A387207 -id:A387207 - 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.

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, 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, 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

Offset: 2

Robin Visser, Aug 21 2025

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.

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.

			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.
For n = 3, every additively indecomposable element in Q(sqrt(A005117(3))) = Q(sqrt(3)) is a totally positive unit, so a(3) = 1.
For n = 4, every additively indecomposable element in Q(sqrt(A005117(4))) = Q(sqrt(5)) is a totally positive unit, so a(4) = 1.
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.

Andreas Dress and Rudolf Scharlau, Indecomposable totally positive numbers in real quadratic orders, J. Number Theory 14 (1982), no. 3, 292-306.
Se Wook Jang and Byeong Moon Kim, A refinement of the Dress-Scharlau theorem, J. Number Theory 158 (2016), 234-243.
Vítězslav Kala, Norms of indecomposable integers in real quadratic fields, J. Number Theory 166 (2016), 193-207.
Vítězslav Kala, Universal quadratic forms and indecomposables in number fields: a survey, Commun. Math. 31 (2023), no. 2, 81-114.
Magdaléna Tinková and Paul Voutier, Indecomposable integers in real quadratic fields, J. Number Theory 212 (2020), 458-482.

Cf. A005117, A035015, A387207.

def a(n):
    D = [d for d in range(2*n) if Integer(d).is_squarefree()][n-1]
    K. = QuadraticField(D); OK = K.ring_of_integers(); ans = 0
    if (D%4==1): cf, d = continued_fraction((a-1)/2), (a+1)/2
    else: cf, d = continued_fraction(a), a
    s = len(cf.period())
    ai = [1]+[c.numer() + c.denom()*d for c in cf.convergents()[:2*s+1]]
    ind = [ai[i]+t*ai[i+1] for i in range(0,2*s+1,2) for t in range(cf[i+1])]
    for i in range(len(ind)):
        for j in range(i):
            if ((ind[i]/ind[j]) in OK) and (OK(ind[i]/ind[j]).is_unit()): break
        else: ans += 1
    return ans

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.

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