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A162156
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Table which contains in row n the mapping of the n-th block of 4 primes to 4 integers with a Diophantine formula.
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1, 31, 29, -11, 12, 434, 430, -60, 48, 1786, 1750, -360, -152, 4206, 4194, -352, -102, 8284, 8276, -378, 60, 13090, 13054, -972, 24, 20798, 20782, -816, 72, 28646, 28630, -960, 24, 41402, 41398, -576, 60, 54418, 54382, -1980, 24, 69122, 69118, -744, -1942, 85944, 85920, -2810, -896
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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COMMENTS
| Diophant's formula takes 4 integers c, b, m and r and maps them onto
four integers b*m-c*r, c*m+b*r, b*m+c*r and c*m-b*r, linked via
(c^2+b^2)*(m^2+r^2) = (b*m-c*r)^2+(c*m+b*r)^2 = (b*m+c*r)^2+(c*m-b*r)^2.
Here, the input are four consecutive primes c=prime(4n-3), b=prime(4n-2),
m=prime(4n-1) and r=prime(4n), and the four quadratic combinations which are
the bases of the squares are placed into the n-th row of the table.
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LINKS
| Eric W. Weisstein, Diophantine Equation, 2nd powers, MathWorld.
Eric W. Weisstein, Fibonacci identity, MathWorld.
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FORMULA
| T(n,1) = prime(4*n-2)*prime(4*n-1)-prime(4*n-3)*p(4*n). T(n,2) = prime(4*n-3)*prime(4*n-1)+prime(4*n-2)*prime(4*n) .
T(n,3) = prime(4*n-2)*prime(4*n-1)+prime(4*n-3)*p(4*n). T(n,4) = prime(4*n-3)*prime(4*n-1)-prime(4*n-2)*prime(4*n).
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EXAMPLE
| For n=3, the primes 23, 29, 31 and 37 are mixed via (23^2+29^2)*(31^2+37^2) = 48^2+1786^2 = 1750^2+360^2 ,
and 48, 1786, 1750 and -360 from the right hand sides fill the third row of the table.
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MAPLE
| A162156 := proc(n, k) c := ithprime(4*n-3) ; b := nextprime(c) ; m := nextprime(b) ; r := nextprime(m) ; op(k, [b*m-c*r, c*m+b*r, b*m+c*r, c*m-b*r] ) ; end: seq(seq(A162156(n, k), k=1..4), n=1..20) ; # R. J. Mathar, Sep 16 2009
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CROSSREFS
| Cf. A000040.
Sequence in context: A066434 A040932 A008685 * A141529 A022987 A023473
Adjacent sequences: A162153 A162154 A162155 * A162157 A162158 A162159
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KEYWORD
| sign,tabf
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AUTHOR
| Juri-Stepan Gerasimov (2stepan(AT)rambler.ru), Jun 26 2009
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EXTENSIONS
| Edited and extended by R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Sep 16 2009
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