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%I #29 May 11 2019 10:40:51
%S 2,6,15,30,48,72,99,154,210,270,360,420,483,598,754,870,990,1155,1260,
%T 1404,1599,1804,2112,2400,2550,2703,2862,3024,3528,4095,4420,4692,
%U 5106,5550,5850,6318,6723,7138,7654,8010,8550,9120,9408,9702,10395,11655
%N Number of lattice points on or inside the rectangle formed by [1 <= x <= (q-1)/2] and [1 <= y <= (p-1)/2], where p = n-th prime, q = (n-1)-st prime.
%C Koshy, p. 499, states "We now employ this geometric approach to establish the lemma. It is due to the German mathematician Ferdinand Eisenstein, a student of Gauss at Berlin" (where the geometric lemma applies to the Law of Quadratic Reciprocity, Koshy, p. 501): "Let p and q be distinct odd primes. Then (p/q)(q/p) = (-1)^[(p-1)/2 * (q-1)/2]." Here (p/q) denotes the Legendre symbol.
%D Thomas Koshy, "Elementary Number Theory with Applications", Harcourt Academic Press; 2002; pp. 498-500.
%H Ben Lynn, <a href="https://crypto.stanford.edu/pbc/notes/numbertheory/gausslemma.html">Gauss' Lemma</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Gauss%27s_lemma_(number_theory)">Gauss's lemma</a>
%F a(n) = (prime(n-1) - 1) * (prime(n) - 1) / 4.
%F a(n) = Sum_{k=1..(p-1)/2} floor(k*q/p) + Sum_{k=1..(q-1)/2} floor(k*p/q).
%e Given the line y = (11/7)*x, the number of lattice points on or inside the rectangle formed by (1 <= y <= 5), (1 <= x <= 3), where p = 11, q = 7; 5 = (p-1)/2, 3 = (q-1)/2; 3*5 = 15.
%e The number of lattice points on or inside the rectangle, (below the line y = (11/7)*x = 8 = Sum_{k=1..(q-1)/2} floor(k*(11/7)) = floor(11*1/7) + floor(11*2/7) + floor(11*3/7) = 1 + 3 + 4 = 8. The number of lattice points on or inside the rectangle above the line y = (11/7)*x = Sum_{k=1..(p-1)/2} floor(k*(7/11)) = floor(7*1/11) + floor(7*2/11) + floor(7*3/11) + floor(7*4/11) + floor(7*5/11) = 0 + 1 + 1 + 2 + 3 = 7.
%e Total number of lattice points inside or on the rectangle = 8 + 7 = 15.
%Y Cf. A087428.
%K nonn
%O 3,1
%A _Gary W. Adamson_, Sep 01 2003
%E Corrected and extended by _Ray Chandler_, Sep 16 2003