A037449 -id:A037449 - cp's OEIS frontend

A003246 Discriminants of real quadratic norm-Euclidean fields (a finite sequence).

5, 8, 12, 13, 17, 21, 24, 28, 29, 33, 37, 41, 44, 57, 73, 76

Offset: 1

Euclidean fields that are not norm-Euclidean, such as Q(sqrt(14)) and Q(sqrt(69)), are not included. Actually, assuming GCH, a real quadratic field is Euclidean if and only if it is a PID (equivalently, if and only if it is a UFD). - Jianing Song, Jun 09 2022

W. J. LeVeque, Topics in Number Theory. Addison-Wesley, Reading, MA, 2 vols., 1956, Vol. 2, p. 57.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
H. M. Stark, An Introduction to Number Theory. Markham, Chicago, 1970, p. 294.

S. R. Finch, Class number theory [Cached copy, with permission of the author]
Erich Kaltofen and Heinrich Rolletschek, Computing greatest common divisors and factorizations in quadratic number fields, Mathematics of Computation 53.188 (1989): 697-720. See page 698.
A. M. Odlyzko, Letters to N. J. A. Sloane Feb 1974
P. Samuel, Unique factorization, Amer. Math. Monthly 75 (1968), 945-952.
Peter J. Weinberger, On Euclidean rings of algebraic integers, Analytic number theory (Proc. Sympos. Pure Math., Vol. XXIV, St. Louis Univ., St. Louis, Mo., 1972), pp. 321-332.
Index entries for sequences related to quadratic fields

Cf. A003174, A003656.

A003174 = {2, 3, 5, 6, 7, 11, 13, 17, 19, 21, 29, 33, 37, 41, 57, 73}; Sort[ NumberFieldDiscriminant /@ Sqrt[A003174]] (* Jean-François Alcover, Jul 18 2012 *)

for(n=1,99,is_A003174(n) && print1(quaddisc(n)",")) \\ M. F. Hasler, Jan 26 2014

Equals A037449(A003174) as a set, not composition of functions (values are sorted by size; it turns out that a(n) is different from A037449(A003174(n)) for all n=1,...,16). - M. F. Hasler, Jan 26 2014

A072903 Numbers m such that the discriminant of the quadratic field Q(sqrt(m)) < m.

Original entry on oeis.org

4, 9, 16, 18, 20, 25, 27, 32, 36, 45, 48, 49, 50, 52, 54, 63, 64, 68, 72, 75, 80, 81, 84, 90, 96, 98, 99, 100, 108, 112, 116, 117, 121, 125, 126, 128, 132, 135, 144, 147, 148, 150, 153, 160, 162, 164, 169, 171, 175, 176, 180, 189, 192, 196, 198, 200, 207, 208, 212

Offset: 1

Benoit Cloitre, Aug 10 2002

Among terms are all the squares for which discriminant = 1.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A037449, A072904.

Select[Range[220], NumberFieldDiscriminant[Sqrt[#]] < # &] (* Amiram Eldar, Jun 24 2022 *)

isok(m) = quaddisc(m) < m; \\ Michel Marcus, Feb 18 2021

a(n) appears to be asymptotic to C*n with C = 3.4... .

A072904 Nonsquares m such that the discriminant of the quadratic field Q(sqrt(m)) < m.

Original entry on oeis.org

18, 20, 27, 32, 45, 48, 50, 52, 54, 63, 68, 72, 75, 80, 84, 90, 96, 98, 99, 108, 112, 116, 117, 125, 126, 128, 132, 135, 147, 148, 150, 153, 160, 162, 164, 171, 175, 176, 180, 189, 192, 198, 200, 207, 208, 212, 216, 224, 228, 234, 240, 242, 243, 244, 245, 250

Offset: 1

Benoit Cloitre, Aug 10 2002

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A037449.

Intersection of A000037 and A072903.

Select[Range[250], !IntegerQ@Sqrt[#] && NumberFieldDiscriminant[Sqrt[#]] < # &] (* Amiram Eldar, Jun 24 2022 *)

isok(m) = !issquare(m) && (quaddisc(m) < m); \\ Michel Marcus, Feb 18 2021

a(n) appears to be asymptotic to C*n with C = 3.4... .

A204993 Negative of the discriminant of quadratic field Q(sqrt(-n)).

Original entry on oeis.org

4, 8, 3, 4, 20, 24, 7, 8, 4, 40, 11, 3, 52, 56, 15, 4, 68, 8, 19, 20, 84, 88, 23, 24, 4, 104, 3, 7, 116, 120, 31, 8, 132, 136, 35, 4, 148, 152, 39, 40, 164, 168, 43, 11, 20, 184, 47, 3, 4, 8, 51, 52, 212, 24, 55, 56, 228, 232, 59, 15, 244, 248, 7, 4, 260

Offset: 1

Artur Jasinski, Jan 27 2012

For the discriminant of the quadratic field Q(sqrt(n)), see A037449.

a(n) is the smallest positive N such that ((-n)/k) = ((-n)/(k mod N)) for every odd k that is coprime to n, where ((-n)/k) is the Jacobi symbol. As we have Dirichlet's theorem on arithmetic progressions, a(n) is also the smallest positive N such that ((-n)/p) = ((-n)/(p mod N)) for every odd prime p that is not a factor of n. - Jianing Song, May 16 2024

Cf. A037449, A007913.

-Table[NumberFieldDiscriminant[Sqrt[-n]], {n, 1, 70}]

a(n) = -quaddisc(-n) \\ Jianing Song, May 16 2024

Let b(n) = A007913(n), then a(n) = b(n) if b(n) == 3 (mod 4) and 4*b(n) otherwise. - Jianing Song, May 16 2024

A072902 Nonprime numbers m such that the discriminant of the quadratic field Q(sqrt(m)) equals m.

