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 1-5 of 5 results.

A014684 In the sequence of positive integers subtract 1 from each prime number.

Original entry on oeis.org

1, 1, 2, 4, 4, 6, 6, 8, 9, 10, 10, 12, 12, 14, 15, 16, 16, 18, 18, 20, 21, 22, 22, 24, 25, 26, 27, 28, 28, 30, 30, 32, 33, 34, 35, 36, 36, 38, 39, 40, 40, 42, 42, 44, 45, 46, 46, 48, 49, 50, 51, 52, 52, 54, 55, 56, 57, 58, 58, 60, 60, 62, 63, 64, 65, 66, 66, 68, 69, 70, 70, 72
Offset: 1

Views

Author

Mohammad K. Azarian

Keywords

Links

Crossrefs

Cf. A000010, A048855, A113638.

Programs

  • Haskell
    a014684 n = n - fromIntegral (a010051 n)
-- Reinhard Zumkeller, Sep 10 2013
  • Magma
    [n - (IsPrime(n) select 1 else 0): n in [1..80]]; // Bruno Berselli, Jul 18 2016
  • Mathematica
    Table[If[PrimeQ[n],n-1,n],{n,100}] (* Harvey P. Dale, Aug 27 2015 *)
  • Python
    from sympy import isprime
def A014684(n): return n-int(isprime(n)) # Chai Wah Wu, Oct 14 2023

Formula

a(n) = A005171(n) + n - 1.
a(n) = phi(n!)/phi((n-1)!). - Vladeta Jovovic, Nov 30 2002
For n > 3: a(n) = A113523(n) = A179278(n). - Reinhard Zumkeller, Jul 08 2010
a(n) = n - A010051(n). - Reinhard Zumkeller, Sep 10 2013

Extensions

More terms from Andrew J. Gacek (andrew(AT)dgi.net)

A179278 Largest nonprime integer <= n.

Original entry on oeis.org

1, 1, 1, 4, 4, 6, 6, 8, 9, 10, 10, 12, 12, 14, 15, 16, 16, 18, 18, 20, 21, 22, 22, 24, 25, 26, 27, 28, 28, 30, 30, 32, 33, 34, 35, 36, 36, 38, 39, 40, 40, 42, 42, 44, 45, 46, 46, 48, 49, 50, 51, 52, 52, 54, 55, 56, 57, 58, 58, 60, 60, 62, 63, 64, 65, 66, 66, 68, 69, 70, 70, 72
Offset: 1

Views

Author

Reinhard Zumkeller, Jul 08 2010

Keywords

Examples

			From _Gus Wiseman_, Dec 04 2024: (Start)
The nonprime integers <= n:
  1  1  1  4  4  6  6  8  9  10  10  12  12  14  15  16
           1  1  4  4  6  8  9   9   10  10  12  14  15
                 1  1  4  6  8   8   9   9   10  12  14
                       1  4  6   6   8   8   9   10  12
                          1  4   4   6   6   8   9   10
                             1   1   4   4   6   8   9
                                     1   1   4   6   8
                                             1   4   6
                                                 1   4
                                                     1
(End)

Crossrefs

Cf. A014684, A113523, A113638, A062298.
For prime we have A007917.
For nonprime we have A179278 (this).
For squarefree we have A070321.
For nonsquarefree we have A378033.
For prime power we have A031218.
For non prime power we have A378367.
For perfect power we have A081676.
For non perfect power we have A378363.
A000040 lists the primes, differences A001223.
A002808 lists the composite numbers, differences A073783.
A018252 lists the nonprimes, differences A065310.
A095195 has row n equal to the k-th differences of the prime numbers.
A113646 gives least nonprime >= n.
A151800 gives the least prime > n, weak version A007918.
A377033 has row n equal to the k-th differences of the composite numbers.

Programs

  • Mathematica
    Array[# - Boole[PrimeQ@ #] - Boole[# == 3] &, 72] (* Michael De Vlieger, Oct 13 2018 *)
Table[Max@@Select[Range[n],!PrimeQ[#]&],{n,30}] (* Gus Wiseman, Dec 04 2024 *)
  • PARI
    a(n) = if (isprime(n), if (n==3, 1, n-1), n); \\ Michel Marcus, Oct 13 2018

Formula

For n > 3: a(n) = A113523(n) = A014684(n);
For n > 0: a(n) = A113638(n). - Georg Fischer, Oct 12 2018
A005171(a(n)) = 1; A010051(a(n)) = 0.
a(n) = A018252(A062298(n)). - Ridouane Oudra, Aug 22 2025

Extensions

Inequality in the name reversed by Gus Wiseman, Dec 05 2024

A256885 a(n) = n*(n + 1)/2 - pi(n), where pi(n) = A000720(n) is the prime counting function.

