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List of values q^5-q^4 as q runs through A000961 (that is, 1, primes, and prime powers).
2

%I #12 Aug 20 2024 23:52:57

%S 0,16,162,768,2500,14406,28672,52488,146410,342732,983040,1336336,

%T 2345778,6156502,9375000,13817466,19803868,27705630,32505856,67469796,

%U 113030440,143589642,224465326,276710448,410305012,702806938,830750460,1056964608,1329973986,1778817670,2044673352

%N List of values q^5-q^4 as q runs through A000961 (that is, 1, primes, and prime powers).

%C Parekh et al. conjecture that q^5-q^4 is the number of nonsingular generalized Weierstrass curves over the field with q elements, for q>1.

%H Param Parekh, Paavan Parekh, Sourav Deb, and Manish K. Gupta, <a href="https://arxiv.org/abs/2310.11768">On the Classification of Weierstrass Elliptic Curves over Z_n</a>, arXiv:2310.11768 [cs.CR], 2023.

%o (Python)

%o from sympy import primepi, integer_nthroot

%o def A371195(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 kmax**4*(kmax-1) # _Chai Wah Wu_, Aug 20 2024

%Y 16*A371193 is a subsequence.

%Y Cf. A000961.

%K nonn

%O 1,2

%A _N. J. A. Sloane_, Mar 14 2024