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# Calendar for Sequence of the Day in March

Template:Sequence of the Day for March 1

A345678: Sequence name

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

More details...

Template:Sequence of the Day for March 2

A555555: Sequence name

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

More details...

Template:Sequence of the Day for March 3

A061278: ${\displaystyle a(n)=5a(n-1)-5a(n-2)+a(n-3)}$ with ${\displaystyle a(1)=1}$.

 { 1, 5, 20, 76, 285, 1065, 3976, 14840, ... }

Gil Broussard noticed that these are "numbers ${\displaystyle n}$ such that ${\displaystyle n(n+1)=\sum _{i=1}^{x}(n+i)[=nx+\sum _{i=1}^{x}i]}$ for some ${\displaystyle x}$," except of course for the first term.

Template:Sequence of the Day for March 4

A179591: Decimal expansion of the surface area of a pentagonal cupola with edge length 1.

16.579749752988...

The pentagonal cupola is the fifth of the Johnson soilds. It has 15 vertices, 25 edges and 12 faces. The formula for its surface are is rather involved compared to the formulas for its volume and circumradius.

Template:Sequence of the Day for March 5

A555555: Sequence name

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

More details...

Template:Sequence of the Day for March 6

A555555: Sequence name

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

More details...

Template:Sequence of the Day for March 7

A555555: Sequence name

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

More details...

Template:Sequence of the Day for March 8

A001316: Gould's sequence: ${\displaystyle a(n)=\sum _{k=0}^{n}{\binom {n}{k}}\mod 2}$

{ 1, 2, 2, 4, 2, 4, 4, 8, 2, 4, 4, 8, 4, 8, 8, 16, 2, ... }

Essentially this counts how many odd entries there are in row ${\displaystyle n}$ of Pascal's triangle, and, like the sequence of row sums, this sequence also consists entirely of powers of 2. Robert Wilson noticed that the first occurrence of ${\displaystyle 2^{k}}$ is when ${\displaystyle n=2^{k}-1}$, while Benoit Cloitre discovered that ${\displaystyle a(n)}$ is the highest power of 2 dividing ${\displaystyle {\binom {2n}{n}}}$.

Template:Sequence of the Day for March 9

A064442: Decimal expansion of the number obtained by interpreting the primes as a continued fraction.

2.31303673643...

That is to say, ${\displaystyle {2+{\cfrac {1}{3+{\cfrac {1}{5+{\cfrac {1}{7+{\cfrac {1}{11+{\cfrac {1}{13+{\cfrac {1}{17+\ddots }}}}}}}}}}}}}\,}$

Template:Sequence of the Day for March 10

A132995: ${\displaystyle a(n)=\gcd \left(\sum _{k=1}^{n}p(k),\prod _{j=1}^{n}p(j)\right)}$, where ${\displaystyle p(k)}$ is the ${\displaystyle k}$-th prime.

{ 2, 1, 10, 1, 14, 1, 2, 77, 10, 3, 10, ..., 2014, 3, 14, ... }

That is, A132995(n) = GCD(A007504(n), A002110(n)).

It may be astonishing that the GCD of sum (A007504) and product (A002110) of the first ${\displaystyle n}$ primes displays such irregular behaviour. But some of the patterns find easy explanations:

• Every second term (${\displaystyle n=2,4,6,\ldots }$) of the sum A007504(n) is odd, so ${\displaystyle a(2k)}$ cannot be even. It turns out that ${\displaystyle a(2k)}$ is often (but not always) equal to 1, for small ${\displaystyle k=1,\ldots ,7}$, then often equal to 3, for ${\displaystyle k=5,\ldots ,23}$.
• Similarly, A007504(n) is even for odd ${\displaystyle n=2k-1=1,3,5,7,\ldots ,}$ therefore ${\displaystyle a(2k-1)}$ is also always even.
• Taking the GCD with the product A002110(n) amounts to have a(n) equal to the product of all those among the first n primes that divide the sum A007504(n).
• For larger terms, it is increasingly probable that A007504(n) has several of the smaller primes as factors. But looking at the very interesting graph of the sequence, it seems that there are infinitely many ${\displaystyle a(n)=1}$, ${\displaystyle a(n)=2}$ and ${\displaystyle a(n)=3}$.

### Trivia

This is the only sequence which has the terms "3, 10" and later "2014", within the displayed 3 lines of data. (It would have been a nice choice for this year's March 14, too, since the 2014 is followed by 3, 14! But there is already another Sequence_of_the_Day_for_March_14 related to π = 3.14...)

A search for "3, 10, __, 2012" yields A212068 and "3, 10, __, 2015" yields A202339 as only result, while no result is found for 2010, 2011, 2013.

Template:Sequence of the Day for March 11

A055265: Smallest positive integer not already in the sequence for which ${\displaystyle a(n)+a(n-1)}$ is prime, with ${\displaystyle a(1)=1}$.

