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

3, 6, 8, 16, 12, 28, 19, 34, 31, 72, 42, 58, 63, 70, 116, 122, 79, 90, 112, 134, 169, 170, 108, 212, 200, 196, 246, 226, 240, 244, 292, 318, 394, 276, 336, 418, 283, 528, 445, 582, 429, 392, 530, 416, 565, 506, 581, 634, 548, 554, 655, 866, 616, 676, 641, 714, 965, 710, 922, 968, 827

Offset: 1

Author

M. F. Hasler, Oct 19 2018

Keywords

• nonn

Comments

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).

Examples

			a(1) = 3 = 2 + 1, where {2, 1} = 4 - {2, 3: primes between 1^2 = 1 and 2^2 = 4}.
a(2) = 6 = 4 + 2, with {4, 2} = 9 - {5, 7: primes between 2^2 = 4 and 3^2 = 9}.
a(3) = 8 = sum of {5, 3} = 16 - {11, 13: primes between 3^2 = 9 and 4^2 = 16}.
a(4) = 16 = sum of {8, 6, 2} = 25 - {17, 19, 23: primes between 4^2 and 5^2 = 25}.
a(5) = 12 = sum of {7, 5} = 36 - {29, 31: primes between 5^2 = 25 and 6^2 = 36}.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Equals A014085 * A000290(.+1) - A108314.

Row sums of A106044 read as a table.

Programs

Maple

N:= 100: # to get a(1)..a(N)
V:= Vector(N):
p:= 1;
do
   p:= nextprime(p);
   n:= floor(sqrt(p));
   if n > N then break fi;
   V[n]:= V[n]+(n+1)^2-p;
od:
convert(V,list); # Robert Israel, Jun 17 2019

PARI

a(n,s=0)={forprime(p=n^2,(n+=1)^2,s+=n^2-p);s}

Formula

a(n) = A014085(n)*A000290(n+1) - A108314(n), where A000290(n) = n^2.