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# Calendar for Sequence of the Day in May

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Template:Sequence of the Day for May 1

A057660: ${\displaystyle \scriptstyle \sum _{k=1}^{n}{\frac {n}{\gcd(n,k)}}.\,}$

 { 1, 3, 7, 11, 21, 21, 43, 43, 61, 63, ... }

Alternatively, we can take the fractions ${\displaystyle \scriptstyle {\frac {1}{n}},\,\ldots ,\,{\frac {n-1}{n}},\,1\,}$ expressed in lowest terms, and add up the denominators. For example, for ${\displaystyle \scriptstyle n\,=\,6\,}$, we have ${\displaystyle \scriptstyle {\frac {1}{6}},{\frac {1}{3}},{\frac {1}{2}},{\frac {2}{3}},{\frac {5}{6}},1\,}$, and 6 + 3 + 2 + 3 + 6 + 1 = 21.

Template:Sequence of the Day for May 2

A555555: Sequence name

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

More details, two sentences maybe, two paragraph tops...

Template:Sequence of the Day for May 3

A182369: Decimal expansion of Jenny's constant ${\displaystyle \scriptstyle \left(7^{(e-{\frac {1}{e}})}-9\right)\,\pi ^{2}\,}$.

 867.53090198...

The first seven digits of this number are the digits of the phone number in the song "867-5309/Jenny" performed by Tommy Tutone.

Template:Sequence of the Day for May 4

A210209: Greatest common divisor of all sums of ${\displaystyle \scriptstyle n\,}$ consecutive Fibonacci numbers.

 { 1, 1, 2, 1, 1, 4, 1, 3, 2, 11, 1, 8, ... }

If you add up any ten consecutive Fibonacci numbers, that sum is a multiple of 11. Or if you add up any fifty-nine consecutive Fibonacci numbers, that sum is a multiple of 1149851 (59 × 19489). However, for most ${\displaystyle \scriptstyle n\,}$, adding up ${\displaystyle \scriptstyle n\,}$ consecutive Fibonacci numbers gives a sequence of coprime numbers (the 1s in this sequence).

Template:Sequence of the Day for May 5

A124091: Decimal expansion of the Fibonacci binary constant ${\displaystyle \scriptstyle \sum _{n=0}^{\infty }{\frac {1}{2^{F_{n}}}}\,}$, where ${\displaystyle \scriptstyle F_{n}\,}$ is the ${\displaystyle \scriptstyle n\,}$th Fibonacci number

 2.41027879720786589...

The binary expansion of this number then of course corresponds to the characteristic function of the Fibonacci numbers (see A010056). This has been known to be a transcendental number, but only for a few decades.

Template:Sequence of the Day for May 6

A555555: Sequence name

 { 5, 6, 20, 12, 7, 6, 5, ... }

More details, two sentences maybe, two paragraph tops...

Template:Sequence of the Day for May 7

A007535: Smallest pseudoprime ${\displaystyle \scriptstyle m\,}$ greater than ${\displaystyle \scriptstyle n\,}$ to base ${\displaystyle \scriptstyle n\,}$: smallest composite number ${\displaystyle \scriptstyle m\,>\,n\,}$ such that ${\displaystyle \scriptstyle n^{m-1}-1\,}$ is divisible by ${\displaystyle \scriptstyle m\,}$.

 { 4, 341, 91, 15, 124, 35, 25, 9, 28, ... }

Here the term "pseudoprime" is derived from Fermat's little theorem (see Fermat pseudoprimes): prime numbers satisfy the specified congruence for any coprime base, but these composite numbers satisfy it for a specific base (and possibly others as well)—and hence the theorem provides a useful but not conclusive test of primality.

Template:Sequence of the Day for May 8

A555555: Sequence name

 { 5, 8, 20, 12, 7, 6, 5, ... }

More details, two sentences maybe, two paragraph tops...

Template:Sequence of the Day for May 9

A555555: Sequence name

 { 5, 9, 20, 12, 7, 6, 5, ... }

More details, two sentences maybe, two paragraph tops...

Template:Sequence of the Day for May 10

A001058: 1-digit numbers in reverse alphabetical order of their names (in English), then 2-digit numbers, etc.

 { 0, 2, 3, 6, 7, 1, 9, 4, 5, 8, 22, 23, 26, 27, 21, ... }

You see, that's zero, two, three, six, seven, one, nine, four, five, eight, then twenty-two, twenty-three, and so on and so forth (we don't say twenty-zero in English...).

Template:Sequence of the Day for May 11

A215889: Decimal expansion of ${\displaystyle \scriptstyle {\sqrt[{3}]{1729}}\,}$.

 { 12.0023143684... }

In his poem for Richard P. Feynman, Elliot Wolf gives this number as "12.002314 to be precise." This number is of course irrational.

