A291440 -id:A291440 - cp's OEIS frontend

A123914 a(n) = prime(n)^2 - prime(n^2). Commutator of (primes, squares) at n.

2, 2, 2, -4, 24, 18, 62, 50, 110, 300, 300, 542, 672, 656, 782, 1190, 1602, 1578, 2052, 2300, 2246, 2780, 3086, 3710, 4772, 5150, 5090, 5442, 5400, 5772, 8556, 9000, 10032, 9980, 12270, 12174, 13328, 14520, 15146, 16430, 17714, 17660, 20604, 20502, 21200

Offset: 1

Jonathan Vos Post, Oct 28 2006

a(4) = -4 is the only negative value. All values are even. Asymptotically a(n) ~ (n log n)^2 - (n^2) log (n^2) = (n^2)*(log n)^2 - 2*(n^2)*(log n) = (n^2)*((log n)^2 - 2*log n) = O((n^2)*(log n)^2) which is the same as the asymptotic of commutator [primes, triangular numbers] at n, or, for that matter, commutator [primes, k-tonal numbers] at n for any k > 2.
For pi(n^2) - pi(n)^2 see A291440. - Jonathan Sondow, Sep 10 2017
Proof that a(n) > 0 for n <> 4: It is known that pi(k^2) >= pi(k)^2 for k <> 7 (see A291440). Take k = prime(n) to get pi(prime(n)^2) >= pi(prime(n))^2 = n^2 for prime(n) <> 7 = prime(4). Thus for n <> 4 there are at least n^2 primes <= prime(n)^2, so prime(n^2) <= prime(n)^2, implying a(n) >= 0. But a prime cannot equal a square, so a(n) > 0 for n <> 4. - Jonathan Sondow, Nov 04 2017

			a(1) = prime(1)^2 - prime(1^2) = prime(1)^2 - prime(1^2) = 4 - 2 = 2.
a(2) = prime(2)^2 - prime(2^2) = prime(2)^2 - prime(2^2) = 9 - 7 = 2.
a(3) = prime(3)^2 - prime(3^2) = prime(3)^2 - prime(3^2) = 25 - 23 = 2.
a(4) = prime(4)^2 - prime(4^2) = prime(4)^2 - prime(4^2) = 49 - 53 = -4.
a(5) = prime(5)^2 - prime(5^2) = prime(5)^2 - prime(5^2) = 121 - 97 = 24.

See A324799 for references. - N. J. A. Sloane, Sep 11 2019

Seiichi Manyama, Table of n, a(n) for n = 1..10000 (terms 1..500 from G. C. Greubel)

Cf. A000040, A000290, A001248, A011757, A262199, A291440.

Main diagonal of A324799.

[NthPrime(n)^2 - NthPrime(n^2): n in [1..60]]; // Vincenzo Librandi, Sep 16 2015

f[n_] := Prime[n]^2 - Prime[n^2]; Array[f, 45] (* Robert G. Wilson v,  Oct 29 2006 *)
Table[(Prime[n])^2 - Prime[n^2], {n,1,300}] (* G. C. Greubel, Sep 15 2015 *)

vector(100, n, prime(n)^2 - prime(n^2)) \\ Altug Alkan, Oct 05 2015

a(n) = A001248(n) - A011757(n).
a(n) = commutator [A000040, A000290] at n.
a(n) = square(prime(n)) - prime(square(n)).
a(n) = A000290(A000040(n)) - A000040(A000290(n)). [corrected by Jonathan Sondow, Sep 10 2017]

More terms from Robert G. Wilson v, Oct 29 2006

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).

A291539 a(n) = PrimePi(n^3) - PrimePi(n) * PrimePi(n^2), where PrimePi = A000720.

