User:José de Jesús Camacho Medina

I was born in Fresnillo Zacatecas Mexico, early I became interested in the Prime Numbers; those mathematical entities that have derided the order and the brightest minds of all times.I studied Engineering subsequently a Masters in Mathematics, to my 29 years I'm dedicated to teaching and investigation.In this site you can find part of my research: http://matematicofresnillense.blogspot.mx/

I love the patterns and the sequences.

Some of my contributions in OEIS.org and other:

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RECURRENCE FORMULA TO TEST IF A NUMBER IS HAPPY.

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${\displaystyle b_{1}}$ the number to test. If  ${\displaystyle b_{f}=1}$ , after some iterations is then considered ${\displaystyle b_{1}}$ is a happy number.

${\displaystyle b_{f}=\sum _{n=0}^{\lfloor ({\frac {\log(b(-1+f))}{\log(10)}})\rfloor }{(-10\,\lfloor ({10}^{-1-n}\,b(-1+f))\rfloor +\lfloor {\frac {b(-1+f)}{{10}^{n}}}\rfloor )}^{2}}$.

You can check: [A007770]

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SEQUENCE FRESNILLENSES NUMBERS (NÚMEROS FRESNILLENSES) .

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Numbers that are equal to the sum of their digits raised to each power from 1 to the number of digits. 1, 2, 3, 4, 5, 6, 7, 8, 9, 90, 336, 4538775, 183670618662, 429548754570, 3508325641459, 3632460407839, 9964270889420, 10256010588126...

For example:

${\displaystyle 9=9^{1}}$

${\displaystyle 90=(9^{1}+0^{1})+(9^{2}+0^{2})}$

${\displaystyle 336=(3^{1}+3^{1}+6^{1})+(3^{2}+3^{2}+6^{2})+(3^{3}+3^{3}+6^{3})}$

You can check: [A240511]

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FORMULA FOR SEQUENCE : NUMBERS THAT CONTAIN ONLY ONE NONZERO DIGIT.

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1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 200, 300, 400, 500, 600, 700, 800, 900, 1000, 2000, 3000, 4000... ${\displaystyle a(n)=(10^{\lfloor (n-1)/9\rfloor })*(n-9*\lfloor (n-1)/9\rfloor )}$

You can check: [A037124]

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FORMULA FOR THE SEQUENCE CONCATENATE OF NATURALS N TIMES (SMARANDACHE SEQUENCE).

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1, 22, 333, 4444, 55555, 666666, 7777777, 88888888, 999999999, 10101010101010101010...

${\displaystyle a(n)=\sum _{i=0}^{n-1}{n*10^{i*(\lfloor \log _{10}n\rfloor +1)}}}$

You can check: [A000461]

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FORMULA FOR THE DIGITAL SUM OF A NUMBER

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${\displaystyle a(n)=\sum _{k=0}^{\lfloor \log _{10}n\rfloor }{\lfloor n/10^{k}\rfloor -10*\lfloor n/10^{k+1}\rfloor }}$

You can check: [A007953]

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FORMULA FOR THE PRODUCT OF DECIMAL DIGITS OF n.

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${\displaystyle a(n)=\prod _{k=0}^{\lfloor \log _{10}n\rfloor }{\lfloor n/10^{k}\rfloor -10*\lfloor n/10^{k+1}\rfloor }}$

You can check: [A007954]

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FORMULA FOR THE SEQUENCE REPDIGIT NUMBERS .

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0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 22, 33, 44, 55, 66, 77, 88, 99, 111, 222, 333, 444, 555, 666, 777, 888, 999, 1111, 2222, 3333...

${\displaystyle a(n)=(n-9*\lfloor (n-1)/9\rfloor )*(10^{\lfloor (n+8)/9\rfloor }-1)/9}$

You can check: [A010785]

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FORMULA FOR THE SEQUENCE NUMBERS n SUCH THAT n EQUALS THE SUM OF ITS DIGITS RAISED TO THE CONSECUTIVE POWERS(1,2,3,...).

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0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 89, 135, 175, 518, 598, 1306, 1676...

