A395950 - OEIS

%I #20 Jun 03 2026 20:36:20

%S 1,29,217,855553,7873231,49256953,3410764331297,754934464052119777,

%T 150038716720069,65876368412941710541,62914758093099533898414523,

%U 2509545192706524274297,1370396881746430171036930471254311,12858298385082857221081852030989769,13656656013407368404931

%N Odd parts of absolute field norms of first derivatives of regularized mock theta limits at primitive 2n-th roots of unity.

%C Equivalently, substitute q = xi_n*(1 - t). The coefficient of t^1 in L_n(xi_n*(1 - t)) is -xi_n*L_n'(xi_n), which has the same absolute field norm as L_n'(xi_n).

%C The terms were computed by exact arithmetic in Q(zeta_(2n)), not by numerical approximation.

%C This is a first-derivative companion to A395116, which gives the odd part of the norm of L_n(xi_n).

%H Kathrin Bringmann and Larry Rolen, <a href="https://doi.org/10.1186/s40687-015-0035-8">Radial limits of mock theta functions</a>, Research in the Mathematical Sciences, Vol. 2, No. 1 (2015), Article 17.

%H Amanda Folsom, Ken Ono, and Robert C. Rhoades, <a href="https://doi.org/10.1017/fmp.2013.3">Mock theta functions and quantum modular forms</a>, Forum of Mathematics, Pi, Vol. 1, Cambridge University Press (2013), e2.

%F a(n) = N_n / 2^v_2(N_n) where N_n = abs(Norm_{Q(xi_n)/Q}(L_n'(xi_n))), L_n(xi) = -4 * Sum_{r=0..n-1} Product_{j=1..r} (1 + xi^j)^2 * xi^(r+1), and xi_n = exp(Pi*I/n).

%e For n = 1, xi_1 = -1. We have L_1(q) = -4*q, so L_1'(-1) = -4. Thus N_1 = 4 and a(1) = 1.

%e For n = 2, a(2) = 29.

%e For n = 3, a(3) = 217 = 7 * 31, and both prime factors are 1 mod 6.

%e For n = 4, the absolute norm of L_4'(xi_4) is 219021568. The 2-adic valuation is 8, so the odd part is 219021568 / 2^8 = 855553 = 257 * 3329.

%e For n = 5, a(5) = 7873231 = 131 * 60101, and both prime factors are 1 mod 10.

%o (SageMath)

%o from sage.all import *

%o def a(n):

%o     K = CyclotomicField(2*n)

%o     xi = K.gen()

%o     R = PowerSeriesRing(K, 't', default_prec=2)

%o     t = R.gen()

%o     q = xi * (1 - t)

%o     inner = R(0)

%o     poch = R(1)

%o     for r in range(n):

%o         if r > 0:

%o             poch *= 1 + q^r

%o         inner += poch^2 * q^(r + 1)

%o     L_taylor = -4 * inner

%o     c1 = L_taylor[1]

%o     N = ZZ(abs(c1.norm(QQ)))

%o     return N // (2^N.valuation(2))

%o print([a(n) for n in range(1, 16)])

%Y Cf. A393567, A394831, A395116, A395117.

%K nonn,new

%O 1,2

%A _Joesph Daniel Burke III_, May 11 2026