Original entry on oeis.org

0, 2, 1, 6, 3, 14, 8, 25, 41, 68, 67, 99, 93, 136, 188, 240, 229, 303, 306, 383, 467, 562, 566, 688, 795, 922, 1066, 1227, 1247, 1421, 1446, 1620, 1826, 2036, 2283, 2511, 2566, 2843, 3115, 3401, 3431, 3746, 3827, 4163, 4526, 4895, 4981, 5369, 5743, 6229, 6712, 7165, 7202, 7743, 8258, 8835, 9453, 9999, 10132, 10736

Offset: 1

Jonathan Sondow, 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) * PrimePi(n)^2 for n > 1.
For PrimePi(n) * PrimePi(n^2) - PrimePi(n)^3, see A291540.
For PrimePi(n^3) - PrimePi(n)^3, see A291538.
For prime(n) * prime(n^2) - prime(n^3), see A291541.

			a(2) = PrimePi(2^3) - PrimePi(2) * PrimePi(2^2) = 4 - 1 * 2 = 2.

Cf. A000720, A123914, A262199, A291440, A291538, A291540, A291541, A291542.

Table[ PrimePi[n^3] - PrimePi[n]*PrimePi[n^2], {n, 60}]

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

a(n) = A000720(n^3) - A000720(n) * A000720(n)^2.
a(n) = A291538(n) - A291540(n).
a(n) ~ (n^3 / log(n))*(1/3  - 1/(2*log(n)^2)) as n tends to infinity.

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).

A294509 a(n) is the least value of pi(nm) - pi(n)pi(m) for any positive m <= n.

Original entry on oeis.org

0, 1, 0, 1, 0, 1, -1, 0, 1, 2, 1, 1, 0, 1, 2, 3, 1, 2, 0, 1, 2, 2, 1, 2, 3, 3, 4, 5, 3, 4, 2, 2, 3, 4, 5, 6, 5, 6, 6, 6, 4, 4, 3, 4, 4, 5, 4, 4, 4, 5, 6, 6, 5, 5, 6, 7, 7, 8, 6, 7, 6, 6, 6, 7, 8, 9, 8, 8, 8, 8, 7, 7, 5, 5, 6, 7, 8, 9, 7, 8, 9, 9, 7, 8, 8, 9, 9, 10, 8, 9, 10, 10, 11, 12, 13, 13, 11, 12, 12, 12, 10, 10, 9, 10, 11

Offset: 1

Jonathan Sondow and Robert G. Wilson v, Nov 06 2017

sign

Least value in the n-th row of the table in A294508.
First occurrence of -1, 0, 1, 2, etc. occurs at n = 7, 1, 2, 10, 16, 27, 28, 36, 56, 58, 66, 88, 93, 94, 95, 125, 130, 145, 147, 148, 156, 190, 206, 207, 215, 216, 218, etc.
Last occurrence of -1, 0, 1, 2, etc. occurs at n = 7, 19, 23, 32, 43, 49, 74, 75, 83, 115, 116, 117, 119, 139, 140, 143, 152, 199, 200, 202, 204, 205, 213, 242, 244, 284, 285, etc.
Conjecture: a(n) <= pi(n*m) - pi(n)*pi(m) for all m > n if n <> 5.

			a(13) = 0 since 0 is the least value in the 13th row of A294508.

Robert Israel, Table of n, a(n) for n = 1..5000

Cf. A000720, A291440, A294508.

f:= n -> min(seq(numtheory:-pi(n*m)-numtheory:-pi(n)*numtheory:-pi(m), m=1..n)):
map(f, [$1..200]); # Robert Israel, Nov 08 2017

t[n_, m_] := PrimePi[n*m] - PrimePi[n]*PrimePi[m]; Min @@@ Table[ t[n, m], {n, 100}, {m, n}]

a(n) = vecmin(vector(n, m, primepi(n*m) - primepi(n)*primepi(m))); \\ Michel Marcus, Nov 08 2017

a(n) = min_{1<=m<=n} A294508(n*(n-1)/2 + m).

a(n) <= A291440(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

A294508 Regular triangular array read by rows: T(n,m) = pi(nm) - pi(n)pi(m) for n > 0 and 0 < m <= n.