1729 is the Hardy-Ramanujan number, the smallest positive integer which is the sum of two cubes in two different ways, ${\displaystyle \scriptstyle 9^{3}+10^{3}\,=\,1^{3}+12^{3}\,}$. 1729 is thus a quasi-cube (i.e 1 more than a cube).

Template:Sequence of the Day for May 12

A555555: Sequence name

 { 5, 12, 20, 12, 7, 6, 5, ... }

More details, two sentences maybe, two paragraph tops...

Template:Sequence of the Day for May 13

A555555: Sequence name

 { 5, 13, 20, 12, 7, 6, 5, ... }

More details, two sentences maybe, two paragraph tops...

Template:Sequence of the Day for May 14

A161914: Gaps between the nontrivial zeros of the Riemann zeta function, rounded to nearest integers.

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

This is assuming the Riemann hypothesis (i.e. that all the nontrivial zeros ${\displaystyle \scriptstyle z\,}$ have ${\displaystyle \scriptstyle \Re (z)\,=\,{\frac {1}{2}}\,}$): the first zero has an imaginary part of about 14.1347 and the second about 21.022, so this difference of 6.887 gets rounded up to 7.

Template:Sequence of the Day for May 15

A555555: Sequence name

 { 5, 14, 20, 12, 7, 6, 5, ... }

More details, two sentences maybe, two paragraph tops...

Template:Sequence of the Day for May 16

A077914: Numbers which can be expressed as the sum of two distinct primes in exactly two ways.

 { 16, 18, 20, 22, 26, 28, 32, 62, 68, ¿ ... ? }

Here we have another sequence believed to be finite and given in full, but we have no proof that 68 is the final term or at least that the sequence is finite. This question is related to the Goldbach conjecture thus: it appears that for most larger even numbers, there are a lot more than two ways to express them as a sum of distinct primes. For example, 70 = 67 + 3 = 59 + 11 = 53 + 17 = 47 + 23 = 41 + 29.

Template:Sequence of the Day for May 17

A555555: Sequence name

 { 5, 17, 20, 12, 7, 6, 5, ... }

More details, two sentences maybe, two paragraph tops...

Template:Sequence of the Day for May 18

A016189: ${\displaystyle \scriptstyle 10^{n}-9^{n},\,n\,\geq \,0.\,}$

 { 0, 1, 19, 271, 3439, 40951, 468559, 5217031, ... }

This sequence can also be computed using a recurrence relation: ${\displaystyle \scriptstyle a(0)\,=\,0;\,a(n)\,=\,9\,a(n-1)+10^{n-1}\,}$. This shows that this sequence answers the question: "How many numbers in ${\displaystyle \scriptstyle \{0,\,\ldots ,\,10^{n}-1\}\,}$ have at least one 9 among their digits?" Up to 10, there is of course just 9. Up to 100 we have 9, 19, 29, ... 89 (that's nine numbers) and 90, 91, 92, ... 99 (that's ten numbers). And up to 1000, we have 900, 901, 902, ... 999 (that's a hundred numbers), and between each multiple of 100 to the next, up to 800, we have nineteen numbers with 9s among their digits, so we can just multiply the previous term by 9 rather than count these numbers one by one, hence the recurrence.

Of course this can also be used for how many numbers (base 10) in ${\displaystyle \scriptstyle \{0,\,\ldots ,\,10^{n}-1\}\,}$ have

• at least one digit greater than or equal to 4: ${\displaystyle \scriptstyle 10^{n}-4^{n},\,n\,\geq \,0;\,}$
• at least one digit greater than or equal to 5: ${\displaystyle \scriptstyle 10^{n}-5^{n},\,n\,\geq \,0;\,}$
• at least one digit greater than or equal to 1: ${\displaystyle \scriptstyle 10^{n}-1^{n},\,n\,\geq \,0.\,}$

This should also work for how many numbers (base ${\displaystyle \scriptstyle b\,}$) in ${\displaystyle \scriptstyle \{0,\,\ldots ,\,b^{n}-1\}\,}$ have ...

Template:Sequence of the Day for May 19

A001142: ${\displaystyle \prod _{k=1}^{n}k^{2k-1-n}}$ or ${\displaystyle \prod _{k=0}^{n}{\binom {n}{k}}}$.

 { 1, 1, 2, 9, 96, 2500, 162000, ... }

The second formula here shows that his sequence is also the products of consecutive horizontal rows of the Pascal triangle. As it happens, these numbers are related to Euler's number ${\displaystyle e}$ thus:

${\displaystyle \lim _{n\to \infty }{\frac {a(n-1)a(n+1)}{a(n)^{2}}}=e.}$

Template:Sequence of the Day for May 20

A555555: Sequence name

 { 5, 21, 20, 12, 7, 6, 5, ... }

More details, two sentences maybe, two paragraph tops...

Template:Sequence of the Day for May 21

A104748: Decimal expansion of the solution to ${\displaystyle \scriptstyle x\,2^{x}\,=\,1\,}$.

