A335135 Number of composite numbers between prime(n)^2 and prime(n + 1)^2 - 1.

3, 11, 18, 57, 39, 98, 61, 141, 265, 104, 351, 268, 148, 314, 520, 594, 208, 678, 486, 258, 806, 573, 918, 1325, 703, 366, 753, 390, 788, 3006, 933, 1443, 503, 2581, 542, 1666, 1734, 1192, 1842, 1917, 644, 3364, 691, 1416, 717, 4457, 4729

Offset: 1

Author

Dimitris Valianatos, May 24 2020

Keywords

• nonn
• look

Examples

			For n = 1, prime(1) = 2 and prime(2) = 3. So the composite numbers between 2^2 = 4 and 3^2 - 1 = 9 - 1 = 8 are 4, 6, and 8, so a(1) = 3.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..2000 from Robert Israel)

Crossrefs

Cf. A000040, A002808, A053683.

Programs

Maple

f:= proc(n) local p,q;
p:= ithprime(n); q:= nextprime(p);
q^2 - p^2 - numtheory:-pi(q^2)+numtheory:-pi(p^2)
end proc:
map(f, [$1..50]); # Robert Israel, Jun 24 2020

Mathematica

Array[#1 - #2 - (PrimePi@ #1 - PrimePi@ #2) & @@ {Prime[# + 1]^2, Prime[#]^2} &, 47] (* Michael De Vlieger, May 24 2020 *)

PARI

forprime(n = 2, 220, s = 0; forcomposite(k = n^2, nextprime(n + 1)^2 - 1, s++); print1(s", "))

Formula

a(n) = prime(n + 1)^2 - prime(n)^2 - (pi(prime(n + 1)^2) - pi(prime(n)^2)).

a(n) = A053683(n+1) - A053683(n). - Michel Marcus, Aug 27 2022