Original entry on oeis.org

0, 1, 1, 2, 1, 0, 2, 2, 1, 2, 3, 1, 0, 2, 0, 3, 2, 1, 3, 1, 2, 4, 2, 0, 1, -1, 1, -1, 4, 2, 1, 3, 0, 3, 0, 2, 4, 3, 1, 3, 2, 4, 2, 4, 6, 4, 4, 2, 4, 3, 5, 3, 6, 8, 9, 5, 3, 1, 4, 1, 3, 1, 3, 5, 9, 5, 5, 4, 1, 5, 2, 5, 3, 4, 8, 10, 7, 9, 6, 3, 0, 3, 0, 3, 0, 3, 6, 7, 4, 6, 3, 6, 3, 1, 4, 1, 5, 1, 5, 6, 10, 6, 9, 6, 8

Offset: 1

Jonathan Sondow and Robert G. Wilson v, Nov 06 2017

Inspired by A291440.
Mincu and Panaitopol (2008) prove that pi(m*n) >= pi(m)*i(n) for all positive m and n except for m = 5, n = 7; m = 7, n = 5; and m = n = 7.
a(i) = -1 for i = 26 and 28, when n = 7 and m = either 5 or 7.
a(i) = 0 for i = 1, 6, 13, 15, 24, 33, 35, 81, 83, 85, 174, 176, 178; when n=m=1; n=m=3; n=5 and m is either 3 or 5; n=7 and m=3; n=8 and m is either 5 or 7; n=13 and m is either 3, 5, or 7; and n=19 with m being either 3, 5 or 7.
First occurrence of k = -1, 0, 1, 2, .., 20, 21, etc. occurs at i = 26, 1, 2, 4, 11, 22, 51, 45, 77, 54, 55, 76, 115, 120, 130, 187, 168, 135, 171, 136, 169, 274, etc.
Last occurrence of k >= -1 occurs at i = 28, 178, 260, 499, 906, 1179, 2704, 2778, 3406, 6558, 6673, 6789, 7024, 9594, 9733, 10156, 11479, 19704, 19903, 20304, 20709, 20913, etc.
Conjecture: min_{1<=m<=n} T(n,m) <= T(n,M) for all M > n if n <> 5.

			a(19) = 3 since 19 = 5*6/2 + 4, so the 19th term is T(6,4) = pi(6*4) - pi(6)*pi(4) = 9 - 3*2 = 3.
Triangular array begins:
   n\ m  1  2  3  4  5  6  7  8  9 10 11 12 13 14
   1  0
   2  1  1
   3  2  1  0
   4  2  2  1  2
   5  3  1  0  2  0
   6  3  2  1  3  1  2
   7  4  2  0  1 -1  1 -1
   8  4  2  1  3  0  3  0  2
   9  4  3  1  3  2  4  2  4  6
  10  4  4  2  4  3  5  3  6  8  9
  11  5  3  1  4  1  3  1  3  5  9  5
  12  5  4  1  5  2  5  3  4  8 10  7  9
  13  6  3  0  3  0  3  0  3  6  7  4  6  3
  14  6  3  1  4  1  5  1  5  6 10  6  9  6  8
  15  6  4  2  5  3  6  3  6  8 11  8 11  8 10 12

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, A291440, A294509.

t[n_, m_] := PrimePi[n*m] - PrimePi[n]*PrimePi[m]; Table[ t[n, m], {n, 13}, {m, n}] // Flatten

T(n,m) = primepi(n*m) - primepi(n)*primepi(m);
tabl(nn) = for (n=1, nn, for (m=1, n, print1(T(n,m), ", ")); print); \\ Michel Marcus, Nov 08 2017

a(n*(n+1)/2) = T(n,n) = A291440(n).

min_{1<=m<=n} a(n*(n-1)/2 + m) = min_{1<=m<=n} T(n,m) = A294509(n).

cp's OEIS Frontend

A123914 a(n) = prime(n)^2 - prime(n^2). Commutator of (primes, squares) at 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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A291539 a(n) = PrimePi(n^3) - PrimePi(n) * PrimePi(n^2), where PrimePi = A000720.

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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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A294509 a(n) is the least value of pi(n*m) - pi(n)*pi(m) for any positive m <= n.

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A294509 a(n) is the least value of pi(nm) - pi(n)pi(m) for any positive m <= n.

A294508 Regular triangular array read by rows: T(n,m) = pi(nm) - pi(n)pi(m) for n > 0 and 0 < m <= n.