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# Calendar for Sequence of the Day in December

Template:Sequence of the Day for December 1

A188892: Numbers ${\displaystyle \scriptstyle n\,}$ such that there is no triangular ${\displaystyle \scriptstyle n\,}$-gonal number greater than 1.

 { 11, 18, 38, 102, 198, 326, 486, ... }

The triangular numbers are essentially the building blocks of the other figurate numbers. Therefore it is rather surprising that there can be sequences of ${\displaystyle \scriptstyle n\,}$-gonal numbers that don't overlap with the sequence of triangular numbers at all (other than 0 and 1.) T. D. Noe has demonstrated that the equation ${\displaystyle \scriptstyle x^{2}+x\,=\,(n-2)y^{2}-(n-4)y\,}$ has no integer solutions ${\displaystyle \scriptstyle x\,\geq \,y\,>\,1\,}$, as conversion to a generalized Pell equation shows that if ${\displaystyle \scriptstyle n\,=\,k^{2}+2\,}$, then the first equation has only a finite number of solutions. From there one can pinpoint those values of ${\displaystyle \scriptstyle n\,}$ that produce no integer solutions greater than 1.

Template:Sequence of the Day for December 2

A1202345: Sequence name.

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

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A076512 / A109395: ${\displaystyle \scriptstyle {\frac {\phi (n)}{n}}\,=\,\prod _{i=1}^{\omega (n)}\left(1-{\frac {1}{p_{i}}}\right)\,}$, where ${\displaystyle \omega (n)}$ is the number of distinct prime factors function.

${\displaystyle \left\{1,{\frac {1}{2}},{\frac {2}{3}},{\frac {1}{2}},{\frac {4}{5}},{\frac {1}{3}},{\frac {6}{7}},{\frac {1}{2}},\cdots \right\}\,}$

The denominator is ${\displaystyle p_{\omega (n)}}$ (the greatest prime factor) if and only if for every ${\displaystyle j<\omega (n)}$, there is a ${\displaystyle k<\omega (n)}$ such that ${\displaystyle p_{j}|(p_{k}-1)}$.

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A029908: Starting with ${\displaystyle \scriptstyle n,\,n\,\geq \,1,\,}$ repeatedly sum prime factors (with multiplicity) until reaching 0 or a fixed point.

 { 0, 2, 3, 4, 5, 5, 7, 5, 5, 7, 11, 7, 13, 5, ... }

The Integer log of n (sum of prime factors of n (with multiplicity)) amounts to replacing multiplication by addition and exponentiation by multiplication in the canonical prime factorization of ${\displaystyle \scriptstyle n\,}$.

Because sopfr(n) <= n (with equality at 4 and the primes), the first appearance of all primes is in the natural order: 2,3,5,7,11,... . [Zak Seidov, Mar 14 2011]

Template:Sequence of the Day for December 5

A197723: Decimal expansion of ${\displaystyle \scriptstyle {\frac {3\pi }{2}}\,}$.

 4.71238898038...

Multiplying a number by ${\displaystyle -i}$ (with ${\displaystyle i}$ being the imaginary unit ${\displaystyle \scriptstyle {\sqrt {-1}}\,}$) is equivalent to rotating it by this amount on the complex plane, which as radians, works out to 270 degrees or 300 gradians.

Template:Sequence of the Day for December 6

A1202345: Sequence name.

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

Template:Sequence of the Day for December 7

A1202345: Sequence name.

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

Template:Sequence of the Day for December 8

A010466: Decimal expansion of ${\displaystyle \scriptstyle {\sqrt {8}}\,}$.

 2.8284271247461900976...

This is the second Lagrange number, used in the Hurwitz irrational number theorem to obtain very good rational approximations of irrational numbers other than multiples of the golden ratio.

Template:Sequence of the Day for December 9

A1202345: Sequence name.

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

Template:Sequence of the Day for December 10

A001190: Wedderburn-Etherington numbers: binary rooted trees (every node has out-degree 0 or 2) with ${\displaystyle n}$ endpoints (and ${\displaystyle 2n-1}$ nodes in all).