{ 1, 2, 3, 4, 7, 6, 5, 8, 9, 10, 13, 16, 15, 14, 17, 12, 11, ... }

Dmitry Kamenetsky wonders if every positive integer occurs, or, in other words, if this sequence is a permutation of the positive integers. According to Robert Wilson, "The answer is almost certainly yes, on probabilistic grounds." But if that was not the case, what would that mean for this sequence? What kind of number would the final term of the sequence be?

Template:Sequence of the Day for March 12

A066680: Badly sieved numbers: as in the sieve of Eratosthenes multiples of unmarked numbers ${\displaystyle p}$ are marked, but only up to ${\displaystyle p^{2}}$.

{ 2, 3, 5, 7, 8, 11, 12, 13, 17, 18, 19, 23, 27, 29, ... }

So 6 gets culled out not for being an even number but for being a multiple of 3 less than 9; and 12 gets to stay because it is greater than both 4 and 9, and not a multiple of 5, 7 or 11.

Template:Sequence of the Day for March 13

A062964: ${\displaystyle \pi }$ in hexadecimal.

3.243F6A8885A3...

It should of course be noted that this is not a conversion from a single or double precision binary floating point value representation of ${\displaystyle \pi }$.

Template:Sequence of the Day for March 14

A062964: ${\displaystyle \pi }$ in hexadecimal.

{ 3, 2, 4, 3, 15, 6, 10, 8, 8, 8, 5, 10, 3, 0, 8, 13, 3, 1, 3, 1, 9, 8, 10, 2, 14, 0, ... }

The use of base 16 is very common for calculations in machines using binary arithmetic. (Hexadecimal digits, usually 0,...,9,A,...,F, are called "nibbles", and bytes are made of two nibbles, 00-FF.)

It happens that base 16 is also very relevant for calculation of digits of ${\displaystyle \pi }$. Indeed, Bailey, Borwein and Plouffe have found the formula

${\displaystyle \pi =\sum _{k=0}^{\infty }\left[{\frac {1}{16^{k}}}\left({\frac {4}{8k+1}}-{\frac {2}{8k+4}}-{\frac {1}{8k+5}}-{\frac {1}{8k+6}}\right)\right].}$

which allows to compute a given base-16 digit of ${\displaystyle \pi }$ without calculating the preceding digits.[1]

Bailey and Crandall conjecture that the terms of this sequence, apart from the first, are given by the formula ${\displaystyle \lfloor 16(x(n)-\lfloor x(n)\rfloor \rfloor }$, where ${\displaystyle x(n)}$ is determined by the recurrence equation ${\displaystyle x(n)=16x(n-1)+{\frac {120n^{2}-89n+16}{512n^{4}-1024n^{3}+712n^{2}-206n+21}}}$ with the initial condition ${\displaystyle x(0)=0}$. They have numerically verified the conjecture for the first 100,000 terms of the sequence.

Template:Sequence of the Day for March 15

A315555: Sequence title

3, 15, 2016, 4, 5, 6, 7, ...

Some remarks.

Template:Sequence of the Day for March 16

A355555: Sequence name

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

More details...

Template:Sequence of the Day for March 17

A355555: Sequence name

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

More details...

Template:Sequence of the Day for March 18

A355555: Sequence name

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

More details...

Template:Sequence of the Day for March 19

A119523: Decimal expansion of the van der Waerden-Ulam binary measure of the primes.

 0.829365019702...

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

The binary expansion of the van der Waerden-Ulam binary measure of the primes being obviously (see A010051)

0.1101010001010001010001000001010000010001010001...

The quaternary (base 4) expansion of the van der Waerden-Ulam binary measure of the primes being (see A??????)

0.31101101101001100101101...

In the quaternary expansion all the digits after the point, with the exception of the initial 3, are 0 or 1 since there is only one even prime.

Template:Sequence of the Day for March 20

A005250: Increasing gaps between primes.

 { 1, 2, 4, 6, 8, 14, 18, 20, 22, 34, 36, 44, 52, ... }

One of the great mysteries of the primes is their distribution, how some of them are so close together and some so far apart. Looking only at the increasing prime gaps does not lessen the mystery any.

Template:Sequence of the Day for March 21

A116700: "Early bird" numbers. (In base 10.)

 { 12, 21, 23, 31, 32, 34, 41, 42, 43, ... }

Concatenate the [base 10 representation of] natural numbers into the string 12345678910111213.... The sequence gives the numbers which occur in the string ahead of their "natural" place. For example, 12 appears at position 14 (the first nine positions are for the numbers 1 to 9, then 10 at position 10 and 11 at position 12), but it also appears at position 1.

The complement of A116700 gives what might be called "punctual bird" numbers (A131881). (Obviously, there can't be "late bird" numbers!)

Template:Sequence of the Day for March 22

A355555: Sequence name

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

More details...

