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A130290
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Number of nonzero quadratic residues modulo the n-th prime.
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19
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1, 1, 2, 3, 5, 6, 8, 9, 11, 14, 15, 18, 20, 21, 23, 26, 29, 30, 33, 35, 36, 39, 41, 44, 48, 50, 51, 53, 54, 56, 63, 65, 68, 69, 74, 75, 78, 81, 83, 86, 89, 90, 95, 96, 98, 99, 105, 111, 113, 114, 116, 119, 120, 125, 128, 131, 134, 135, 138, 140, 141, 146, 153, 155, 156, 158
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OFFSET
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1,3
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COMMENTS
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Row lengths for formatting A063987 as a table: The number of nonzero quadratic residues modulo a prime p equals floor(p/2), or (p-1)/2 if p is odd. The number of squares including 0 is (p+1)/2, if p is odd (rows prime(i) of A096008 formatted as a table). In fields of characteristic 2, all elements are squares. For any m > 0, floor(m/2) is the number of even positive integers less than or equal to m, so a(n) also equals the number of even positive integers less than or equal to the n-th prime. For all n > 0, A130290(n+1) = A005097(n) = A102781(n+1) = A102781(n+1) = A130291(n+1)-1 = A111333(n+1)-1 = A006254(n)-1.
a(1)+2 and, for n >= 2, a(n)+1 is the smallest k such that there exists 0 < k_1 < k with the condition k_1^2 == k^2 (mod prime(n)).
Indeed, for n >= 2, if prime(n) = 4*t+1 then k = 2*t+1 = a(n)+1, since (2*t+1)^2 == (2*t)^2 (mod prime(n)) and there cannot be a smaller value of k; if prime(n) = 4*t-1, then k = 2*t = a(n)+1, since (2*t)^2 == (2*t-1)^2 (mod prime(n)). (End)
a(n) is the number of pairs (a,b) such that a + b = prime(n) with 1 <= a <= b. - Nicholas Leonard, Oct 02 2022
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LINKS
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FORMULA
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a(n) = floor( A000040(n)/2 ) = #{ even positive integers <= A000040(n) }
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EXAMPLE
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a(1)=1 since the only nonzero element of Z/2Z equals its square.
a(3)=2 since 1=1^2=(-1)^2 and 4=2^2=(-2)^2 are the only nonzero squares in Z/5Z.
a(1000000) = 7742931 = (prime(1000000)-1)/2.
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MAPLE
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MATHEMATICA
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PROG
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(Python)
from sympy import prime
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CROSSREFS
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Cf. A102781 (Number of even numbers less than the n-th prime), A063987 (quadratic residues modulo the n-th prime), A006254 (Numbers n such that 2n-1 is prime), A111333 (Number of odd numbers <= n-th prime), A000040 (prime numbers), A130291.
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KEYWORD
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easy,nonn
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AUTHOR
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STATUS
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approved
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