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.

Previous Showing 11-19 of 19 results.

A053683 Number of nonprimes <= prime(n)^2.

Original entry on oeis.org

2, 5, 16, 34, 91, 130, 228, 289, 430, 695, 799, 1150, 1418, 1566, 1880, 2400, 2994, 3202, 3880, 4366, 4624, 5430, 6003, 6921, 8246, 8949, 9315, 10068, 10458, 11246, 14252, 15185, 16628, 17131, 19712, 20254, 21920, 23654, 24846, 26688, 28605
Offset: 1

Views

Author

Labos Elemer, Feb 16 2000

Keywords

Examples

			Up to 25 = prime(3)^2, the 16 nonprimes are 1, 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25.

Links

Crossrefs

Cf. A000879.

Programs

  • Mathematica
    a[n_] := Prime[n]^2 - PrimePi[Prime[n]^2]; Array[a, 50] (* Amiram Eldar, Feb 16 2025 *)
  • PARI
    a(n) = prime(n)^2 - primepi(prime(n)^2); \\ Michel Marcus, Oct 28 2013

Formula

a(n) = A000040(n)^2 - A000879(n). - Jean-Christophe Hervé, Oct 27 2013

A089609 Prime number of primes between squares of consecutive primes; or, primes occurring in A050216.

Original entry on oeis.org

2, 5, 11, 47, 163, 89, 463, 479, 199, 107, 241, 151, 709, 151, 599, 313, 547, 211, 613, 859, 863, 241, 1217, 1091, 827, 311, 967, 1327, 691, 1109, 1123, 829, 389, 821, 857, 431, 1301, 433, 1451, 1933, 3449, 5701, 1753, 4663, 563, 3557, 4253, 1867, 4447
Offset: 0

Views

Author

Cino Hilliard, Dec 30 2003

Keywords

Comments

For small values of n, these numbers exhibit higher and lower values as n increases. Conjecture: There exists an n such that seq(n1) is < seq(n1+1) for all n1 >= n.

Links

Crossrefs

Cf. A000879, A050216.

Programs

  • Mathematica
    Select[PrimePi[#[[2]]]-PrimePi[#[[1]]]&/@Partition[Prime[Range[500]]^2,2,1],PrimeQ] (* Harvey P. Dale, May 15 2022 *)
  • PARI
    \ prime number of primes between squares. pbetweensq(n) = { for(x=1,n, c=0; for(y=prime(x)^2, prime((x+1))^2, if(isprime(y),c++) ); if(isprime(c),print1(c",")) ) }

Extensions

Edited by Ray Chandler, Jan 05 2004

A181962 Numbers not of the form pi(p) + pi(sqrt(p)) for some prime p.

Original entry on oeis.org

3, 6, 12, 19, 35, 45, 68, 80, 108, 156, 173, 231, 276, 297, 344, 425, 504, 537, 628, 695, 726, 833, 909, 1024, 1188, 1278, 1321, 1409, 1452, 1553, 1908, 2008, 2174, 2224, 2524, 2583, 2766, 2953, 3082, 3281, 3477, 3554, 3911, 3989, 4134, 4210, 4674, 5154, 5323
Offset: 1

Views

Author

Vladimir Shevelev, Apr 06 2012

Keywords

Comments

Or places of squares in A000430.

Examples

			12 is in the sequence, since pi(23)+pi(sqrt(23))=9+2=11, while pi(29)+pi(sqrt(29))=10+3=13.
Also 12 is in the sequence since A000430(12)=25 is not prime.

Crossrefs

Cf. A000430, A000040, A000720, A000879.

Programs

  • Maple
    a:= n-> numtheory[pi](ithprime(n)^2)+n:
seq(a(n), n=1..50);  # Alois P. Heinz, Feb 21 2025
  • Mathematica
    t = Table[PrimePi[n] + PrimePi[Sqrt[n]], {n, Prime[Range[10000]]}]; Complement[Range[t[[-1]]], t] (* T. D. Noe, Apr 09 2012 *)
  • Python
    from sympy import primepi, prime
def A181962(n): return primepi(prime(n)**2)+n # Chai Wah Wu, Feb 18 2025

Formula

a(n) = pi(prime(n)^2) + n = A000879(n) + n. - Chai Wah Wu, Feb 18 2025

A217019 First differences of A128301.

Original entry on oeis.org

2, 6, 8, 23, 16, 34, 24, 50, 89, 36, 115, 80, 49, 101, 149, 190, 58, 202, 140, 79, 231, 159, 270, 371, 193, 103, 216, 113, 212, 804, 260, 391, 151, 667, 148, 431, 472, 318, 486, 507, 153, 870, 169, 365, 185, 1145, 1214, 405, 207, 434, 614, 218, 1135, 703, 721
Offset: 1

Views

Author

Zak Seidov, Sep 24 2012

Keywords

Comments

The sequence is of chaotic behavior and unbound. Exactly as in A050216 (see comment by T. D. Noe over there), the lines in the graph correspond to prime gaps of 2, 4, 6,...

Links

Crossrefs

Cf. A000879, A050216, A128301, A165144.
Cf. A001248 (primes squared), A001358 (semiprimes).

Formula

a(n) = A165144(n) + 1. - Flávio V. Fernandes, Nov 19 2020

A348836 a(n) is the number of primes <= prime(n)^2/2.

