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A240370 Positive integers n such that every element in the ring of integers modulo n can be written as the sum of two squares modulo n. 2
1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 25, 26, 29, 30, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 46, 47, 50, 51, 53, 55, 57, 58, 59, 61, 62, 65, 66, 67, 69, 70, 71, 73, 74, 75, 77, 78, 79, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97, 101, 102, 103, 105, 106, 107, 109, 110, 111, 113, 114, 115, 118, 119, 122, 123, 125, 127, 129, 130, 131, 133, 134, 137, 138, 139, 141, 142, 143, 145, 146, 149, 150, 151, 154, 155, 157, 158, 159, 161, 163, 165, 166, 167, 169 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Numbers n such that, if p^2 divides n for any prime p, then p = 1 mod 4.

Equivalently, squarefree numbers times A004613.

Thus, numbers k such that A065338(A057521(k)) = 1. - Antti Karttunen, Jun 21 2014

Different from A193304: terms 169, 289, 338, 507, 578, 841, 845, 867, ... are here but not in A193304. - Michel Marcus, Jun 20 2014

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..10000

Joshua Harrington, Lenny Jones, and Alicia Lamarche, Representing Integers as the Sum of Two Squares in the Ring Z_n, arXiv:1404.0187 [math.NT], Apr 01 2014 and, J. Int. Seq. 17 (2014) # 14.7.4.

EXAMPLE

In Z_7, 0^2 + 0^2 = 0, 1^2 + 0^2 = 1, 1^2 + 1^2 = 2, 3^2 + 1^2 = 3, 2^2 + 0^2 = 4, 2^2 + 1^2 = 5, 3^2 + 2^2 = 6. Therefore 7 is in the sequence.

In Z_8, there is no way to express 3 as a sum of two squares. Therefore 8 is not in the sequence.

MATHEMATICA

rad[n_] := Times @@ First /@ FactorInteger[n];

a57521[n_] := n/Denominator[n/rad[n]^2];

a65338[n_] := a65338[n] = If[n==1, 1, Mod[p = FactorInteger[n][[1, 1]], 4]* a65338[n/p]];

Select[Range[200], a65338[a57521[#]] == 1&] (* Jean-Fran├žois Alcover, Sep 22 2018, after Antti Karttunen *)

PROG

(PARI) is(n)=my(f=factor(n)); for(i=1, #f~, if(f[i, 2]>1 && f[i, 1]%4>1, return(0))); 1

(PARI) isok(n) = { if (n < 2, return (0)); if ((n % 4) == 0, return (0)); forprime(q = 2, n, if (((q % 4) == 3) && ((n % q) == 0) && ((n % q^2) == 0), return (0)); ); return (1); } \\ Michel Marcus, Jun 08 2014

(Scheme, with Antti Karttunen's IntSeq-library)

(define A240370 (MATCHING-POS 1 1 (lambda (k) (= 1 (A065338 (A057521 k))))))

;; Antti Karttunen, Jun 21 2014

CROSSREFS

The subsequence A240109 is a version not allowing 0.

Different from A193304.

Complement of A053443. Subsequence of A192450.

Cf. A004613, A057521, A065338.

Sequence in context: A248792 A064594 A325511 * A193304 A076144 A005117

Adjacent sequences:  A240367 A240368 A240369 * A240371 A240372 A240373

KEYWORD

nonn

AUTHOR

Charles R Greathouse IV, Apr 04 2014

STATUS

approved

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Last modified October 15 19:25 EDT 2019. Contains 328037 sequences. (Running on oeis4.)