A024698 -id:A024698 - cp's OEIS frontend

A366502 Let q = A246655(n) for n >= 2, then a(n) = (q - Kronecker(-4,q))/4 - 1.

0, 0, 0, 1, 1, 1, 2, 2, 3, 3, 4, 5, 5, 6, 6, 7, 7, 8, 9, 10, 11, 11, 12, 14, 14, 15, 16, 17, 17, 19, 19, 20, 21, 23, 24, 25, 26, 26, 27, 29, 30, 31, 31, 32, 33, 34, 36, 37, 38, 40, 41, 41, 42, 44, 44, 47, 47, 48, 49, 52

Offset: 2

Jianing Song, Oct 12 2023

If q is not a power of 2, then a(n) is the number of pairs of consecutive nonzero squares in the finite field F_q. In other words, a(n) is the number of solutions to x^((q-1)/2) = (x+1)^((q-1)/2) = 1 in F_q. This can be proved by generalizing the argument of Jack D'Aurizio to the Math Stack Exchange question "Existence of Consecutive Quadratic residues" to the case of F_q.

			a(5) = 1 because there is one pair of consecutive nonzero squares in the finite field F_q with q = A246655(5) = 7, namely {1, 2}.
a(7) = 1 because there is one pair of consecutive nonzero squares in the finite field F_q with q = A246655(7) = 9, namely {1, 2} (note that 2 = -1 = i^2 in F_9 = F_3(i)).

Jianing Song, Table of n, a(n) for n = 2..10000
Mathematics Stack Exchange, Existence of Consecutive Quadratic residues.

Cf. A246655, A101455 ({kronecker(-4,n)}), A024698.

A015518(n)-1 and A003463(n)-1 are respectively the number of consecutive nonzero squares in F_{3^n} and in F_{5^n}.

lim_A366502(N) = for(n=3, N, if(isprimepower(n), print1((n - kronecker(-4,n))/4 - 1, ", ")))

from sympy import primepi, integer_nthroot, kronecker_symbol
def A366502(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x): return int(n+x-sum(primepi(integer_nthroot(x,k)[0]) for k in range(1,x.bit_length())))
    return ((m:=bisection(f,n,n))-kronecker_symbol(-4,m)>>2)-1 # Chai Wah Wu, Jan 19 2025

A290636 The arithmetic function v_3(n,4).

Original entry on oeis.org

1, 0, 2, 1, 3, 2, 4, 2, 5, 3, 6, 3, 7, 3, 8, 4, 9, 5, 10, 6, 11, 6, 12, 6, 13, 6, 14, 7, 15, 8, 16, 9, 17, 10, 18, 9, 19, 9, 20, 10, 21, 11, 22, 11, 23, 12, 24, 14, 25, 12, 26, 13, 27, 15, 28, 15, 29, 15, 30, 15, 31, 18, 32, 16, 33, 17, 34, 18, 35

Offset: 2

Robert Price, Aug 13 2017

nonn

J. Butterworth, Examining the arithmetic function v_g(n,h). Research Papers in Mathematics, B. Bajnok, ed., Gettysburg College, Vol. 8 (2008).

Robert Israel, Table of n, a(n) for n = 2..10000
Bela Bajnok, Additive Combinatorics: A Menu of Research Problems, arXiv:1705.07444 [math.NT], May 2017. See Table in Section 1.6.1.

Cf. A211316 (equals v_1(n,3)).
Cf. A289435 - A289441.
Cf. A024698.

f:= n -> max(map(t -> (floor((t - 1 - igcd(t, 3))/4) + 1)*n/t, numtheory:-divisors(n))):
map(f, [$2..100]); # Robert Israel, Aug 16 2017

v[g_, n_, h_] := (d = Divisors[n]; Max[(Floor[(d - 1 - GCD[d, g])/h] + 1)*n/d]); Table[v[3, n, 4], {n, 2, 70}]

For n >= 3, f(prime(n)) = A024698(n-1). - Robert Israel, Aug 16 2017

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A366502 Let q = A246655(n) for n >= 2, then a(n) = (q - Kronecker(-4,q))/4 - 1.

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A290636 The arithmetic function v_3(n,4).

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