 0.6411857445...

Jonathan Sondow has shown not only that the solution is the infinite power tower ${\displaystyle \scriptstyle {\frac {1}{2}}^{{\frac {1}{2}}^{{\frac {1}{2}}^{.^{.^{.}}}}}\,}$, but also that it is a transcendental number.

Template:Sequence of the Day for May 22

A100943: Decimal expansion of the infinitely nested radical ${\displaystyle \scriptstyle {\sqrt {e+{\sqrt {e+{\sqrt {e+\ldots }}}}}}\,}$.

 2.222870229721...

This number is the solution to the equation ${\displaystyle \scriptstyle y^{2}-y-e\,=\,0\,}$.

Template:Sequence of the Day for May 23

A083876: Smallest Fermat pseudoprime to bases 2 through ${\displaystyle \scriptstyle n\,}$th prime ${\displaystyle \scriptstyle p_{n}.\,}$

 { 341, 1105, 1729, 29341, 29341, 162401, 252601, ... }

One of the less heralded properties of 1729 is that it is pseudoprime to bases 2, 3 and 5. However, it is not pseudoprime to base 7. There are numbers smaller than 1729 that are pseudoprime to base 7 and also to some other smaller bases, but the smallest pseudoprime to bases 2, 3, 5 and 7 is 29341 (which, quite coincidentally, has its three least significant digits in decimal equal to the smallest base 2 pseudoprime).

Template:Sequence of the Day for May 24

A164102: Decimal expansion of ${\displaystyle \scriptstyle 2\pi ^{2}\,}$.

 19.739208802...

A lot of us have quite enough trouble with just three dimensions. The hypersurface "area" of a unit hypersphere in four dimensions (i.e. a 3-sphere) is ${\displaystyle \scriptstyle 2\pi ^{2}\,}$ (length unit)3. The "volume" of the contained hyperball (i.e. a 4-ball) is ${\displaystyle \scriptstyle {\frac {\pi ^{2}}{2}}\,}$ (length unit)4. Compare it with the 3-dimensional unit ball: the surface area of a unit sphere in three dimensions (i.e. a 2-sphere) is ${\displaystyle \scriptstyle 4\pi \,}$ (length unit)2. The volume of the contained ball (i.e. a 3-ball) is ${\displaystyle \scriptstyle {\frac {4\pi }{3}}\,}$ (length unit)3.

For the "volume" of hyperballs in different dimensions see: The Volume of a Hypersphere.

Template:Sequence of the Day for May 25

A202300: Decimal expansion of the real root of ${\displaystyle \scriptstyle x^{3}+2x^{2}+10x-20\,}$.

 1.36880810782137...

Today, we could get a thousand digits of this number in any base we wanted just by pushing a few buttons. But in Fibonacci's day, it was quite an achievement to get this number correct to five sexagesimal places, ${\displaystyle \scriptstyle 1+{\frac {22}{60}}+{\frac {7}{60^{2}}}+{\frac {42}{60^{3}}}+{\frac {33}{60^{4}}}+{\frac {4}{60^{5}}}+{\frac {40}{60^{6}}}\,}$, approximately 1.3688081078532235 in decimal, which as you can see matches the correct answer to ten decimal places.

Template:Sequence of the Day for May 26

A555555: Sequence name

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

More details, two sentences maybe, two paragraph tops...

Template:Sequence of the Day for May 27

A166532: Decimal expansion of ${\displaystyle \scriptstyle {\left(e^{\pi {\sqrt {163}}}\right)}^{6}\,}$.

 3274516666390792...

The integer part of this near integer is 105 digits long. ${\displaystyle \scriptstyle e^{\pi {\sqrt {163}}}\,}$ is Ramanujan's constant.

Template:Sequence of the Day for May 28

A555555: Sequence name

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

More details, two sentences maybe, two paragraph tops...

Template:Sequence of the Day for May 29

A555555: Sequence name

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

More details, two sentences maybe, two paragraph tops...

Template:Sequence of the Day for May 30

A053003: Simple continued fraction for Gauß's constant ${\displaystyle \scriptstyle {\frac {2}{\pi }}\int _{0}^{1}{\frac {1}{\sqrt {1-x^{4}}}}dx\,}$

${\displaystyle 1+{\frac {1}{5+{\cfrac {1}{21+{\cfrac {1}{3+{\cfrac {1}{\ddots \qquad {}}}}}}}}}\,}$

This is the reciprocal of the arithmetic-geometric mean of 1 and ${\displaystyle \scriptstyle {\sqrt {2}}\,}$. It was on May 30, 1799 that Carl Friedrich Gauß discovered the integral for this number shown above. (For its decimal expansion, see A014549.)

Template:Sequence of the Day for May 31

A555555: Sequence name

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

More details, two sentences maybe, two paragraph tops...