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A240109 Positive integers n such that every element in the ring of integers modulo n can be written as the sum of two nonzero squares modulo n. 2
10, 13, 17, 26, 29, 30, 34, 37, 39, 41, 50, 51, 53, 58, 61, 65, 70, 73, 74, 78, 82, 85, 87, 89, 91, 97, 101, 102, 106, 109, 110, 111, 113, 119, 122, 123, 125, 130, 137, 143, 145, 146, 149, 150, 157, 159, 169, 170, 173, 174, 178, 181, 182, 183, 185, 187, 190, 193, 194, 195, 197 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

LINKS

Giovanni Resta and Charles R Greathouse IV, Table of n, a(n) for n = 1..10000 (first 5000 terms from Giovanni Resta)

Joshua Harrington, Lenny Jones, and Alicia Lamarche, Representing integers as the sum of two squares in the ring Z_n, arXiv:1404.0187 (2014) and J. Int. Seq. 17 (2014) # 14.7.4.

EXAMPLE

13 is a member since 0=1^2+5^2, 1=2^2+6^2, 2=1^2+1^2, 3=2^2+5^2, 4=1^2+4^2, 5=1^2+2^2, 6=3^2+6^2, 7=2^2+4^2, 8=2^2+2^2, 9=5^2+6^2, 10=1^2+3^2, 11=1^2+6^2, and 12=3^2+4^2 mod 13.

5 is not a member since there are no nonzero x and y such that x^2 + y^2 = 4 (mod 5).

MATHEMATICA

ok[n_] := Block[{t = Union@ Select[Mod[ Range[n]^2, n], # > 0 &], f = Range[n] 0}, Do[ f[[1 + Mod[t[[i]] + t[[j]], n]]]++, {i, Length@t}, {j, i}]; Position[f, 0] == {}]; Select[Range[2, 200], ok] (* Giovanni Resta, Apr 01 2014 *)

PROG

(PARI) is(n)=my(f=factor(n), P=#select(k->k%4==1, f[, 1])); if(P==0, return(0)); for(i=1, #f~, if(f[i, 2]>1 && f[i, 1]%4>1, return(0))); P>1 || n%2==0 || n%5 || n%125==0 \\ Charles R Greathouse IV, Apr 04 2014

CROSSREFS

Sequence in context: A282108 A306035 A335016 * A163652 A081642 A174274

Adjacent sequences:  A240106 A240107 A240108 * A240110 A240111 A240112

KEYWORD

nonn

AUTHOR

Lenny Jones, Mar 30 2014

STATUS

approved

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Last modified July 12 18:03 EDT 2020. Contains 335666 sequences. (Running on oeis4.)