A359129 - OEIS

%I #13 Aug 20 2024 23:52:44

%S 0,211341312,20560831566912,67802350642790400,35817806390625000000,

%T 450782974156649555296512,19045158721552047314829312,

%U 516964372056378442547769600,143027806714329275383382337600,15411735887347424297802263464512

%N q^12*(q^8+q^4+1)*(q^6-1)*(q^2-1) as q runs through the prime powers A000961.

%C For n>1, the order of the twisted Chevalley group (3)D_4(q).

%D R. W. Carter, Simple Groups of Lie Type, Wiley 1972, Chap. 14, p. 262.

%D J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker and R. A. Wilson, ATLAS of Finite Groups. Oxford Univ. Press, 1985 [for best online version see https://oeis.org/wiki/Welcome#Links_to_Other_Sites], p. xvi.

%o (Python)

%o from sympy import primepi, integer_nthroot

%o def A359129(n):

%o     def f(x): return int(n-2+x-sum(primepi(integer_nthroot(x,k)[0]) for k in range(1,x.bit_length())))

%o     kmin, kmax = 1,2

%o     while f(kmax) >= kmax:

%o         kmax <<= 1

%o     while True:

%o         kmid = kmax+kmin>>1

%o         if f(kmid) < kmid:

%o             kmax = kmid

%o         else:

%o             kmin = kmid

%o         if kmax-kmin <= 1:

%o             break

%o     return ((m:=kmax**2)*(m*(m*(m*(m*(m*(m*(m-1)+1)-2)+2)-2)+1)-1)+1)*m**6 # _Chai Wah Wu_, Aug 20 2024

%Y Cf. A000961, A037253.

%K nonn

%O 1,2

%A _N. J. A. Sloane_, Dec 28 2022