 { 1, 1, 1, 2, 3, 6, 11, 23, 46, 98, ... }

This is also the number of ways to place ${\displaystyle n}$ stars in a single bound stable hierarchical multiple star system; i.e. taking only the configurations from A003214 where all stars are included in single outer parentheses.

Template:Sequence of the Day for December 11

A188859: Decimal expansion of ${\displaystyle \scriptstyle 2-\log 4\,}$.

 0.613705638880109...

This is the limit (as ${\displaystyle n}$ increases without bound) of the probability that ${\displaystyle \scriptstyle n{\bmod {m}}\,}$ is less than ${\displaystyle \scriptstyle {\frac {m}{2}}\,}$, with ${\displaystyle m}$ chosen uniformly at random from ${\displaystyle \scriptstyle \{1,\,\ldots ,\,n\}\,}$. (As usual, ${\displaystyle \scriptstyle 0\,\leq \,n{\bmod {m}}\,<\,m\,}$.)

Template:Sequence of the Day for December 12

A033880: Abundance of ${\displaystyle n}$.

 { ..., –14, 0, –28, 12, –30, –1, –18, ... }

The negative ${\displaystyle a(n)}$ in this listing correspond to deficient numbers, the zeros correspond to perfect numbers and the positives correspond to abundant numbers. I'm choosing to start the listing here from 27 in order to get a better mix of abundant, perfect and deficient numbers.

Template:Sequence of the Day for December 13

A033879: Deficiency of ${\displaystyle n}$.

 { ..., 14, 0, 28, –12, 30, 1, 18, ... }

The negative ${\displaystyle a(n)}$ in this listing correspond to abundant numbers, the zeros correspond to perfect numbers and the positives correspond to deficient numbers. I'm choosing to start the listing here from 27 in order to get a better mix of abundant, perfect and deficient numbers.

Template:Sequence of the Day for December 14

A1202345: Sequence name.

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

Template:Sequence of the Day for December 15

A1202345: Sequence name.

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

Template:Sequence of the Day for December 16

A054245: Beethoven's Symphony No. 5 in C minor.

 { 5, 5, 5, 3, 4, 4, 4, 2, 5, 5, 5, 3, 6, 6, 6, 5, 10, 10, 10, 8, 5, 5, 5, ... }

Contrary to popular belief, the first movement of Beethoven's Fifth Symphony has an eight-note motif, not four notes. Here of course we have just the principal melody line, without the bass line and inner voices. Before "polyphonic" ring tones became available, this is how you would hear it as a ring tone.

Compare:

Beethoven - Symphony No. 5 in C Minor (1) (Symphony no. 5 in C minor Op. 67 — 1. Allegro con brio)
http://oeis.org/A054245/listen (Pitch offset of 40 works somewhat better, sort of...)

Today is Beethoven's birthday, by the way.

Template:Sequence of the Day for December 17

A1202345: Sequence name.

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

Template:Sequence of the Day for December 18

A1202345: Sequence name

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

Template:Sequence of the Day for December 19

A1202345: Sequence name

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

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A119524: Decimal expansion of the van der Waerden-Ulam binary measure of the composites.

0.17063498029777...

The formula is ${\displaystyle \sum _{k=1}^{\infty }{\frac {k-1-\pi (k)}{2^{k}}}}$, where ${\displaystyle \pi (n)}$ is the prime counting function. The primes have a larger measure than the composites because they dominate the smaller integers.

Template:Sequence of the Day for December 21

A081244: Named periods in the Mayan/mesoamerican calendars.

1, 20, 260, 360, 365, 7200, 18980, 144000, 2880000, 57600000, 1152000000, 23040000000

The periods are named kin, winal, Tzolkin year, tun, Haab year, katun, Calendar Round, baktun, pictun, calabtun, kinchiltun, alautun. With approximately 63 million years, the alautun is probably the longest named period in any calendar. December 21, 2012 marked the end of the 13th baktun, causing some concern over the widely misunderstood prophesied descent of the god of war. But the tablet inscribed with that prophecy is incomplete, so we don't know what, if anything, the Mayans expected the god of war to do upon his descent.

