequence of the Day for February 1

A174375: n^2 - XOR(n^2, n)

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

Plotting the points of ${\displaystyle a(n)}$ versus ${\displaystyle n}$ up to a power of 2 approximates a Sierpiński gasket.

Template:Sequence of the Day for February 2

A196244: Number of distinct values taken by r^r^...^r where r = 1/2 (with ${\displaystyle n}$ r's and parentheses inserted in all possible ways).

{ 1, 1, 2, 4, 9, 20, 47, ... }

In the case of say, ${\displaystyle n=4}$, we're considering the expressions${\displaystyle {\frac {1}{2}}^{({\frac {1}{2}}^{({\frac {1}{2}}^{\frac {1}{2}})})}}$, ${\displaystyle {\frac {1}{2}}^{({({\frac {1}{2}}^{\frac {1}{2}})}^{\frac {1}{2}})}}$, ${\displaystyle {({({\frac {1}{2}}^{\frac {1}{2}})}^{\frac {1}{2}})}^{\frac {1}{2}},{({\frac {1}{2}}^{\frac {1}{2}})}^{({\frac {1}{2}}^{\frac {1}{2}})}}$ and ${\displaystyle {({\frac {1}{2}}^{({\frac {1}{2}}^{\frac {1}{2}})})}^{\frac {1}{2}}}$, and whether or not these expressions result in distinct values or not.

Template:Sequence of the Day for February 3

A059801: Numbers ${\displaystyle n}$ such that ${\displaystyle 4^{n}-3^{n}}$ is prime.

{ 2, 3, 7, 17, 59, 283, 311, 383, 499, ... }

For example, ${\displaystyle 4^{7}-3^{7}=16384-2187=14197}$ and that is a prime number.

Template:Sequence of the Day for February 4

A116697: ${\displaystyle a(n)=-a(n-1)-a(n-3)+a(n-4)}$.

{ 1, –2, 2, –2, 5, –9, 13, –20, ... }

Most recurrence relations use the last few consecutive terms (usually two) to determine the next term. But in this one, ${\displaystyle a(n-2)}$ is actually ignored, and ${\displaystyle a(n-3)}$ and ${\displaystyle a(n-4)}$ are used. There's something else that is interesting about this sequence: read the odd-indexed terms and see if that reminds you of anything.

Template:Sequence of the Day for February 5

A157218: Number of ways to write the ${\displaystyle n}$-th positive odd integer in the form ${\displaystyle p+2^{x}+7(2^{y})}$ with ${\displaystyle p}$ a prime congruent to 1 mod 6 and ${\displaystyle x}$, ${\displaystyle y}$ positive integers.

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

In 2009, Zhi-Wei Sun conjectured that ${\displaystyle a(n)>0}$ for all ${\displaystyle n>17}$; in other words, any odd integer greater than 34 can be written as the sum of a prime congruent to 1 mod 6, a positive power of 2 and seven times a positive power of 2. Sun verified the conjecture for odd integers below ${\displaystyle 5\times 10^{7}}$, and Qing-Hu Hou continued the verification for odd integers below ${\displaystyle 1.5\times 10^{8}}$ (on Sun's request). Compare the conjecture with R. Crocker's result that there are infinitely many positive odd integers not of the form ${\displaystyle p+2^{x}+2^{y}}$ with ${\displaystyle p}$ an odd prime and ${\displaystyle x}$, ${\displaystyle y}$ positive integers.

Template:Sequence of the Day for February 6

A144226: Prime numbers containing equal number of odd and even digits.

{ 23, 29, 41, 43, 47, 61, 67, 83, 89, 1009, 1021, 1049, ... }

It seems 'obvious' that this sequence contains only a thin fraction of the primes, but can this be proven? More precisely, is

${\displaystyle \liminf a_{n}/p_{n}=+\infty }$?

Template:Sequence of the Day for February 7

A002025: Smaller of amicable pair.

{ 220, 1184, 2620, 5020, 6232, 10744, 12285, ... }

These numbers are of course abundant numbers.

Template:Sequence of the Day for February 8

A234567: Sequence name

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

More details...

Template:Sequence of the Day for February 9

A234567: Sequence name

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

More details...

Template:Sequence of the Day for February 10

A060590: Numerator of the expected time to finish a random tower of Hanoi problem with ${\displaystyle n}$ disks using optimal moves.

{ 2, 2, 14, 10, 62, 42, 254, ... }

The formula for this sequence is ${\displaystyle {\frac {2(2^{n}-1)(2-(-1)^{n}}{3}}}$.

Template:Sequence of the Day for February 11

A234567: Sequence name

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

More details...

Template:Sequence of the Day for February 12

A002046: Larger number ${\displaystyle \scriptstyle n\,}$ of amicable pair ${\displaystyle \scriptstyle (m,\,n)\,}$ of numbers.

