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. (Formerly M2221)

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

1,2

Comments

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

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

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

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]

Formula

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

Example

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

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, 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 *)

Prog

(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

Crossrefs

Cf. A002144, A002973, A005098, A261858.

Sequence in context: A348161 A334194 A046730 * A324896 A029652 A238952

Adjacent sequences: A002969 A002970 A002971 * A002973 A002974 A002975

Keyword

nonn

Author

N. J. A. Sloane

Extensions

Better description from Jud McCranie, Mar 05 2003

Status

approved