Original entry on oeis.org

1, 2, 5, 9, 17, 23, 34, 41, 56, 81, 92, 124, 146, 157, 185, 222, 270, 283, 334, 368, 386, 444, 481, 548, 635, 681, 703, 753, 780, 832, 1013, 1068, 1160, 1192, 1345, 1376, 1472, 1577, 1648, 1752, 1864, 1899, 2090, 2129, 2197, 2240, 2492, 2749, 2837, 2881, 2976, 3107, 3159, 3389, 3540
Offset: 1

Views

Author

Michel Marcus, Nov 01 2021

Keywords

Links

Crossrefs

First column of A283235.
Cf. A000040, A000720, A000879, A001248.

Programs

  • Mathematica
    a[n_] := PrimePi[Prime[n]^2/2]; Array[a, 55] (* Amiram Eldar, Nov 01 2021 *)
  • PARI
    a(n) = primepi(prime(n)^2/2);

Formula

a(n) = A000720(prime(n)^2/2).

A062773 Index of the smallest prime which follows square of n-th prime.

Original entry on oeis.org

3, 5, 10, 16, 31, 40, 62, 73, 100, 147, 163, 220, 264, 284, 330, 410, 488, 520, 610, 676, 706, 812, 887, 1001, 1164, 1253, 1295, 1382, 1424, 1524, 1878, 1977, 2142, 2191, 2490, 2548, 2730, 2916, 3044, 3242, 3437, 3513, 3869, 3946, 4090, 4165, 4628
Offset: 1

Views

Author

Labos Elemer, Jul 18 2001

Keywords

Examples

			100th prime, 541 immediately follows 529, square of 9th prime, a(9)=100.

Links

Crossrefs

Cf. A054270, A054271, A000879.

Programs

  • Mathematica
    PrimePi[NextPrime[#]]&/@(Prime[Range[50]]^2) (* Harvey P. Dale, Apr 12 2023 *)
  • PARI
    a(n) = { primepi(prime(n)^2) + 1 } \\ Harry J. Smith, Aug 10 2009

Formula

a(n) = pi( nextprime( prime(n)^2 ) ).
a(n) = A000720(A062772(n)). - Michel Marcus, Jun 24 2014

Extensions

Offset changed from 0 to 1 by Harry J. Smith, Aug 10 2009

A054273 Number of primes p in the interval prime(n+1) <= p < prime(n+1)^2 such that A002110(n)+p is prime.

Original entry on oeis.org

2, 6, 10, 19, 23, 29, 25, 38, 42, 35, 56, 54, 45, 60, 67, 84, 66, 76, 94, 98, 95, 92, 108, 108, 107, 129, 127, 128, 127, 152, 160, 152, 145, 173, 153, 156, 183, 214, 208, 212, 201, 220, 220, 219, 222, 248, 255, 241, 252, 265, 265, 252, 280, 276, 291, 292
Offset: 1

Views

Author

Labos Elemer, May 05 2000

Keywords

Examples

			n=3, prime(4)=7, prime(4)^2=49; 3rd primorial number = 30; in interval [7,49] 12 primes p occur of which 10 are such that 30+p is prime, namely 30+{7,11,13,17,23,29,31,37,41,43} = {37,41,...,73}, "post-primorial primes", while two primes 19 and 47 yield 49, 77 which are composites. So a(3)=10.

Crossrefs

Cf. A000879, A001248, A000040, A005235, A002110.

A054409 Maximal difference between consecutive primes in range [prime(n), prime(n)^2].

Original entry on oeis.org

1, 2, 4, 6, 8, 14, 14, 14, 14, 18, 20, 34, 34, 34, 34, 34, 34, 34, 34, 34, 34, 34, 34, 34, 34, 36, 36, 36, 36, 36, 44, 44, 44, 44, 52, 52, 52, 52, 52, 52, 72, 72, 72, 72, 72, 72, 72, 72, 72, 72, 72, 72, 72, 72, 72, 72, 72, 72, 72, 72, 72, 72, 72, 72, 72, 72, 72, 72, 72, 72, 72
Offset: 1

Views

Author

Labos Elemer, May 09 2000

Keywords

Examples

			n=6, prime(6)=13, prime(6)^2=169, the maximal difference in this interval is 127-113=14, so a(6)=14

Crossrefs

Cf. A000040, A000879, A001223, A001248, A002110.

A276824 a(n) = p-s, where s = Sum_{i=1..n} prime(i) and p = primepi(prime(n)^2).

Original entry on oeis.org

0, -1, -1, -2, 2, -2, 3, -5, -1, 17, 2, 22, 25, 2, 1, 28, 47, 18, 41, 36, -7, 20, 12, 37, 103, 91, 30, 10, -57, -70, 157, 125, 153, 63, 213, 120, 145, 168, 129, 154, 170, 65, 230, 114, 61, -63, 189, 445, 386, 239, 172, 203, 52, 239, 257, 268
Offset: 1

Views

Author

Dimitris Valianatos, Sep 27 2016

Keywords

Examples

			n= 10, prime(10) = 29, s = Sum_{i=1,10} prime(i) = 2+3+5+7+11+13+17+19+23+29 = 129,
p= primepi(29^2) = primepi(841) = 146. So a(10) = p-s = 146 - 129 = 17.

Crossrefs

Cf. A000879, A007504.

Programs

  • PARI
    a(n) = primepi(prime(n)^2) - sum(k=1, n, prime(k)); \\ Michel Marcus, Oct 17 2016

Formula

a(n) = A000879(n) - A007504(n).
Previous Showing 11-19 of 19 results.

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