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# Calendar for Sequence of the Day in April

Template:Sequence of the Day for April 1

D000019: Integers overprinted with themselves

You write the number twice, then do a flip vertical on the second instance and move it over the first instance.

Template:Sequence of the Day for April 2

A108911: Difference between ${\displaystyle n}$ and the sum of the factorials of its decimal digits.

{ 0, 0, –3, –20, –115, –714, –5033, –40312, –362871, 8, ... }

${\displaystyle a(n)=0}$ only for base 10 factorions.

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A116582: Numbers from Bhargava's 33 theorem.

{ 1, 3, 5, 7, 11, 15, 33 }

According to Bhargava's 33 theorem, an integral quadratic form represents all odd numbers if it represents the terms of this sequence.

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A555555: Sequence name

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

More details...

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A555555: Sequence name

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

More details...

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A455555: Sequence name

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

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

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A038179: Result of second stage of the sieve of Eratosthenes.

{ 2, 3, 5, 7, 11, 13, 17, 19, 23, 25, 29, 31, 35, 37, 41, 43, 47, 49, ... }

As you can see, a lot of composites have been culled out. Actually, just most even numbers and all multiples of 3 save 3 itself. Powers of larger primes still remain at this point, as well as other multiples of primes that are not also multiples of 2 or 3.

Alexander Povolotsky astutely observed that "terms of this sequence [after 2] are equal to the result of the expression ${\displaystyle {\sqrt {4!(k+1)+1}}}$—but only when this expression yields integral values."

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A455555: Sequence name

{ 4, 8, 20, 12, 7, 6, 5, ... }

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

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A455555: Sequence name

{ 4, 9, 20, 12, 7, 6, 5, ... }

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

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A203907: Successor function for Conway's PRIMEGAME.

{ 55, 15, 165, 30, 275, 45, 1, 60, 495, ... }

This sequence demonstrates that Conway's prime producing "machine" never stops working. Whatever integer is input, one of two things will happen: either the machine will take us on a roller-coaster ride towards a power of 2 with a prime exponent, or get stuck on an infinite loop between two values. But never will it happen that the machine receives an integer and refuses to do anything to it.

Template:Sequence of the Day for April 11

A027427: Number of distinct products ${\displaystyle ij}$ with ${\displaystyle 0\leq i.

{ 1, 2, 4, 7, 11, 14, 20, 25, 32, ... }

So essentially we take multiplication tables, crop out any multiplicands greater than ${\displaystyle n}$, discard squares and duplicate results and then tally up the remaining results.

Template:Sequence of the Day for April 12

A065442: Decimal expansion of the Erdős-Borwein constant ${\displaystyle \sum _{k=1}^{\infty }{\frac {1}{2^{k}-1}}}$

1.60669515241529...

In 1948, Paul Erdős proved this number is irrational. We can't say for sure if the related number ${\displaystyle \sum _{k=1}^{\infty }{\frac {\chi _{p}(2^{k}-1)}{2^{k}-1}}}$ (see A173898) is also irrational. (${\displaystyle \chi _{p}(n)}$ is the characteristic function of the primes).

Template:Sequence of the Day for April 13

A455555: Sequence name

{ 4, 13, 20, 12, 7, 6, 5, ... }

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

Template:Sequence of the Day for April 14

A003325: Numbers that are the sum of 2 positive cubes.

{ 2, 9, 16, 28, 35, 54, 65, 72, ... }

The positive cubes are not necessarily distinct (hence some terms are of the form ${\displaystyle 2n^{3}}$). Here it is necessary to specify "positive cubes" where it was not necessary with numbers that are the sum of 2 squares (since obviously ${\displaystyle n^{2}\in \mathbb {Z} ^{+}}$ as long as ${\displaystyle n\neq 0}$). With cubes we could very well have ${\displaystyle 37=4^{3}+(-3)^{3}}$, to give just one example.

A conjecture that still stands is that this sequence and the sequence of numbers of the form ${\displaystyle n^{3}\pm 3}$ have infinitely many terms in common, yet the only known example is ${\displaystyle 4^{3}+4^{3}=5^{3}+3=128}$.

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A057466: ${\displaystyle {\sqrt[{9}]{10e^{8}}}}$

{ 3.141598280658... }

Most infinite sums and products for ${\displaystyle \pi }$ converge very slowly and can require thousands of terms to get just three or four digits. So it is a bit of a wonder that an expression so simple as this gets five digits right right off the bat (five digits is good enough for many practical applications).

