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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.
(Formerly M0135)
15
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,3
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]
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]
FORMULA
a(n) = Min(A173331(n), A002144(n) - A173331(n)) / 2. [Reinhard Zumkeller, Feb 16 2010]
EXAMPLE
The 3rd prime of the form 4i+1 is 17 = 1^2 + 4^2, so a(3) = 4/2 = 2.
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, 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 *)
PROG
(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
CROSSREFS
Sequence in context: A124458 A199538 A324337 * A071476 A071499 A039953
KEYWORD
nonn
AUTHOR
EXTENSIONS
Better description from Jud McCranie, Mar 05 2003
STATUS
approved

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Last modified June 25 17:15 EDT 2024. Contains 373706 sequences. (Running on oeis4.)