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 A178153 Difference between the numbers of quadratic residues (mod p) less than p/2 and greater than p/2, where p=prime(n). 4
 1, 0, 1, 3, 0, 0, 3, 3, 0, 3, 0, 0, 3, 5, 0, 9, 0, 3, 7, 0, 5, 9, 0, 0, 0, 5, 9, 0, 0, 5, 15, 0, 9, 0, 7, 0, 3, 11, 0, 15, 0, 13, 0, 0, 9, 9, 7, 15, 0, 0, 15, 0, 21, 0, 13, 0, 11, 0, 0, 9, 0, 9, 19, 0, 0, 9, 0, 15, 0, 0, 19, 9, 0, 9, 17, 0, 0, 0, 0, 27, 0, 21, 0, 15, 15, 0, 0, 0, 7, 21, 25, 7, 27, 9, 21, 0 (list; graph; refs; listen; history; text; internal format)
 OFFSET 2,4 COMMENTS When prime(n)=1 (mod 4), then a(n)=0. When prime(n)=3 (mod 4), then a(n)>0. When prime(n)=3 (mod 8) and prime(n)>3, then 3 divides a(n). See Borevich and Shafarevich. The nonzero terms of this sequence are closely related to A002143, the class number of primes p=3 (mod 4). Same as difference between the numbers of quadratic residues and nonresidues (mod p) less than p/2, where p=prime(n). - Jonathan Sondow, Oct 30 2011 REFERENCES Z. I. Borevich and I. R. Shafarevich, Number Theory, Academic Press, NY, 1966, p. 346. H. Davenport, Multiplicative Number Theory, Graduate Texts in Math. 74, 2nd ed., Springer, 1980, p. 51. LINKS R. J. Mathar, Table of n, a(n) for n = 2..2066 MathOverflow, Most squares in the first half-interval FORMULA a(n) = A178151(n) - A178152(n). a(n) = sum(j=1..(p-1)/2, (j|p)), where p = prime(n) and (j|p) = +/-1 is the Legendre symbol. - Jonathan Sondow, Oct 30 2011 EXAMPLE The quadratic residues of 19, the 8th prime, are 1, 4, 5, 6, 7, 9, 11, 16, 17. Hence a(8)=6-3=3. MAPLE A178153 := proc(n) local r, a, p; p := ithprime(n) ; a := 0 ; for r from 1 to p/2 do if numtheory[legendre](r, p) =1 then a := a+1 ; end if; end do: for r from ceil(p/2) to p-1 do if numtheory[legendre](r, p) =1 then a := a-1 ; end if; end do: a; end proc: # R. J. Mathar, Feb 10 2017 MATHEMATICA Table[p=Prime[n]; Length[Select[Range[(p-1)/2], JacobiSymbol[ #, p]==1&]] - Length[Select[Range[(p+1)/2, p-1], JacobiSymbol[ #, p]==1&]], {n, 2, 100}] Table[p = Prime[n]; Sum[ JacobiSymbol[a, p], {a, 1, (p-1)/2}], {n, 2, 100}] (* Jonathan Sondow, Oct 30 2011 *) CROSSREFS Cf. A178154 (without the zero terms). Sequence in context: A230652 A230661 A178952 * A267794 A282499 A216194 Adjacent sequences: A178150 A178151 A178152 * A178154 A178155 A178156 KEYWORD nonn AUTHOR T. D. Noe, May 21 2010 STATUS approved

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Last modified November 28 05:31 EST 2022. Contains 358407 sequences. (Running on oeis4.)