A002973 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.

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, 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, 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

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn

Comments

a(n) is odd iff x^2 + y^2 == 5 (mod 8). [Vladimir Shevelev, Jul 12 2009]

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]

Examples

			The 3rd prime of the form 4i+1 is 17 = 1^2 + 4^2, so a(3) = 4/2 = 2.
		

References

• 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).

Links

• Rainer Rosenthal, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
• S. R. Finch, Powers of Euler's q-Series, arXiv:math/0701251 [math.NT], 2007.
• E. Kogbetliantz and A. Krikorian, Handbook of First Complex Prime Numbers, Gordon and Breach, NY, 1971. [Annotated scans of a few pages]

Crossrefs

Cf. A002144, A002972, A005098.

Programs

Mathematica

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, pJean-François Alcover, Feb 26 2016 *)

PARI

\\ use function decomp2sq from A002972
forprime (p=5, 1000, if (p%4==1, print1(select(x->!(x%2),decomp2sq(p))[1]/2,", "))) \\ Hugo Pfoertner, Aug 27 2022

Formula

a(n) = Min(A173331(n), A002144(n) - A173331(n)) / 2. [Reinhard Zumkeller, Feb 16 2010]

Extensions

Better description from Jud McCranie, Mar 05 2003