Template:Sequence of the Day for April 16

A029635: The "Lucas" triangle

 2 1 2 1 3 2 1 4 5 2 1 5 9 7 2 1 6 14 16 9 2 1 7 20 30 25 11 2

This variant of Pascal's triangle gives the Lucas numbers in a manner analogous to how the original triangle gives the Fibonacci numbers. If you follow the diagonals indicated by the color-coding, you will see that they add up to the Lucas numbers as listed in A000032.

Template:Sequence of the Day for April 17

A004090: Sum of base 10 digits of Fibonacci numbers.

 { 1, 1, 2, 3, 5, 8, 4, 3, 7, 10, 17, 9, 8, 17, 7, 24, 22, 19, ... }

Not to be confused with A030132, digital roots of the Fibonacci numbers: the two sequences start diverging from 55 (the tenth Fibonacci number) onwards.

Template:Sequence of the Day for April 18

A173898: Decimal expansion of sum of the reciprocals of the Mersenne primes.

 0.51645417894...

Even though we don't know whether there are any more Mersenne primes than the forty-odd we know, we do know that this number is less that the sum of reciprocals of all the Mersenne numbers (the Erdős-Borwein constant, see A065442). However, unlike that other number, we don't know if it's irrational: disproving that would require a proof of the finiteness of the Mersenne primes.

Template:Sequence of the Day for April 19

A092287: ${\displaystyle \prod _{j=1}^{n}\prod _{k=1}^{n}\gcd(j,k),\ n\geq 0.}$

 { 1, 1, 2, 6, 96, 480, 414720, 2903040, ... }

Peter Bala conjectures that the order of the primes in the prime factorization of ${\displaystyle a(n)}$ is given by the formula

${\displaystyle \operatorname {ord} _{p}\ a(n)=\sum _{k=1}^{\lfloor \log _{p}(n)\rfloor }\left\lfloor {\frac {n}{p^{k}}}\right\rfloor ^{2}=\left\lfloor {\frac {n}{p}}\right\rfloor ^{2}+\left\lfloor {\frac {n}{p^{2}}}\right\rfloor ^{2}+\left\lfloor {\frac {n}{p^{3}}}\right\rfloor ^{2}+\cdots .}$

Charles R Greathouse IV proved Bala's conjecture very recently.

Comparing this with the de Polignac–Legendre formula for the prime factorization of ${\displaystyle n!}$, i.e.

${\displaystyle \operatorname {ord} _{p}\ n!=\sum _{k=1}^{\lfloor \log _{p}(n)\rfloor }\left\lfloor {\frac {n}{p^{k}}}\right\rfloor =\left\lfloor {\frac {n}{p}}\right\rfloor +\left\lfloor {\frac {n}{p^{2}}}\right\rfloor +\left\lfloor {\frac {n}{p^{3}}}\right\rfloor +\cdots ,}$

this suggests that ${\displaystyle a(n)}$ can be considered as a generalization of the factorial numbers (the product between braces is obviously 1 if ${\displaystyle n}$ is noncomposite)

${\displaystyle {\frac {a(n)}{n!}}=\left(\prod _{k=1}^{n-1}\gcd(n,k)\right)^{2}{\frac {a(n-1)}{(n-1)!}},\quad n\geq 1.}$

Recurrence:

${\displaystyle a(0):=1;\ a(n):=n\left(\prod _{k=1}^{n-1}\gcd(n,k)\right)^{2}a(n-1),\quad n\geq 1.}$

Formula:

${\displaystyle a(n)=n!\left(\prod _{j=1}^{n}\prod _{k=1}^{j-1}\gcd(j,k)\right)^{2},\quad n\geq 0.}$

Template:Sequence of the Day for April 20

A020995: Numbers ${\displaystyle \scriptstyle n\,}$ such that sum of digits of ${\displaystyle \scriptstyle F_{n}\,}$ is ${\displaystyle \scriptstyle n\,}$.

{ 0, 1, 5, 10, 31, 35, 62, 72, 175, 180, 216, 251, 252, 360, 494, 504, 540, 946, 1188, 2222, ¿ ... ? }

The 180th Fibonacci number is 18547707689471986212190138521399707760, and

1 + 8 + 5 + 4 + 7 + 7 + 0 + 7 + 6 + 8 + 9 + 4 + 7 + 1 + 9 + 8 + 6 + 2 + 1 + 2 + 1 + 9 + 0 + 1 + 3 + 8 + 5 + 2 + 1 + 3 + 9 + 9 + 7 + 0 + 7 + 7 + 6 + 0 = 180.

