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A002972
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a(n) is the odd member of {x,y}, where x^2 + y^2 is the n-th prime of the form 4i+1.
(Formerly M2221)
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16
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1, 3, 1, 5, 1, 5, 7, 5, 3, 5, 9, 1, 3, 7, 11, 7, 11, 13, 9, 7, 1, 15, 13, 15, 1, 13, 9, 5, 17, 13, 11, 9, 5, 17, 7, 17, 19, 1, 3, 15, 17, 7, 21, 19, 5, 11, 21, 19, 13, 1, 23, 5, 17, 19, 25, 13, 25, 23, 1, 5, 15, 27, 9, 19, 25, 17, 11, 5, 25, 27, 23, 29, 29, 25, 23, 19, 29, 13, 31, 31
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OFFSET
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1,2
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COMMENTS
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It appears that the terms in this sequence are the absolute values of the terms in A046730. - Gerry Myerson, Dec 02 2010
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REFERENCES
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E. Kogbetliantz and A. Krikorian, Handbook of First Complex Prime Numbers, Gordon and Breach, NY, 1971, p. 243.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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FORMULA
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EXAMPLE
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The 2nd prime of the form 4i+1 is 13 = 2^2 + 3^2, so a(2)=3.
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MATHEMATICA
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pmax = 1000; odd[p_] := Module[{k, m}, 2m+1 /. ToRules[Reduce[k>0 && m >= 0 && (2k)^2 + (2m+1)^2 == p, {k, m}, Integers]]]; For[n=1; p=5, p<pmax, p = NextPrime[p], If[Mod[p, 4] == 1, a[n] = odd[p]; Print["a(", n, ") = ", a[n]]; n++]]; Array[a, n-1] (* Jean-François Alcover, Feb 26 2016 *)
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PROG
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(PARI) decomp2sq(p) = {my (m=(p-1)/4, r, x, limit=ceil(sqrt(p))); if (p>4 && denominator(m)==1, forprime (c=2, oo, if (!issquare(Mod(c, p)), r=c; break)); x=lift (Mod(r, p)^m); until (p<limit, r=p%x; p=x; x=r); if(p^2+x^2==4*m+1, [p, x], [0, 0]), [0, 0])};
forprime (p=5, 1000, if (p%4==1, print1(select(x->x%2, decomp2sq(p))[1], ", "))) \\ Hugo Pfoertner, Aug 27 2022
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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