cp's OEIS Frontend

A387203 Number of additively indecomposable elements in the real quadratic field Q(sqrt(D)) up to multiplication by totally positive units, where D = A005117(n) is the n-th squarefree number.

%I A387203 #9 Aug 28 2025 00:43:03
%S A387203 2,1,1,2,2,6,3,3,2,1,5,7,1,6,2,10,5,2,8,4,2,1,7,6,4,11,2,13,8,2,7,7,4,
%T A387203 7,20,9,11,2,9,8,19,2,6,6,21,20,1,2,2,18,9,9,16,3,21,12,3,12,2,27,11,
%U A387203 10,18,3,34,13,17,2,8,23,12,5,18,2,22,11,24,15,26,15,6,22,27,2,31,4,2
%N A387203 Number of additively indecomposable elements in the real quadratic field Q(sqrt(D)) up to multiplication by totally positive units, where D = A005117(n) is the n-th squarefree number.
%C A387203 For any totally real field K, an additively indecomposable element of K is a totally positive element in the maximal order of K which cannot be written as the sum of two totally positive integral elements of K.  Here, an element x of K is totally positive if all conjugates of x are positive real numbers.
%C A387203 Let K = Q(sqrt(D)) be a real quadratic field. By studying the continued fraction expansion of sqrt(D), Dress and Scharlau classified all additively indecomposable elements of K and showed that every such indecomposable element has its norm bounded by the discriminant of K.
%H A387203 Andreas Dress and Rudolf Scharlau, <a href="https://doi.org/10.1016/0022-314X(82)90064-6">Indecomposable totally positive numbers in real quadratic orders</a>, J. Number Theory 14 (1982), no. 3, 292-306.
%H A387203 Se Wook Jang and Byeong Moon Kim, <a href="https://doi.org/10.1016/j.jnt.2015.06.003">A refinement of the Dress-Scharlau theorem</a>, J. Number Theory 158 (2016), 234-243.
%H A387203 Vítězslav Kala, <a href="https://doi.org/10.1016/j.jnt.2016.02.022">Norms of indecomposable integers in real quadratic fields</a>, J. Number Theory 166 (2016), 193-207.
%H A387203 Vítězslav Kala, <a href="https://doi.org/10.46298/cm.10896">Universal quadratic forms and indecomposables in number fields: a survey</a>, Commun. Math. 31 (2023), no. 2, 81-114.
%H A387203 Magdaléna Tinková and Paul Voutier, <a href="https://doi.org/10.1016/j.jnt.2019.11.005">Indecomposable integers in real quadratic fields</a>, J. Number Theory 212 (2020), 458-482.
%e A387203 For n = 2, every additively indecomposable element in Q(sqrt(A005117(2))) = Q(sqrt(2)) is of the form u or u*(2 + sqrt(2)), for some totally positive unit u. Thus a(2) = 2.
%e A387203 For n = 3, every additively indecomposable element in Q(sqrt(A005117(3))) = Q(sqrt(3)) is a totally positive unit, so a(3) = 1.
%e A387203 For n = 4, every additively indecomposable element in Q(sqrt(A005117(4))) = Q(sqrt(5)) is a totally positive unit, so a(4) = 1.
%e A387203 For n = 5, every additively indecomposable element in Q(sqrt(A005117(5))) = Q(sqrt(6)) is of the form u or u*(3 + sqrt(6)), for some totally positive unit u. Thus a(5) = 2.
%o A387203 (SageMath)
%o A387203 def a(n):
%o A387203     D = [d for d in range(2*n) if Integer(d).is_squarefree()][n-1]
%o A387203     K.<a> = QuadraticField(D); OK = K.ring_of_integers(); ans = 0
%o A387203     if (D%4==1): cf, d = continued_fraction((a-1)/2), (a+1)/2
%o A387203     else: cf, d = continued_fraction(a), a
%o A387203     s = len(cf.period())
%o A387203     ai = [1]+[c.numer() + c.denom()*d for c in cf.convergents()[:2*s+1]]
%o A387203     ind = [ai[i]+t*ai[i+1] for i in range(0,2*s+1,2) for t in range(cf[i+1])]
%o A387203     for i in range(len(ind)):
%o A387203         for j in range(i):
%o A387203             if ((ind[i]/ind[j]) in OK) and (OK(ind[i]/ind[j]).is_unit()): break
%o A387203         else: ans += 1
%o A387203     return ans
%Y A387203 Cf. A005117, A035015, A387207.
%K A387203 nonn,new
%O A387203 2,1
%A A387203 _Robin Visser_, Aug 21 2025