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

3, 4, 6, 11, 10, 24, 26, 34, 26, 33, 50, 67, 72, 46, 70, 109, 96, 132, 122, 153, 132, 145, 174, 229, 208, 175, 194, 287, 232, 244, 338, 267, 276, 345, 374, 239, 392, 396, 424, 390, 484, 373, 514, 563, 618, 424, 654, 821, 442, 557, 890, 814, 668, 741, 580, 642, 990, 811, 982, 968, 772

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: (p1 - n^2) + ... + (pK - n^2). Obviously this equals (sum of these primes) - (number of these primes) * n^2.

Links

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

Crossrefs

Cf. A014085, A000290, A108314, A106044.

Row sums of A056892, read as a table.

Programs

Maple

R:= NULL: p:= 2: n:= 1: t:= 0:
while n <= 100 do
    t:= t + p-n^2;
    p:= nextprime(p);
    if p > (n+1)^2 then
     R:= R, t; t:= 0; n:= n+1;
    fi:
od:
R; # Robert Israel, Dec 17 2024

PARI

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

Formula

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