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A320687 Sum of differences of the larger square and primes between two squares.

%I A320687 #11 Jun 17 2019 18:02:10
%S A320687 3,6,8,16,12,28,19,34,31,72,42,58,63,70,116,122,79,90,112,134,169,170,
%T A320687 108,212,200,196,246,226,240,244,292,318,394,276,336,418,283,528,445,
%U A320687 582,429,392,530,416,565,506,581,634,548,554,655,866,616,676,641,714,965,710,922,968,827
%N A320687 Sum of differences of the larger square and primes between two squares.
%C A320687 Consider the primes p1,...,pK between two squares n^2 and (n+1)^2, and take the sum of the differences (listed as A106044): ((n+1)^2 - p1) + ... + ((n+1)^2 - pK).
%H A320687 Robert Israel, <a href="/A320687/b320687.txt">Table of n, a(n) for n = 1..10000</a>
%F A320687 a(n) = A014085(n)*A000290(n+1) - A108314(n), where A000290(n) = n^2.
%e A320687 a(1) = 3 = 2 + 1, where {2, 1} = 4 - {2, 3: primes between 1^2 = 1 and 2^2 = 4}.
%e A320687 a(2) = 6 = 4 + 2, with {4, 2} = 9 - {5, 7: primes between 2^2 = 4 and 3^2 = 9}.
%e A320687 a(3) = 8 = sum of {5, 3} = 16 - {11, 13: primes between 3^2 = 9 and 4^2 = 16}.
%e A320687 a(4) = 16 = sum of {8, 6, 2} = 25 - {17, 19, 23: primes between 4^2 and 5^2 = 25}.
%e A320687 a(5) = 12 = sum of {7, 5} = 36 - {29, 31: primes between 5^2 = 25 and 6^2 = 36}.
%p A320687 N:= 100: # to get a(1)..a(N)
%p A320687 V:= Vector(N):
%p A320687 p:= 1;
%p A320687 do
%p A320687    p:= nextprime(p);
%p A320687    n:= floor(sqrt(p));
%p A320687    if n > N then break fi;
%p A320687    V[n]:= V[n]+(n+1)^2-p;
%p A320687 od:
%p A320687 convert(V,list); # _Robert Israel_, Jun 17 2019
%o A320687 (PARI) a(n,s=0)={forprime(p=n^2,(n+=1)^2,s+=n^2-p);s}
%Y A320687 Equals A014085 * A000290(.+1) - A108314.
%Y A320687 Row sums of A106044 read as a table.
%K A320687 nonn
%O A320687 1,1
%A A320687 _M. F. Hasler_, Oct 19 2018