A291539 -id:A291539 - cp's OEIS frontend

A291440 a(n) = pi(n^2) - pi(n)^2, where pi(n) = A000720(n).

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

Jonathan Sondow, Aug 23 2017

sign

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

			a(7) = pi(7^2) - pi(7)^2 = 15 - 4^2 = -1.

Robert Israel, Table of n, a(n) for n = 1..10000
Gabriel Mincu and Laurentiu Panaitopol, Properties of some functions connected to prime numbers, J. Inequal. Pure Appl. Math., 9 No. 1 (2008), Art. 12.

Cf. A000720, A123914, A262199, A291538, A291539, A291540, A291541, A291542.

[#PrimesUpTo(n^2)-#PrimesUpTo(n)^2: n in [1..80]]; // Vincenzo Librandi, Aug 26 2017

seq(numtheory:-pi(n^2)-numtheory:-pi(n)^2, n=1..100); # Robert Israel, Aug 25 2017

Table[PrimePi[n^2] - PrimePi[n]^2, {n, 80}]

a(n) = primepi(n^2) - primepi(n)^2; \\ Michel Marcus, Sep 10 2017

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)

A291538 a(n) = PrimePi(n^3) - PrimePi(n)^3, where PrimePi = A000720.

Original entry on oeis.org

0, 3, 1, 10, 3, 20, 4, 33, 65, 104, 92, 144, 111, 184, 260, 348, 313, 422, 370, 495, 635, 786, 728, 904, 1092, 1291, 1498, 1731, 1707, 1961, 1897, 2181, 2486, 2806, 3152, 3490, 3466, 3851, 4267, 4685, 4653, 5111, 5045, 5549, 6066, 6617, 6541, 7124, 7723, 8359, 9007, 9685, 9650, 10383, 11106, 11859, 12669, 13487, 13498, 14374

Offset: 1

Jonathan Sondow, following a suggestion from Altug Alkan, Aug 25 2017

nonn

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)^3 for n > 1. Indeed, by A291539 and A291540, PrimePi(n^3) > PrimePi(n) * PrimePi(n^2) > PrimePi(n)^3 for n > 7.
For prime(n)^3 - prime(n^3), see A262199.
For PrimePi(n^2) - PrimePi(n)^2, see A291440.

			a(3) = PrimePi(3^3) - PrimePi(3)^3 = 9 - 2^3 = 1.

Cf. A000720, A123914, A262199, A291440, A291539, A291540, A291541, A291542.

Table[ PrimePi[n^3] - PrimePi[n]^3, {n, 60}]

a(n) = primepi(n^3) - primepi(n)^3; \\ Michel Marcus, Sep 10 2017

a(n) = A000720(n^3) - A000720(n)^3.
a(n) = A291539(n) + A291540(n).
a(n) ~ (n^3 / log(n))*(1/3  - 1/log(n)^2) as n tends to infinity.

A291542 a(n) = prime(n)^3 - prime(n) * prime(n^2).

Original entry on oeis.org

4, 6, 10, -28, 264, 234, 1054, 950, 2530, 8700, 9300, 20054, 27552, 28208, 36754, 63070, 94518, 96258, 137484, 163300, 163958, 219620, 256138, 330190, 462884, 520150, 524270, 582294, 588600, 652236, 1086612, 1179000, 1374384, 1387220, 1828230, 1838274, 2092496, 2366760, 2529382, 2842390

Offset: 1

Jonathan Sondow, Aug 25 2017

sign

Same as prime(n) * A123914(n). See A123914 for other comments and formulas.
All terms are even.
For prime(n)^3 - prime(n^3) see A262199.
For prime(n) * prime(n^2) - prime(n^3) see A291541.
For PrimePi(n) * PrimePi(n^2) - PrimePi(n)^3 see A291540.

			a(4) = prime(4)^3 - prime(4) * prime(16) = 7^3 - 7*53 = -28.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000040, A000720, A123914, A262199, A291440, A291538, A291539, A291540, A291541.

seq(ithprime(n)^3 - ithprime(n)*ithprime(n^2), n=1..100); # Robert Israel, Sep 11 2017

Table[ Prime[n]^3 - Prime[n] * Prime[n^2], {n, 40}]

a(n) = prime(n)^3 - prime(n) * prime(n^2); \\ Michel Marcus, Sep 10 2017

a(n) + A291541(n) = A262199(n).

A291540 a(n) = PrimePi(n) * PrimePi(n^2) - PrimePi(n)^3, where PrimePi = A000720.

