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

1,1

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).

Links

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

Formula

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

Example

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}.

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

Prog

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

Crossrefs

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

Row sums of A106044 read as a table.

Sequence in context: A117148 A343272 A305595 * A340494 A039567 A032912

Adjacent sequences: A320684 A320685 A320686 * A320688 A320689 A320690

Keyword

nonn

Author

M. F. Hasler, Oct 19 2018

Status

approved