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%I #11 Sep 10 2017 14:58:26
%S 0,2,1,6,3,14,8,25,41,68,67,99,93,136,188,240,229,303,306,383,467,562,
%T 566,688,795,922,1066,1227,1247,1421,1446,1620,1826,2036,2283,2511,
%U 2566,2843,3115,3401,3431,3746,3827,4163,4526,4895,4981,5369,5743,6229,6712,7165,7202,7743,8258,8835,9453,9999,10132,10736
%N a(n) = PrimePi(n^3) - PrimePi(n) * PrimePi(n^2), where PrimePi = A000720.
%C All terms are positive except a(1) = 0, by the PNT with error term for large n and computation for smaller n. In particular, PrimePi(n^3) > PrimePi(n) * PrimePi(n)^2 for n > 1.
%C For PrimePi(n) * PrimePi(n^2) - PrimePi(n)^3, see A291540.
%C For PrimePi(n^3) - PrimePi(n)^3, see A291538.
%C For prime(n) * prime(n^2) - prime(n^3), see A291541.
%F a(n) = A000720(n^3) - A000720(n) * A000720(n)^2.
%F a(n) = A291538(n) - A291540(n).
%F a(n) ~ (n^3 / log(n))*(1/3 - 1/(2*log(n)^2)) as n tends to infinity.
%e a(2) = PrimePi(2^3) - PrimePi(2) * PrimePi(2^2) = 4 - 1 * 2 = 2.
%t Table[ PrimePi[n^3] - PrimePi[n]*PrimePi[n^2], {n, 60}]
%o (PARI) a(n) = primepi(n^3) - primepi(n) * primepi(n^2); \\ _Michel Marcus_, Sep 10 2017
%Y Cf. A000720, A123914, A262199, A291440, A291538, A291540, A291541, A291542.
%K nonn
%O 1,2
%A _Jonathan Sondow_, Aug 25 2017
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