Template:Sequence of the Day for December 22

A1202345: Sequence name

{ 12, 22, 2012, 3, 4, 5, 6, 7, ... }

Template:Sequence of the Day for December 23

A038772: Numbers not divisible by any of their base 10 digits.

 { 23, 27, 29, 34, 37, 38, 43, ... }

All prime numbers with more than one digit and none of those digits 1 are in this sequence.

Template:Sequence of the Day for December 24

A1202345: Sequence name

{ 12, 24, 2012, 3, 4, 5, 6, 7, ... }

Template:Sequence of the Day for December 25
{ 1, 4, 10, 20, 35, 56, 84, 120, 165, 220, 286, 364, ... }

On the first day of Christmas, my true love gave me a partridge in a pear tree. The next day, she gave me two turtle doves and another partridge in a pear tree. And the next day, three French hens, another two turtle doves and yet another partridge in a pear tree. By the seventh day, I had a serious storage problem.

If there was a way to encase all these gifts into small balls with a radius of half a unit, a triangular frame with each side twelve units long would be sufficient to organize all these gifts into a nice, little tetrahedron, since it just so happens that the running total of gifts received each day is a tetrahedral number. But of course the whole issue of storage is moot because the gifts are symbols of various concepts relevant to the celebration of Christmas.

Template:Sequence of the Day for December 26

A1202345: Sequence name

{ 12, 26, 2012, 3, 4, 5, 6, 7, ... }

Template:Sequence of the Day for December 27

A151542: Generalized pentagonal numbers: ${\displaystyle \scriptstyle 12n+{\frac {3n(n-1)}{2}},\,n\,\geq \,0.\,}$

 { 0, 12, 27, 45, 66, 90, 117, 147, 180, 216, 255, 297, 342, 390, 441, 495, 552, ... }

Originally, pentagonal numbers are those of the form ${\displaystyle \scriptstyle {\frac {3n(n\pm 1)}{2}}\,}$.

Those with the "+" sign are sometimes referred to as the "second" pentagonal numbers. The generalized pentagonal numbers ${\displaystyle \scriptstyle bn+{\frac {3n(n-1)}{2}}\,}$, for ${\displaystyle \scriptstyle b\,}$ = 1 through 12, form sequences A000326, A005449, A045943, A115067, A140090, A140091, A059845, A140672, A140673, A140674, A140675, A151542.

One might wonder whether it is a pure coincidence to find many "remarkable" numbers in this sequence: multiples of 11 (as 66, 297, 495,...) and other numbers with doubled digits (117, 255 = 2^8-1 and its reversal 552, 441 and 882, 1116, 1200, 1377, 1566 and its "permutation" 1665, 2205, 2322, 2442, 3225, 3366, ...), the pair 147 and 741; 216 = 6^3, etc...

Template:Sequence of the Day for December 28

A1202345: Sequence name

{ 12, 28, 2012, 3, 4, 5, 6, 7, ... }

Template:Sequence of the Day for December 29

A1202345: Sequence name

{ 12, 29, 2012, 3, 4, 5, 6, 7, ... }

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A202138/A203363: The numbers of Conway's PRIMEGAME

${\displaystyle {\frac {17}{91}},{\frac {78}{85}},{\frac {19}{51}},{\frac {23}{38}},{\frac {29}{33}},{\frac {77}{29}},{\frac {95}{23}},}$ ${\displaystyle {\frac {77}{19}},{\frac {1}{17}},{\frac {11}{13}},{\frac {13}{11}},{\frac {15}{2}},{\frac {1}{7}},55}$

It's a bit of a roller-coaster, but these numbers generate prime numbers.

Template:Sequence of the Day for December 31

A1202345: Sequence name

{ 12, 21, 2011, 3, 4, 5, 6, 7, ... }