OFFSET
1,3
COMMENTS
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.
From Vladimir Shevelev, Jun 18 2016: (Start)
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
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Quadratic Residue
Wikipedia, Quadratic Residue
FORMULA
EXAMPLE
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.
MAPLE
A130290 := proc(n): if n =1 then 1 else (A000040(n)-1)/2 fi: end: A000040 := proc(n): ithprime(n) end: seq(A130290(n), n=1..55); # Johannes W. Meijer, Oct 25 2012
MATHEMATICA
Quotient[Prime[Range[66]], 2] (* Vladimir Joseph Stephan Orlovsky, Sep 20 2008 *)
PROG
(PARI) A130290(n) = prime(n)>>1
(Magma) [Floor((NthPrime(n))/2): n in [1..60]]; // Vincenzo Librandi, Jan 16 2013
(Python)
from sympy import prime
def A130290(n): return prime(n)//2 # Chai Wah Wu, Jun 04 2022
CROSSREFS
Essentially the same as A005097.
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.
Appears in A217983. - Johannes W. Meijer, Oct 25 2012
KEYWORD
easy,nonn
AUTHOR
M. F. Hasler, May 21 2007
STATUS
approved