A072903 -id:A072903 - cp's OEIS frontend

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

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... .

A186129 Numbers that can be partitioned into four parts s, t, u, v such that s+k = t-k = u*k = v/k for some k > 1.

Original entry on oeis.org

18, 27, 32, 36, 45, 48, 50, 54, 63, 64, 72, 75, 80, 81, 90, 96, 98, 99, 100, 108, 112, 117, 125, 126, 128, 135, 144, 147, 150, 153, 160, 162, 171, 175, 176, 180, 189, 192, 196, 198, 200, 207, 208, 216, 224, 225, 234, 240, 242, 243, 245, 250, 252, 256, 261

Offset: 1

Manuel Valdivia, Feb 13 2011

nonn

Equivalently, solutions n to a*(b+1)^2 = b*n with a > b >= 2.
The general rule to obtain such a partition is to start with any number b > 1 and one of its multiples a = k*b (k > 1 and a < n) and let s = a-b, t = a+b, u = a/b and v = a*b.
Sequence appears to be a subsequence of A013929, of A046790, and of A072903.

			18 = 2+6+2+8; for k=2 we have 2+2 = 6-2 = 2*2 = 8/2 = 4, hence 18 is a term.
45 = 8+12+5+20; for k=2 we have 8+2 = 12-2 = 5*2 = 20/2 = 10, hence 45 is a term.

José Estalella, Ciencia Recreativa. Gustavo Gili - Editor. Barcelona, 1918, pp. 5-6.

Klaus Brockhaus, Table of n, a(n) for n = 1..1178 (terms <= 5000)

Cf. A000041, A013929, A046790, A072903.

[ n: n in [1..300] | exists{ b: b in [2..n] | exists{ a: a in [b+1..n div 4] | n*b eq a*(b+1)^2 } } ]; // Klaus Brockhaus, Feb 15 2011

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

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A186129 Numbers that can be partitioned into four parts s, t, u, v such that s+k = t-k = u*k = v/k for some k > 1.

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