 { 284, 1210, 2924, 5564, 6368, 10856, 14595, ... }

Two integers ${\displaystyle \scriptstyle m\,}$ and ${\displaystyle \scriptstyle n\,}$ are amicable numbers (amicable pair of numbers) if ${\displaystyle \scriptstyle \sigma (m)-m\,=\,n\,}$ and ${\displaystyle \scriptstyle \sigma (n)-n\,=\,m\,}$.

These larger numbers of amicable pairs are of course deficient numbers, while the smallest numbers of amicable pairs are abundant numbers. One might say that the pair is mutually perfect (so to speak) since the abundancy of the smaller number cancels out the deficiency of the larger number, i.e.

${\displaystyle (\sigma (m)+\sigma (n))-2(m+n)=0.\,}$

Template:Sequence of the Day for February 13

A066340: Fermat's triangle: ${\displaystyle T(n,m)=m^{\phi (n)}\mod n}$ for ${\displaystyle 1\leq m.

1
1   1
1   0   1
1   1   1   1
1   4   3   4   1
1   1   1   1   1   1
1   0   1   0   1   0   1
1   1   0   1   1   0   1   1
1   6   1   6   5   6   1   6   1
1   1   1   1   1   1   1   1   1   1
1   4   9   4   1   0   1   4   9   4   1
1   1   1   1   1   1   1   1   1   1   1   1
1   8   1   8   1   8   7   8   1   8   1   8   1

Per Fermat's little theorem, the rows consisting of all 1s correspond to prime numbers.

Template:Sequence of the Day for February 14
{ 220, 284, 1184, 1210, 2620, 2924, 5020, ... }

Notice how for example ${\displaystyle \sigma (220)-220=284}$ and ${\displaystyle \sigma (284)-284=220}$.

Template:Sequence of the Day for February 15
{ 561, 1105, 1729, 2465, 2821, 6601, 8911, ... }

Ghatage and Scott prove using Fermat's little theorem that ${\displaystyle (a+b)^{n}=a^{n}+b^{n}\mod n}$ (the freshman's dream) exactly when ${\displaystyle n}$ is a prime or a Carmichael number.

Template:Sequence of the Day for February 16

A072508: Decimal expansion of Backhouse's constant.

1.456074948582689...

Given ${\displaystyle x}$ being the real zero of ${\displaystyle 1+\sum _{k=1}^{\infty }p_{k}x^{k}}$, Backhouse's constant is ${\displaystyle {\frac {-1}{x}}}$. It almost seems like ${\displaystyle x}$ should also be a constant named after a person.

Template:Sequence of the Day for February 17

A005936: Pseudoprimes to base 5.

{ 4, 124, 217, 561, 781, 1541, 1729, ... }

If both numbers ${\displaystyle q}$ and ${\displaystyle 2q-1}$ are primes, then ${\displaystyle n=q(2q-1)}$ is a pseudoprime to base 5 if and only if ${\displaystyle q}$ is of the form ${\displaystyle 10k+1}$.

Template:Sequence of the Day for February 18

A195264 Iterate x -> A080670(x) starting at ${\displaystyle n}$ until reach 1 or a prime; or –1 if a prime is never reached.

{ 1, 2, 3, 211, 5, 23, 7, 23, 2213, ... }

For example, 9 = 3^2 -> 32 = 2^5 -> 25 = 5^2 -> 52 = 2^2*13 -> 2213, which is prime, so a(9) = 2213. It remains an open question whether the –1 is ever needed, with the smallest value, 20, having so far been tested past 70 iterations.

Template:Sequence of the Day for February 19
Go is to Western chess what philosophy is to double-entry accounting. — Rodney William Whitaker ("Trevanian"), Shibumi

A089071: Number of liberties a big eye of size ${\displaystyle n}$ gives in the game of Go.

 { 1, 2, 3, 5, 8, 12, 17, 23, 30, 38, 47, ... }

A 5-space big eye can be almost filled in 4 moves, after which one takes and has a 4-space big eye (5 liberties) left. This gives a total of 4+5 moves for the opponent and 1 for oneself, for de facto 8 liberties.

Template:Sequence of the Day for February 20

A205601: Goldbach's problem extended to division: number of decompositions of ${\displaystyle 2n}$ into the floor of unordered reciprocals of two primes, ${\displaystyle \lfloor q/p\rfloor =2n}$, where ${\displaystyle p<2n.

{ 0, 1, 3, 5, 4, 5, 10, 5, 10, ... }

Essentially, what we are counting here is this: for each prime ${\displaystyle p<2n}$, how many primes are there in the range ${\displaystyle 2pn? As ${\displaystyle 2n}$ gets larger, the available primes get larger and correspondingly so does ${\displaystyle 2n(p+1)-2pn}$. Consider for example, 8, which can't be represented as ${\displaystyle \lfloor q/3\rfloor }$ with the constraints specified here, but both ${\displaystyle \lfloor 41/5\rfloor }$ and ${\displaystyle \lfloor 43/5\rfloor }$ give the desired result.

Template:Sequence of the Day for February 21

A088751: Decimal expansion of ${\displaystyle x}$, the real root of the equation ${\displaystyle 0=1+\sum _{k=1}^{\infty }p_{k}x^{k}}$, where${\displaystyle p_{n}}$ is the ${\displaystyle n}$th prime.

