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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)
14
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

T. D. Noe, Table of n, a(n) for n=1..1000

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-Fran├žois Alcover, Feb 26 2016 *)

CROSSREFS

Cf. A002144, A002972.

Sequence in context: A195079 A124458 A199538 * A071476 A071499 A039953

Adjacent sequences:  A002970 A002971 A002972 * A002974 A002975 A002976

KEYWORD

nonn

AUTHOR

N. J. A. Sloane.

EXTENSIONS

Better description from Jud McCranie, Mar 05 2003

STATUS

approved

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Last modified March 27 06:50 EDT 2017. Contains 284144 sequences.