Original entry on oeis.org

1, 2, 4, 8, 12, 18, 24, 32, 41, 51, 61, 73, 85, 99, 114, 130, 146, 164, 182, 202, 223, 245, 267, 291, 316, 342, 369, 397, 425, 455, 485, 517, 550, 584, 619, 655, 691, 729, 768, 808, 848, 890, 932, 976, 1021, 1067, 1113, 1161, 1210, 1260, 1311, 1363, 1415, 1469, 1524
Offset: 1

Views

Author

Wesley Ivan Hurt, Apr 11 2015

Keywords

Comments

Number of lattice points (x,y) in the region 1 <= x <= n, 1 <= y <= n - A010051(n); see example.
This sequence gives the row sums of the triangle A257232. - Wolfdieter Lang, Apr 21 2015

Examples

			10 .                             x
9  .                          x  x
8  .                       x  x  x
7  .                    .  x  x  x
6  .                 x  x  x  x  x
5  .              .  x  x  x  x  x
4  .           x  x  x  x  x  x  x
3  .        .  x  x  x  x  x  x  x
2  .     .  x  x  x  x  x  x  x  x
1  .  x  x  x  x  x  x  x  x  x  x
0  .__.__.__.__.__.__.__.__.__.__.
   0  1  2  3  4  5  6  7  8  9  10

Links

Crossrefs

Cf. A000217, A000720, A010051, A014684, A113638, A257232.

Programs

  • Haskell
    a256885 n = a000217 n - a000720 n  -- Reinhard Zumkeller, Apr 21 2015
  • Magma
    [n*(n + 1)/2 - #PrimesUpTo(n): n in [1..60] ]; // Vincenzo Librandi, Apr 12 2015
  • Maple
    with(numtheory)[pi]: A256885:=n->n*(n+1)/2-pi(n): seq(A256885(n), n=1..100);
  • Mathematica
    Table[n (n + 1)/2 - PrimePi[n], {n, 1, 50}]
  • PARI
    vector(80, n, n*(n+1)/2 - primepi(n)) \\ Michel Marcus, Apr 13 2015

Formula

a(n) = A000217(n) - A000720(n).
a(n) - a(n-1) = A014684(n), n >= 2.
a(n) = Sum_{i=1..n} A014684(i).
a(n) = 1 + Sum_{i=2..n}(i - A000720(i) + A000720(i-1)).

Extensions

Edited, following the hint by Reinhard Zumkeller to change the offset. - Wolfdieter Lang, Apr 22 2015

A113636 In the sequence of positive integers add 1 to each nonprime number.

Original entry on oeis.org

2, 2, 3, 5, 5, 7, 7, 9, 10, 11, 11, 13, 13, 15, 16, 17, 17, 19, 19, 21, 22, 23, 23, 25, 26, 27, 28, 29, 29, 31, 31, 33, 34, 35, 36, 37, 37, 39, 40, 41, 41, 43, 43, 45, 46, 47, 47, 49, 50, 51, 52, 53, 53, 55, 56, 57, 58, 59, 59, 61, 61, 63, 64, 65, 66, 67, 67, 69, 70, 71, 71, 73
Offset: 1

Views

Author

Cino Hilliard, Jan 15 2006

Keywords

Comments

This is the complement of sequence A014683.
Möbius transform of A380449(n). - Wesley Ivan Hurt, Jun 21 2025

Crossrefs

Cf. A005171, A014683, A014684, A113523, A113638, A179278, A380449.

Programs

  • Mathematica
    Array[# + Boole[! PrimeQ@ #] &, 72] (* Michael De Vlieger, Nov 05 2020 *)
  • PARI
    a(n) = if (!isprime(n), n+1, n); \\ Michel Marcus, Nov 06 2020

Formula

a(n) = A014684(n) + 1. - Bill McEachen, Nov 01 2020
From Wesley Ivan Hurt, Jun 21 2025: (Start)
a(n) = n + c(n), where c = A005171.
a(n) = Sum_{d|n} A380449(d) * mu(n/d). (End)

Extensions

Offset 1 from Michel Marcus, Nov 06 2020

A113637 In the sequence of positive integers subtract 1 from each nonprime number.

Original entry on oeis.org

0, 2, 3, 3, 5, 5, 7, 7, 8, 9, 11, 11, 13, 13, 14, 15, 17, 17, 19, 19, 20, 21, 23, 23, 24, 25, 26, 27, 29, 29, 31, 31, 32, 33, 34, 35, 37, 37, 38, 39, 41, 41, 43, 43, 44, 45, 47, 47, 48, 49, 50, 51, 53, 53, 54, 55, 56, 57, 59, 59, 61, 61, 62, 63, 64, 65, 67, 67, 68, 69, 71, 71, 73
Offset: 1

Views

Author

Cino Hilliard, Jan 15 2006

Keywords

Crossrefs

Cf. A113636, A113638.

Programs

  • Mathematica
    Table[If[PrimeQ[n],n,n-1],{n,73}] (* James C. McMahon, Jul 07 2024 *)
  • PARI
    a(n) = if (isprime(n), n, n-1); \\ Michel Marcus, Jul 08 2024
Showing 1-5 of 5 results.

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