–0.68677783446063...

This is the inverse of Backhouse's constant (A072508), a number that is somewhat harder to compute than this one.

Template:Sequence of the Day for February 22

A008604: Multiples of 22.

{ 22, 44, 66, 88, 110, 132, 154, ... }

For small composite numbers we can devise simple "composite" divisibility tests: if a number is even and divisible by 11, then it is also divisible by 22 (see, we are combining the divisibility test for 2 with the divisibility test for 11). Is there a "prime" divisibility test for 22?

Template:Sequence of the Day for February 23

A002322: Reduced totient function ${\displaystyle \psi (n)}$: least ${\displaystyle k}$ such that ${\displaystyle x^{k}=1\mod n}$ for all ${\displaystyle x}$ coprime to ${\displaystyle n}$

{ 1, 1, 2, 2, 4, 2, 6, 2, 6, 4, 10, 2, 12, 6, 4, 4, 16, ... }

This is also the largest period of repeating digits of ${\displaystyle {\frac {1}{n}}}$ written in different bases.

Template:Sequence of the Day for February 24

A109671: ${\displaystyle a(1)=1}$; thereafter, ${\displaystyle a(2n)=a(n)}$, ${\displaystyle a(2n+1)}$ is the smallest positive number such that ${\displaystyle |a(2n+1)-a(2n-1)|=a(n)}$.

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

This is a variant of the semi-Fibonacci numbers (A030067). The sequence is self-describing: the sequence of the absolute differences between odd-indexed terms is the sequence itself. It appears that the record values form sequence A038754 and occur at indices of the form ${\displaystyle 2^{k}-1}$. It remains an open question whether or not the sequence contains every positive integer (cf. A169741).

Template:Sequence of the Day for February 25

A175607: Largest number ${\displaystyle x}$ such that the greatest prime factor of ${\displaystyle x^{2}-1}$ is the ${\displaystyle n}$th prime.

{ 3, 17, 161, 8749, 19601, 246401, 672281, ... }

For any prime ${\displaystyle p}$, there are finitely many ${\displaystyle x}$ such that ${\displaystyle x^{2}-1}$ has p as its largest prime factor. For every prime ${\displaystyle p}$, is there some ${\displaystyle x}$ where the greatest prime factor of ${\displaystyle x^{2}-1}$ is ${\displaystyle p}$? Yes, there is. As mentioned by Luca and Najman, this problem is closely related to the one in A002071.

Template:Sequence of the Day for February 26

A003064: Smallest number with addition chain of length ${\displaystyle n}$.

{ 1, 2, 3, 5, 7, 11, 19, 29, 47, 71, 127, ... }

What could be simpler than a sequence about addition? And yet, obtaining terms of this sequence beyond ${\displaystyle a(24)}$ has involved many different people working independently of each other.

Template:Sequence of the Day for February 27

A030133: ${\displaystyle a(n)}$ is the sum of base 10 digits of ${\displaystyle a(n-2)+a(n-1)}$

{ 2, 1, 3, 4, 7, 2, 9, ... }

I don't know of many recurrence relations involving base 10 digits (or digits in any base, for that matter). That alone makes this sequence interesting.

Template:Sequence of the Day for February 28

A006345: Linus sequence

{ 1, 2, 1, 1, 2, 2, 1, 2, 1, 1, 2, 1, 2, 2, 1, 1, 2, 1, 1, 1, 2, 2, 1, 2, 1, 1, 2, 2, 1, 1, ... }

${\displaystyle a(n)}$ "breaks the pattern" by avoiding the longest doubled suffix.

The initial part of the sequence was described in a Peanuts comic strip where Linus is trying to figure out the answers to a true-false test by assuming the teacher ordered the answers such that there would be no sequential 'pattern'.

LEAP YEARS ONLY
|Template:Sequence of the Day for February 29

A090651: Perpetual calendar sequence

{ 3, 4, 5, 13, 1, 2, 3, 11, 6, 7, ... }

There are 14 basic year calendars, 7 for normal years and 7 for leap years. This sequence identifies the calendars for years 1901 through 2099, when it reinitializes because 2100 is not a leap year. Note that ${\displaystyle \scriptstyle a(n)\,=\,1\,}$ for years starting on a Sunday, 2 for years starting on a Monday, and so on to 7; 8 for leap years starting on a Sunday, 9 for leap years starting on Monday, and so on to 14. Since this year, 2016, is a leap year — requiring the addition of an extra day, today — starting on a Friday, ${\displaystyle \scriptstyle a(2016)\,=\,13\,}$.

SWEDEN, 1712 ONLY
|Template:Sequence of the Day for February 30

A123456: Ludwig van Beethoven, Bagatelle No. 25, "Für Elise".

 { –20, 56, 55, 56, 55, 56, 51, … }

At David Applegate's suggestion, 20 has been subtracted from each entry to make this sound better with the default settings used by the OEIS Midi player. See A144488 for the old version.