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 1-4 of 4 results.

A060967 Number of squared primes <= 2^n.

Original entry on oeis.org

0, 0, 1, 1, 2, 3, 4, 5, 6, 8, 11, 14, 18, 24, 31, 42, 54, 72, 97, 128, 172, 229, 309, 418, 564, 760, 1028, 1393, 1900, 2585, 3512, 4792, 6542, 8952, 12251, 16777, 23000, 31579, 43390, 59631, 82025, 112957, 155611, 214516, 295947, 408493, 564163, 779638
Offset: 0

Views

Author

Labos Elemer, May 09 2001

Keywords

Examples

			For n = 8, the squared primes not exceeding 2^8 = 256 are 4, 9, 25, 49, 121, 169, so a(8) = 6.

Links

Crossrefs

Cf. A000720, A001248, A007053, A017910, A036386, A060969, A060970, A060971, A122121.

Programs

  • Mathematica
    Table[ PrimePi[ Floor[ 2^(g/2)//N ] ], {g, 1, 75} ]
  • PARI
    a(n) = { primepi(sqrtint(2^n)) } \\ Harry J. Smith, Jul 15 2009

Formula

a(2*n) = A007053(n). - Amiram Eldar, Jul 10 2024
a(n) = A000720(A017910(n)). - Amiram Eldar, Mar 22 2025

Extensions

a(0) prepended by Harry J. Smith, Jul 15 2009

A036386 Number of prime powers (p^2, p^3, ...) <= 2^n.

Original entry on oeis.org

0, 1, 2, 4, 7, 9, 13, 16, 20, 26, 31, 40, 50, 61, 78, 93, 119, 150, 189, 242, 310, 400, 525, 684, 900, 1190, 1581, 2117, 2836, 3807, 5136, 6948, 9425, 12811, 17437, 23788, 32517, 44512, 60971, 83640, 114899, 157948, 217336, 299360, 412635, 569193, 785753, 1085319, 1500140, 2074794, 2870849, 3974425, 5504966
Offset: 1

Views

Author

Labos Elemer

Keywords

Examples

			For n = 6, there are 9 prime powers not exceeding 2^6 = 64: 4, 8, 9, 16, 25, 27, 32, 49, 64, so a(6) = 9.
For n = 25, a(25) = 900, pi(5792) + pi(322) + pi(76) + pi(32) + pi(17) + pi(11) + pi(8) + pi(6) + pi(5) + pi(4) + pi(4) + pi(3) + pi(3) + pi(3) + pi(2) + pi(2) + pi(2) + pi(2) + pi(2) + pi(2) + pi(2) + pi(2) + pi(2) + pi(2) + pi(1) = 760 + 66 + 21 + 11 + 7 + 5 + 4 + 3 + 3 + 2 + 2 + 2 + 2 + 2 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 0 = 900.

Links

Crossrefs

Cf. A000720, A007053, A029837, A036378-A036390, A060967, A060969, A060970, A060971, A122121, A246547.

Programs

  • Mathematica
    f[n_] := Length@ Union@ Flatten@ Table[ Prime[j]^k, {k, 2, n + 1}, {j, PrimePi[2^(n/k)]}]; Array[f, 46] (* Robert G. Wilson v, Jul 08 2011 *)
  • Python
    from sympy import primepi, integer_nthroot
def A036386(n):
    m = 1<Chai Wah Wu, Jan 23 2025

Formula

a(n) = Sum_{j=2..n+1} pi(floor(2^(n/j))). The summation starts with squares (j=2); for arbitrary range (=y), the y^(1/j) argument has to be used.

Extensions

More terms from Labos Elemer, May 07 2001
Terms a(47) and beyond from Hiroaki Yamanouchi, Nov 15 2016

A060969 Number of cubes of primes <= 2^n.

Original entry on oeis.org

0, 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 6, 8, 9, 11, 12, 15, 18, 22, 26, 31, 37, 46, 54, 66, 79, 97, 117, 141, 172, 209, 257, 309, 376, 457, 564, 687, 842, 1028, 1266, 1549, 1900, 2327, 2861, 3512, 4323, 5320, 6542, 8072, 9936, 12251, 15104, 18640, 23000, 28428
Offset: 0

Views

Author

Labos Elemer, May 09 2001

Keywords

Examples

			For n = 10, the cubes of primes not exceeding 2^10 = 1024 are 8, 27, 125, 343, so a(10) = 4.

Links

Crossrefs

Cf. A000720, A007053, A017979, A030078, A036386, A060967, A060970, A060971.

Programs

  • Mathematica
    Table[ PrimePi[ Floor[ 2^(g/3)//N ] ], {g, 0, 90} ]

Formula

a(3*n) = A007053(n). - Chai Wah Wu, Jan 23 2025
a(n) = A000720(A017979(n)). - Amiram Eldar, Mar 22 2025

Extensions

Missing a(0)=0 inserted by Sean A. Irvine, Jan 09 2023

A060971 Number of fifth powers of primes <= 2^n.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 4, 4, 4, 5, 6, 6, 7, 8, 9, 9, 11, 11, 13, 15, 16, 18, 21, 23, 25, 29, 31, 34, 39, 44, 47, 54, 62, 68, 76, 86, 97, 107, 122, 137, 154, 172, 193, 217, 244, 275, 309, 349, 393, 442, 499, 564, 635, 712, 807, 914, 1028, 1163, 1315, 1482
Offset: 0

Views

Author

Labos Elemer, May 09 2001

Keywords

Examples

			For n = 10: the 5th powers of primes not exceeding 2^10 = 1024 are 32 and 243, so a(10) = 2.

Links

Crossrefs

Cf. A000720, A007053, A018117, A036386, A050997, A060967, A060969, A060970.

Programs

  • Mathematica
    Table[ PrimePi[ Floor[ 2^(g/5)//N ] ], {g, 0, 150} ]

Formula

a(5*n) = A007053(n). - Chai Wah Wu, Jan 23 2025
a(n) = A000720(A018117(n)). - Amiram Eldar, Mar 22 2025

Extensions

Missing a(0)=0 inserted by Sean A. Irvine, Jan 09 2023
Showing 1-4 of 4 results.

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