Original entry on oeis.org

1, 8, 12, 21, 24, 28, 33, 40, 44, 56, 57, 60, 65, 69, 76, 77, 85, 88, 92, 93, 104, 105, 120, 124, 129, 133, 136, 140, 141, 145, 152, 156, 161, 165, 168, 172, 177, 184, 185, 188, 201, 204, 205, 209, 213, 217, 220, 221, 232, 236, 237, 248, 249, 253, 264, 265, 268

Offset: 1

Benoit Cloitre, Aug 10 2002

A subset of the nonprime numbers (A018252).

Positive fundamental discriminants (A003658) that are not Pythagorean primes (A002144). - Paul Muljadi, Mar 30 2008

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A037449.

Cf. A002144, A018252, A003658.

FundamentalDiscriminantQ[d_] := Module[{m, mod = Mod[d, 4]}, If[mod > 1, Return[False]]; If[mod == 1, Return[ SquareFreeQ[d] && d != 1]]; m = d/4; Return[ SquareFreeQ[m] && Mod[m, 4] > 1]]; Join[{1}, Select[ Range[270], !PrimeQ[#] && FundamentalDiscriminantQ[#]& ]] (* Jean-François Alcover, Jun 05 2012, after Eric W. Weisstein *)

isok(m) = !isprime(m) && (quaddisc(m) == m); \\ Michel Marcus, Feb 18 2021

a(n) appears to be asymptotic to C*n with C = 3.91... .

A236546 Discriminant of A048981(n) (= squarefree integers for which the quadratic field Q[sqrt(D)] is norm-Euclidean).

Original entry on oeis.org

-11, -7, -3, -8, -4, 8, 12, 5, 24, 28, 44, 13, 17, 76, 21, 29, 33, 37, 41, 57, 73

Offset: 1

M. F. Hasler, Jan 28 2014

sign

Note that here, values are not sorted by size, but in direct correspondence with terms of A048981. This is in contrast to A003246, where the same positive values are listed by size.

Cf. A003174, A003246, A037449, A048981, A204993, A187776.

for(D=-11,73,is_A048981(D)&&print1(quaddisc(D)","))

PARI
```
A236546 = n->quaddisc(A048981(n))
```

a(n) = A037449(A048981(n)) for n>5, a(n) = -A204993(-A048981(n)) for n <= 5.

A072900 Discriminant of quadratic field Q(sqrt(prime(n))) where prime(n) is the n-th prime.

Original entry on oeis.org

8, 12, 5, 28, 44, 13, 17, 76, 92, 29, 124, 37, 41, 172, 188, 53, 236, 61, 268, 284, 73, 316, 332, 89, 97, 101, 412, 428, 109, 113, 508, 524, 137, 556, 149, 604, 157, 652, 668, 173, 716, 181, 764, 193, 197, 796, 844, 892, 908, 229, 233, 956, 241, 1004, 257, 1052

Offset: 1

Benoit Cloitre, Aug 10 2002

a(n) = A037449(A000040(n)).

If[Mod[#,4]==1,#,4#]&/@Prime[Range[60]] (* Harvey P. Dale, Jul 26 2015 *)

PARI
```
a(n)=if(n<0,0,coredisc(prime(n)))
```

a(n) = if prime(n)=4k+1 then prime(n) else prime(n)*4.

a(n) = (3*floor((prime(n) mod 4)/2) + 1)*prime(n). - Reinhard Zumkeller, Aug 20 2002

A072901 Composite numbers n such that the discriminant of the quadratic field Q(sqrt(n)) equals 4n.

Original entry on oeis.org

6, 10, 14, 15, 22, 26, 30, 34, 35, 38, 39, 42, 46, 51, 55, 58, 62, 66, 70, 74, 78, 82, 86, 87, 91, 94, 95, 102, 106, 110, 111, 114, 115, 118, 119, 122, 123, 130, 134, 138, 142, 143, 146, 154, 155, 158, 159, 166, 170, 174, 178, 182, 183, 186, 187, 190, 194, 195

Offset: 1

Benoit Cloitre, Aug 10 2002

Conjecture: All terms are squarefree. Example: 6=2*3; 15=3*5; 30=2*3*5; 154=2*7*11; 195=3*5*13. - Vincenzo Librandi, Aug 08 2010 and Michel Marcus, Nov 26 2013

If prime numbers were accepted, then sequence A230375 would have been obtained. - Michel Marcus, Nov 26 2013

Wikipedia, Quadratic field

Cf. A037449.

isok(n) = !isprime(n) && (quaddisc(n) == 4*n); \\ Michel Marcus, Nov 26 2013

cp's OEIS Frontend

A003246 Discriminants of real quadratic norm-Euclidean fields (a finite sequence).

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A072903 Numbers m such that the discriminant of the quadratic field Q(sqrt(m)) < m.

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A072904 Nonsquares m such that the discriminant of the quadratic field Q(sqrt(m)) < m.

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A204993 Negative of the discriminant of quadratic field Q(sqrt(-n)).

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A072902 Nonprime numbers m such that the discriminant of the quadratic field Q(sqrt(m)) equals m.

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A236546 Discriminant of A048981(n) (= squarefree integers for which the quadratic field Q[sqrt(D)] is norm-Euclidean).

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A072900 Discriminant of quadratic field Q(sqrt(prime(n))) where prime(n) is the n-th prime.

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A072901 Composite numbers n such that the discriminant of the quadratic field Q(sqrt(n)) equals 4n.

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