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A320688 Sum of the square excess A056892 of the primes between two squares.

%I A320688 #14 Dec 19 2024 23:40:08
%S A320688 3,4,6,11,10,24,26,34,26,33,50,67,72,46,70,109,96,132,122,153,132,145,
%T A320688 174,229,208,175,194,287,232,244,338,267,276,345,374,239,392,396,424,
%U A320688 390,484,373,514,563,618,424,654,821,442,557,890,814,668,741,580,642,990,811,982,968,772
%N A320688 Sum of the square excess A056892 of the primes between two squares.
%C A320688 Consider the primes p1,...,pK between two squares n^2 and (n+1)^2, and take the sum of the differences: (p1 - n^2) + ... + (pK - n^2). Obviously this equals (sum of these primes) - (number of these primes) * n^2.
%H A320688 Robert Israel, <a href="/A320688/b320688.txt">Table of n, a(n) for n = 1..10000</a>
%F A320688 a(n) = A108314(n) - A014085(n)*A000290(n), where A000290(n) = n^2.
%p A320688 R:= NULL: p:= 2: n:= 1: t:= 0:
%p A320688 while n <= 100 do
%p A320688     t:= t + p-n^2;
%p A320688     p:= nextprime(p);
%p A320688     if p > (n+1)^2 then
%p A320688      R:= R, t; t:= 0; n:= n+1;
%p A320688     fi:
%p A320688 od:
%p A320688 R; # _Robert Israel_, Dec 17 2024
%o A320688 (PARI) a(n,s=0)={forprime(p=n^2,(n+1)^2,s+=p-n^2);s}
%Y A320688 Cf. A014085, A000290, A108314, A106044.
%Y A320688 Row sums of A056892, read as a table.
%K A320688 nonn
%O A320688 1,1
%A A320688 _M. F. Hasler_, Oct 19 2018