cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-7 of 7 results.

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

Original entry on oeis.org

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

Views

Author

Jonathan Sondow, Aug 23 2017

Keywords

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

Examples

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

Links

Crossrefs

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

Programs

  • Magma
    [#PrimesUpTo(n^2)-#PrimesUpTo(n)^2: n in [1..80]]; // Vincenzo Librandi, Aug 26 2017
  • 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}]
  • PARI
    a(n) = primepi(n^2) - primepi(n)^2; \\ Michel Marcus, Sep 10 2017

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)

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

Original entry on oeis.org

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

Views

Author

Jonathan Vos Post, Oct 28 2006

Keywords

Comments

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

Examples

			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.

References

Links

Crossrefs

Cf. A000040, A000290, A001248, A011757, A262199, A291440.
Main diagonal of A324799.

Programs

  • Magma
    [NthPrime(n)^2 - NthPrime(n^2): n in [1..60]]; // Vincenzo Librandi, Sep 16 2015
  • Mathematica
    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 *)
  • PARI
    vector(100, n, prime(n)^2 - prime(n^2)) \\ Altug Alkan, Oct 05 2015

Formula

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]

Extensions

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

Views

Author

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

Keywords

Comments

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.

Examples

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

Crossrefs

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

Programs

  • Mathematica
    Table[ PrimePi[n^3] - PrimePi[n]^3, {n, 60}]
  • PARI
    a(n) = primepi(n^3) - primepi(n)^3; \\ Michel Marcus, Sep 10 2017

Formula

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

Views

Author

Jonathan Sondow, Aug 25 2017

Keywords

Comments

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.

Examples

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

Links

Crossrefs

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

Programs

  • Maple
    seq(ithprime(n)^3 - ithprime(n)*ithprime(n^2), n=1..100); # Robert Israel, Sep 11 2017
  • Mathematica
    Table[ Prime[n]^3 - Prime[n] * Prime[n^2], {n, 40}]
  • PARI
    a(n) = prime(n)^3 - prime(n) * prime(n^2); \\ Michel Marcus, Sep 10 2017

Formula

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

Views

Author

Jonathan Sondow, Aug 25 2017

Keywords

Comments

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.

Examples

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

Crossrefs

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

Programs

  • Mathematica
    Table[ PrimePi[n^3] - PrimePi[n]*PrimePi[n^2], {n, 60}]
  • PARI
    a(n) = primepi(n^3) - primepi(n) * primepi(n^2); \\ Michel Marcus, Sep 10 2017

Formula

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

Views

Author

Jonathan Sondow, Aug 25 2017

Keywords

Comments

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.

Examples

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

Crossrefs

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

Programs

  • Mathematica
    Table[ PrimePi[n] * PrimePi[n^2] - PrimePi[n]^3, {n, 65}]
  • PARI
    a(n) = primepi(n) * primepi(n^2) - primepi(n)^3; \\ Michel Marcus, Sep 10 2017

Formula

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

Views

Author

Jonathan Sondow, Aug 25 2017

Keywords

Comments

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.

Examples

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

Crossrefs

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

Programs

  • Mathematica
    Table[ Prime[n] * Prime[n^2] - Prime[n^3], {n, 40}]
  • PARI
    a(n) = prime(n) * prime(n^2) - prime(n^3); \\ Michel Marcus, Sep 10 2017

Formula

A291542(n) + a(n) = A262199(n).
Showing 1-7 of 7 results.

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