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# 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:

RECURRENCE FORMULA TO TEST IF A NUMBER IS HAPPY.

${\displaystyle b_{1}}$ 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]

SEQUENCE FRESNILLENSES NUMBERS (NÚMEROS FRESNILLENSES) .

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]

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

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]

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

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]

FORMULA FOR THE DIGITAL SUM OF A NUMBER

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

You can check: [A007953]

FORMULA FOR THE PRODUCT OF DECIMAL DIGITS OF n.

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

You can check: [A007954]

FORMULA FOR THE SEQUENCE REPDIGIT NUMBERS .

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]

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

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]

FORMULA FOR THE SEQUENCE OF NARCISSISTIC NUMBERS

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]

FORMULA FOR THE SEQUENCE OF NUMBERS WRITTEN IN BASE 2.

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]

FORMULA FOR THE SEQUENCE READ n BACKWARDS

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]

FORMULA DIGITAL ROOT OF n

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]

FORMULA OF CONGRUENT TO 0 OR 1 MOD 5.

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]

SEQUENCE WITH MANY PRIME NUMBERS AND ZEROS

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]

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

SEQUENCE OF SUM OF THE FIRST 10^n PRIMES.

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

For i>=0.

You can check: [A099824]

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

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

MAGIC TRIANGLES WITH PRIMES NUMBER

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

You can check: [2]

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

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

You can check: [A252043]