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%I #24 Aug 27 2022 03:16:04
%S 3,11,18,57,39,98,61,141,265,104,351,268,148,314,520,594,208,678,486,
%T 258,806,573,918,1325,703,366,753,390,788,3006,933,1443,503,2581,542,
%U 1666,1734,1192,1842,1917,644,3364,691,1416,717,4457,4729
%N Number of composite numbers between prime(n)^2 and prime(n + 1)^2 - 1.
%H Robert Israel, <a href="/A335135/b335135.txt">Table of n, a(n) for n = 1..2000</a>
%F a(n) = prime(n + 1)^2 - prime(n)^2 - (pi(prime(n + 1)^2) - pi(prime(n)^2)).
%F a(n) = A053683(n+1) - A053683(n). - _Michel Marcus_, Aug 27 2022
%e 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.
%p f:= proc(n) local p,q;
%p p:= ithprime(n); q:= nextprime(p);
%p q^2 - p^2 - numtheory:-pi(q^2)+numtheory:-pi(p^2)
%p end proc:
%p map(f, [$1..50]); # _Robert Israel_, Jun 24 2020
%t Array[#1 - #2 - (PrimePi@ #1 - PrimePi@ #2) & @@ {Prime[# + 1]^2, Prime[#]^2} &, 47] (* _Michael De Vlieger_, May 24 2020 *)
%o (PARI) forprime(n = 2, 220, s = 0; forcomposite(k = n^2, nextprime(n + 1)^2 - 1, s++); print1(s", "))
%Y Cf. A000040, A002808, A053683.
%K nonn,look
%O 1,1
%A _Dimitris Valianatos_, May 24 2020
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