Template:Sequence of the Day for March 23

A355555: Sequence name

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

More details...

Template:Sequence of the Day for March 24

A355555: Sequence name

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

More details...

Template:Sequence of the Day for March 25

A018834: Numbers ${\displaystyle n}$ such that decimal expansion of ${\displaystyle n^{2}}$ contains ${\displaystyle n}$ as a substring.

{ 5, 6, 10, 25, 50, 60, 76, 100, ... }

Most of the terms of this sequence are rather predictable. All the powers of 10 should be in the sequence. Review of the squares of the numbers 1 to 9 suggests that there should be at least a few terms ending in 5 or 6. So it is a pleasant surprise to find 3792 as a term: its square is 14379264. The substring doesn't have to be at the end, as with multiples of 5 or 6, nor at the beginning, as with the powers of 10.

Template:Sequence of the Day for March 26

A033308: Decimal expansion of the Copeland-Erdős constant: concatenate the base 10 representations of the prime numbers ${\displaystyle p_{i}}$ as a number in the unit interval,

${\displaystyle \sum _{i=1}^{\infty }p_{i}\,10^{-\left(\sum _{j=1}^{i}(1+\lfloor \log _{10}p_{j}\rfloor )\right)}=\sum _{i=1}^{\infty }p_{i}\,10^{-\left(i+\sum _{j=1}^{i}\lfloor \log _{10}p_{j}\rfloor \right)},}$

or

${\displaystyle \sum _{i=1}^{\infty }p_{i}\,10^{-\left(\sum _{j=1}^{i}\lceil \log _{10}(1+p_{j})\rceil \right)},}$

giving

 0.23571113171923293137...

In 1946, A. H. Copeland and Paul Erdős proved that this is a normal number.

March 26, 2013 marks the 100th anniversary of the birth of Paul Erdős.

Template:Sequence of the Day for March 27

A202955: Decimal expansion of ${\displaystyle \scriptstyle \pi {\uparrow \uparrow }4\,\equiv \,\pi ^{\pi ^{\pi ^{\pi }}}\,}$

 90802224553906177697...

This number, ${\displaystyle 9.080222\ldots \times 10^{666262452970848503}}$, is even larger than ${\displaystyle \scriptstyle e{\uparrow \uparrow }4\,\equiv \,e^{e^{e^{e}}}\,}$ (see A085667), boasting 666262452970848504 (instead of 1656521) digits to the left of the decimal point.

See more about tetration, the 4th operation.

Template:Sequence of the Day for March 28

A355555: Sequence name

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

More details...

Template:Sequence of the Day for March 29

A154293: Integers of the form: ${\displaystyle \sum _{i=1}^{k}{\frac {i}{6}}={\frac {1}{6}}+{\frac {1}{3}}+{\frac {1}{2}}+{\frac {2}{3}}+{\frac {5}{6}}+\ldots }$

{ 1, 6, 11, 13, 20, 35, 46, ... }

For example, ${\displaystyle 13={\frac {1}{6}}+{\frac {1}{3}}+{\frac {1}{2}}+{\frac {2}{3}}+{\frac {5}{6}}}$${\displaystyle +1+{\frac {7}{6}}+{\frac {4}{3}}+{\frac {3}{2}}+{\frac {5}{3}}+{\frac {11}{6}}+2}$. (If you type the right hand side of that equation into WolframAlpha, it will respond with, among other things, a pie chart representation).

Template:Sequence of the Day for March 30

A121023: Multiples of 3 containing a 3 in their decimal representation.

{ 3, 30, 33, 36, 39, 63, 93, 123, 132, 135, 138, 153, 183, 213, 231, ... }

The graph of this sequence is (roughly) self-similar: it has the same appearance when the scale is multiplied by 10.

The same can be generalized to multiples of any other number d which have d as a substring in their decimal (or other?) expansion. Sequences A011531 (numbers having the digit 1) and A121022 through A121040 cover this for d = 1, ..., 20.

Another generalization is that of numbers containing some or all of their divisors as substrings in their decimal expansion, or simply the digits thereof. Relevant sequences include:

• A092911: All divisors can be formed using the digits of the number
• A239058: All divisors are a substring of the decimal expansion, which is the union of primes having a digit 1 (A208270) and the more challenging
• A239060: Non-prime numbers having all divisors are a substring of the decimal expansion (of which the 4th term is not yet known!)

Template:Sequence of the Day for March 31

A214620: Numbers ${\displaystyle n}$ such that at least one other integer ${\displaystyle m}$ exists with the same smallest prime factor, same largest prime factor, and same set of binary digits as ${\displaystyle n}$.

{ 18, 24, 36, 42, 48, 56, 70, 72, 84, 90, 96, 98, ... }

This sequence contains a bold conjecture: the lower natural density of this sequence is at least ${\displaystyle {\frac {1}{2}}}$. Can anyone prove or disprove this?