Original entry on oeis.org

0, 1, 0, 4, 0, 6, -4, 8, 24, 36, 25, 45, 18, 48, 72, 108, 84, 119, 64, 112, 168, 224, 162, 216, 297, 369, 432, 504, 460, 540, 451, 561, 660, 770, 869, 979, 900, 1008, 1152, 1284, 1222, 1365, 1218, 1386, 1540, 1722, 1560, 1755, 1980, 2130, 2295, 2520, 2448, 2640, 2848, 3024, 3216, 3488, 3366, 3638, 3510, 3744, 4050, 4320, 4572

Offset: 1

Jonathan Sondow, Aug 25 2017

sign

All terms are positive except a(1) = a(3) = a(5) = 0 and a(7) = -4, by the PNT with error term for large n and computation for smaller n. In particular, PrimePi(n) * PrimePi(n^2) > PrimePi(n)^3, for n > 7.
For PrimePi(n^3) - PrimePi(n) * PrimePi(n^2), see A291539.
For PrimePi(n^3) - PrimePi(n)^3, see A291538.
For prime(n)^3 - prime(n) * prime(n^2), see A291542.

			a(7) = PrimePi(7) * PrimePi(7^2) - PrimePi(7)^3 = 4 * 15 - 4^3 = -4.

Cf. A000720, A123914, A262199, A291440, A291538, A291539, A291541, A291542.

Table[ PrimePi[n] * PrimePi[n^2] - PrimePi[n]^3, {n, 65}]

a(n) = primepi(n) * primepi(n^2) - primepi(n)^3; \\ Michel Marcus, Sep 10 2017

A291539(n) + a(n) = A291538(n).

A291541 a(n) = prime(n) * prime(n^2) - prime(n^3).

Original entry on oeis.org

2, 2, 12, 60, 376, 642, 1550, 2238, 4118, 7770, 9534, 15846, 21966, 26490, 35750, 46934, 63204, 73164, 94248, 112812, 128922, 161128, 185576, 225062, 278260, 315108, 347596, 393898, 426998, 478078, 614064, 682998, 769800, 827466, 962166, 1036806, 1156866, 1286448, 1390878, 1534754

Offset: 1

Jonathan Sondow, Aug 25 2017

nonn

All terms are even.
For prime(n)^3 - prime(n^3) see A262199.
For prime(n)^3 - prime(n) * prime(n^2) see A291542.
For PrimePi(n^3) - PrimePi(n) * PrimePi(n^2) see A291539.

			a(2) = prime(2) * prime(4) - prime(8) = 3*7 - 19 = 2.

Cf. A000720, A123914, A262199, A291440, A291538, A291539, A291540, A291542.

Table[ Prime[n] * Prime[n^2] - Prime[n^3], {n, 40}]

a(n) = prime(n) * prime(n^2) - prime(n^3); \\ Michel Marcus, Sep 10 2017

A291542(n) + a(n) = A262199(n).

A304483 a(n) = pi(n)*pi(2n), where pi is A000720: the prime counting function.

Original entry on oeis.org

0, 2, 6, 8, 12, 15, 24, 24, 28, 32, 40, 45, 54, 54, 60, 66, 77, 77, 96, 96, 104, 112, 126, 135, 135, 135, 144, 144, 160, 170, 198, 198, 198, 209, 209, 220, 252, 252, 252, 264, 286, 299, 322, 322, 336, 336, 360, 360, 375, 375, 390, 405, 432, 448, 464, 464, 480, 480, 510

Offset: 1

Jason Kimberley, May 13 2018

Jason Kimberley, Table of n, a(n) for n = 1..10000

Cf. A304484(n) = A033270(n)*A033270(2n).

Cf. A291440, A291538, A291539, A291540,

A000720:=func;
A304483:=funcA000720(n)*A000720(2*n)>;
[A304483(n):n in[1..59]];

Array[PrimePi[#] PrimePi[2 #] &, 59] (* Michael De Vlieger, May 27 2018 *)

a(n) = primepi(n) * primepi(2*n); \\ Michel Marcus, May 27 2018

cp's OEIS Frontend

A291440 a(n) = pi(n^2) - pi(n)^2, where pi(n) = A000720(n).

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A291538 a(n) = PrimePi(n^3) - PrimePi(n)^3, where PrimePi = A000720.

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A291542 a(n) = prime(n)^3 - prime(n) * prime(n^2).

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A291540 a(n) = PrimePi(n) * PrimePi(n^2) - PrimePi(n)^3, where PrimePi = A000720.

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A291541 a(n) = prime(n) * prime(n^2) - prime(n^3).

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A304483 a(n) = pi(n)*pi(2n), where pi is A000720: the prime counting function.

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Magma

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