For example:

${\displaystyle 2427=2^{1}+4^{2}+2^{3}+7^{4}.}$

Let

${\displaystyle a(n)=\sum _{x=0}^{\lfloor (\log _{10}n)\rfloor }({\lfloor n/10^{x}\rfloor -10*\lfloor n/10^{x+1}\rfloor })^{\lfloor \log _{10}n+1\rfloor -x}-n}$

If a(n)=0, then 'n' is a number of this sequence

You can check: [A032799]

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FORMULA FOR THE SEQUENCE OF NARCISSISTIC NUMBERS

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1, 2, 3, 4, 5, 6, 7, 8, 9, 153, 370, 371, 407, 1634, 8208...

For example:

${\displaystyle 153=1^{3}+5^{3}+3^{3}}$

Let

${\displaystyle a(x)=\sum _{n=0}^{\lfloor (\log _{10}x)\rfloor }({\lfloor x/10^{n}\rfloor -10*\lfloor x/10^{n+1}\rfloor })^{\lfloor \log _{10}x+1\rfloor }-x}$

If a(x)=0, then 'x' is a Narcissistic number

You can check: [A005188]

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FORMULA FOR THE SEQUENCE OF NUMBERS WRITTEN IN BASE 2.

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0, 1, 10, 11, 100, 101, 110, 111, 1000, 1001, 1010, 1011, 1100, 1101, 1110, 1111..

${\displaystyle a(n)=\sum _{k=0}^{\lfloor (\log _{2}n)\rfloor }({\lfloor (n/2^{k})\mod 2}\rfloor )*(10^{k})}$

You can check: [A007088]

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FORMULA FOR THE SEQUENCE READ n BACKWARDS

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0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 11, 21, 31, 41, 51, 61, 71, 81, 91, 2, 12...

${\displaystyle a(n)=\sum _{k=0}^{\lfloor (\log _{10}n)\rfloor }({\lfloor n/10^{k}\rfloor -10*\lfloor n/10^{k+1}\rfloor })*10^{\lfloor \log _{10}n\rfloor -k}}$

You can check: [A004086]

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FORMULA DIGITAL ROOT OF n

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0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 2...

${\displaystyle a(n)=n-9*\lfloor (n-1)/9\rfloor }$

You can check: [A010888]

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FORMULA OF CONGRUENT TO 0 OR 1 MOD 5.

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0, 1, 5, 6, 10, 11, 15, 16, 20, 21, 25, 26, 30, 31, 35, 36, 40, 41, 45, 46, 50, 51, 55...

${\displaystyle a(n)=\left|(n\mod {10})-(n^{2}\mod {10})\right|}$

You can check: [A008851]

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SEQUENCE WITH MANY PRIME NUMBERS AND ZEROS

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1, 2, 0, 0, 0, 0, 0, 0, 0, 0, 11, 0, 0, 0, 0, 0, 0, 0, 19, 0, 0, 0, 0, 0, 0, 0, 0, 0, 29, 0, 31, 0, 0, 0, 0, 0, 0, 0, 0, 0, 41...

${\displaystyle a(n)=n*\lfloor (\gcd {(n,Fibonacci((-1)^{n}+n))}\mod ({1+n}))/n\rfloor }$

You can check: [A245515]

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FORMULA SIEVE OF ERATOSTHENES WITH FUNCTIONS MOD AND GCD (PRIME NUMBERS)

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The Formula Produces Prime Numbers and Zeros , check this article:[1]

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SEQUENCE OF SUM OF THE FIRST 10^n PRIMES.

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${\displaystyle a(n)=\sum _{n=1}^{10^{i}}{P(n)}}$

For i>=0.

You can check: [A099824]

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FORMULA FOR CALCULATING PI APPROXIMATED WITH SIX DIGITS OF PRECISION INCLUDED GOLDEN RATIO AND EULER NUMBER

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${\displaystyle \pi \simeq {\frac {1872}{1953125}}+{\frac {1}{13}}{\sqrt {379*e*\phi }}}$

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MAGIC TRIANGLES WITH PRIMES NUMBER

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${\displaystyle {\begin{bmatrix}{0}&{0}&{0}&{17}&{0}&{0}&{0}\\{0}&{0}&{3}&{0}&{5}&{0}&{0}\\{0}&{53}&{0}&{0}&{0}&{41}&{0}\\{13}&{0}&{43}&{0}&{7}&{0}&{23}\end{bmatrix}}}$

You can check: [2]

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SEQUENCE a(n) IS THE CONCATENATION OF FIRST n TERMS

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1, 12, 123, 1234, 12345, 123456, 1234567, 12345678, 123456789, 1234567891, 12345678910, 123456789101, 1234567891011,...

You can check: [A252043]