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A291440
a(n) = pi(n^2) - pi(n)^2, where pi(n) = A000720(n).
10
0, 1, 0, 2, 0, 2, -1, 2, 6, 9, 5, 9, 3, 8, 12, 18, 12, 17, 8, 14, 21, 28, 18, 24, 33, 41, 48, 56, 46, 54, 41, 51, 60, 70, 79, 89, 75, 84, 96, 107, 94, 105, 87, 99, 110, 123, 104, 117, 132, 142, 153, 168, 153, 165, 178, 189, 201, 218, 198, 214, 195, 208, 225, 240, 254, 270, 248, 263, 280, 293, 275, 290, 264, 281, 298, 316, 338, 352, 327, 350
OFFSET
1,4
COMMENTS
The only zero values are a(1) = a(3) = a(5) = 0. The only negative value is a(7) = -1. In particular, pi(n^2) > pi(n)^2 for n > 7. These can be proved by the PNT with error term for large n and computation for smaller n.
For prime(n)^2 - prime(n^2), see A123914.
For pi(n^3) - pi(n)^3, see A291538.
Mincu and Panaitopol (2008) prove that pi(m*n) >= pi(m)*pi(n) for all positive m and n except for m = 5, n = 7; m = 7, n = 5; and m = n = 7. This implies for m = n that a(n) >= 0 if n <> 7. - Jonathan Sondow, Nov 03 2017
Diagonal of the triangular array A294508. - Jonathan Sondow and Robert G. Wilson v, Nov 08 2017
LINKS
Gabriel Mincu and Laurentiu Panaitopol, Properties of some functions connected to prime numbers, J. Inequal. Pure Appl. Math., 9 No. 1 (2008), Art. 12.
FORMULA
a(n) = A000720(n^2) - A000720(n)^2.
a(n) ~ (n^2 / log(n))*(1/2 - 1/log(n)) as n tends to infinity, by the PNT.
From Jonathan Sondow and Robert G. Wilson v, Nov 08 2017: (Start)
a(n) = A294508(n*(n+1)/2).
a(n) >= A294509(n). (End)
EXAMPLE
a(7) = pi(7^2) - pi(7)^2 = 15 - 4^2 = -1.
MAPLE
seq(numtheory:-pi(n^2)-numtheory:-pi(n)^2, n=1..100); # Robert Israel, Aug 25 2017
MATHEMATICA
Table[PrimePi[n^2] - PrimePi[n]^2, {n, 80}]
PROG
(Magma) [#PrimesUpTo(n^2)-#PrimesUpTo(n)^2: n in [1..80]]; // Vincenzo Librandi, Aug 26 2017
(PARI) a(n) = primepi(n^2) - primepi(n)^2; \\ Michel Marcus, Sep 10 2017
KEYWORD
sign
AUTHOR
Jonathan Sondow, Aug 23 2017
STATUS
approved