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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. 3
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
The asymptotic density of this sequence is 3/(8*K^2) = (3/4) * A243379 = 0.64208..., where K is the Landau-Ramanujan constant (A064533). - Amiram Eldar, Dec 19 2020
LINKS
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-François Alcover, Sep 22 2018, after Antti Karttunen *)
Select[Range[200], AllTrue[FactorInteger[#], Mod[First[#1], 4] == 1 || Last[#1] == 1 &] &] (* Amiram Eldar, Dec 19 2020 *)
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.
Sequence in context: A358976 A064594 A325511 * A193304 A333634 A348499
KEYWORD
nonn
AUTHOR
STATUS
approved

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Last modified March 28 03:28 EDT 2024. Contains 371235 sequences. (Running on oeis4.)