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%I M0135 #48 Oct 29 2023 01:49:54
%S 1,1,2,1,3,2,1,3,4,4,2,5,5,4,2,5,3,1,5,6,7,1,4,2,8,5,7,8,1,6,7,8,9,4,
%T 9,5,3,10,10,7,6,10,2,5,11,10,5,7,10,12,4,12,9,8,2,11,3,6,13,13,11,1,
%U 13,10,6,11,13,14,7,5,9,2,3,8,10,12,5,14,2,3,14,11,15,16,16,5,15,1,8,11
%N a(n) is half of the even member of {x,y}, where x^2 + y^2 is the n-th prime of the form 4i+1.
%C a(n) is odd iff x^2 + y^2 == 5 (mod 8). [_Vladimir Shevelev_, Jul 12 2009]
%C A002972(n)^2 + 4*a(n)^2 = A002144(n); A002331(n+1) = Min(A002972(n),2*a(n)) and A002330(n+1) = Max(A002972(n),2*a(n)). [_Reinhard Zumkeller_, Feb 16 2010]
%D E. Kogbetliantz and A. Krikorian, Handbook of First Complex Prime Numbers, Gordon and Breach, NY, 1971, p. 243.
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Rainer Rosenthal, <a href="/A002973/b002973.txt">Table of n, a(n) for n = 1..10000</a> (first 1000 terms from T. D. Noe)
%H S. R. Finch, <a href="https://arxiv.org/abs/math/0701251">Powers of Euler's q-Series</a>, arXiv:math/0701251 [math.NT], 2007.
%H E. Kogbetliantz and A. Krikorian, <a href="/A002970/a002970.pdf">Handbook of First Complex Prime Numbers</a>, Gordon and Breach, NY, 1971. [Annotated scans of a few pages]
%F a(n) = Min(A173331(n), A002144(n) - A173331(n)) / 2. [_Reinhard Zumkeller_, Feb 16 2010]
%e The 3rd prime of the form 4i+1 is 17 = 1^2 + 4^2, so a(3) = 4/2 = 2.
%t pmax = 1000; k[p_] := Module[{k, m}, k /. 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] = k[p]; Print["a(", n, ") = ", a[n]]; n++]]; Array[a, n-1] (* _Jean-François Alcover_, Feb 26 2016 *)
%o (PARI) \\ use function decomp2sq from A002972
%o forprime (p=5, 1000, if (p%4==1, print1(select(x->!(x%2),decomp2sq(p))[1]/2,", "))) \\ _Hugo Pfoertner_, Aug 27 2022
%Y Cf. A002144, A002972, A005098.
%K nonn
%O 1,3
%A _N. J. A. Sloane_
%E Better description from _Jud McCranie_, Mar 05 2003
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