

A320687


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


1



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
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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)^2p;
od:
convert(V, list); # Robert Israel, Jun 17 2019


PROG

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


CROSSREFS

Equals A014085 * A000290(.+1)  A108314.
Row sums of A106044 read as a table.
Sequence in context: A165298 A117148 A305595 * A340494 A039567 A032912
Adjacent sequences: A320684 A320685 A320686 * A320688 A320689 A320690


KEYWORD

nonn


AUTHOR

M. F. Hasler, Oct 19 2018


STATUS

approved