Is this sequence finite? There are conflicting opinions on the matter. And even if it is finite, that does not guarantee that 2222 is the final term.

A plot of digit sums of Fibonacci numbers. The red line is ${\displaystyle \scriptstyle y\,=\,(c-1)x\,}$ where ${\displaystyle \scriptstyle c\,=\,{\frac {9}{20}}\log _{10}\phi \,}$.

The following graph by Robert Israel, suggested by David Wilson, suggests the sequence is finite.

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A555555: Sequence name

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

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

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A555555: Sequence name

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

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

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A412357: Sequence name

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

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

Template:Sequence of the Day for April 24

A187677: Primes of the form ${\displaystyle \scriptstyle 8n^{2}+6n-1\,}$.

 221 223 225 227 229 231 233 235 219 145 147 149 151 153 155 157 217 143 85 87 89 91 93 95 215 141 83 41 43 45 47 49 213 139 81 39 13 15 17 19 211 137 79 37 11 1 3 21 209 135 77 35 9 7 5 23

In a variant of the Ulam spiral in which only odd numbers are entered, some primes still line up along some diagonals but not others. Without the even numbers, primes can also line up in horizontal and diagonal lines. This sequence comes from an upwards vertical line which starts with 13.

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A180251: Decimal expansion of ${\displaystyle \scriptstyle {\frac {6}{5}}\phi ^{2}\,}$

 3.14164078649987...

This pretty good approximation to ${\displaystyle \scriptstyle \pi \,}$, discovered in an attempt to link the Fibonacci numbers to that mysterious constant, is off by only 0.00004813291... (+ 0.000015321181... %)

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A412357: Sequence name

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

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

Template:Sequence of the Day for April 27

A024816: Antisigma(n): ${\displaystyle \scriptstyle {\overline {\sigma }}(n)\,\equiv \,\sum _{i\nmid n}i\,}$

 { 0, 0, 2, 3, 9, 9, 20, 21, 32, 37, ... }

For primes ${\displaystyle \scriptstyle p\,}$, ${\displaystyle \scriptstyle {\overline {\sigma }}(p)\,=\,t_{\{p-1\}}-1\,=\,{\frac {p\,(p-1)}{2}}-1,\,}$ where ${\displaystyle \scriptstyle t_{\{p-1\}}\,}$ is the ${\displaystyle \scriptstyle (p-1)\,}$th triangular number.

Compare with Euler's cototient function ${\displaystyle \scriptstyle {\overline {\varphi }}(n)\,}$.

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A181284: Decimal expansion of ${\displaystyle \scriptstyle {\sqrt {{\frac {9}{121}}100^{30}-{\frac {1208}{121}}}}\,}$

 2727272727...272727272727 .272727...2727089696969...

(The first ellipsis omits four instances, out of fifteen, of the digit pairs 27, the second omits nine, out of fourteen.)

This is an example of a so-called zebra irrational number.

Most irrational numbers in the OEIS are small and in the unit interval (between 0 and 1). This is one is an interesting exception to that trend.

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A555555: Sequence name

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

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

Template:Sequence of the Day for April 30

A038458: Decimal expansion of Smarandache's constant.

 0.567148130202...

Given two consecutive primes ${\displaystyle \scriptstyle p\,}$ and ${\displaystyle \scriptstyle q\,}$, what is the smallest positive real number ${\displaystyle \scriptstyle x\,}$ such that ${\displaystyle \scriptstyle q^{x}-p^{x}\,=\,1\,}$? If ${\displaystyle \scriptstyle p\,=\,113\,}$ and ${\displaystyle \scriptstyle q\,=\,127\,}$, then the answer is Smarandache's constant. (We verify that ${\displaystyle \scriptstyle 113^{x}\,\approx \,14.60157}$ and ${\displaystyle \scriptstyle 127^{x}\,\approx \,15.60157}$.) Sukanto Bhattacharya conjectures that this number is irrational.

Of course if ${\displaystyle \scriptstyle p\,=\,2\,}$ and ${\displaystyle \scriptstyle q\,=\,3\,}$, the answer ${\displaystyle \scriptstyle x\,=\,1\,}$ is not quite so interesting.