A002972 a(n) is the odd member of {x,y}, where x^2 + y^2 is the n-th prime of the form 4i+1.

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

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn

Comments

It appears that the terms in this sequence are the absolute values of the terms in A046730. - Gerry Myerson, Dec 02 2010

"the n-th prime of the form 4i+1" is A005098(n). - Rainer Rosenthal, Aug 24 2022

Examples

			The 2nd prime of the form 4i+1 is 13 = 2^2 + 3^2, so a(2)=3.
		

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]
• Stan Wagon, Editor's Corner: The Euclidean Algorithm Strikes Again, The American Mathematical Monthly, vol. 97, no. 2, 1990, pp. 125-29. [Description of efficient decomposition algorithm implemented in PARI program]

Crossrefs

Cf. A002144, A002973, A005098, A261858.

Programs

Mathematica

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, pJean-François Alcover, Feb 26 2016 *)

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 (px%2,decomp2sq(p))[1],", "))) \\ Hugo Pfoertner, Aug 27 2022

Formula

a(n) = Min(A173330(n), A002144(n) - A173330(n)). - Reinhard Zumkeller, Feb 16 2010

a(n)^2 + 4*A002973(n)^2 = A002144(n); A002331(n+1) = Min(a(n),2*A002973(n)) and A002330(n+1) = Max(a(n),2*A002973(n)). - Reinhard Zumkeller, Feb 16 2010

(a(n) - 1)/2 = A208295(n), n >= 1. - Wolfdieter Lang, Mar 03 2012

a(A267858(k)) == 1 (mod 4), k >= 1. - Wolfdieter Lang, Feb 18 2016

Extensions

Better description from Jud McCranie, Mar 05 2003