A014684 -id:A014684 - cp's OEIS frontend

A179278 Largest nonprime integer <= n.

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

Reinhard Zumkeller, Jul 08 2010

			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)

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.

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 *)

a(n) = if (isprime(n), if (n==3, 1, n-1), n); \\ Michel Marcus, Oct 13 2018

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

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

A113638 In the sequence of nonnegative integers subtract 1 from each prime number.

Original entry on oeis.org

0, 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: 0

Cino Hilliard, Jan 15 2006

Cf. A014684.

Table[If[PrimeQ[n],n-1,n],{n,0,80}] (* Harvey P. Dale, Sep 10 2017 *)

from sympy import isprime
def A113638(n): return n-int(isprime(n)) # Chai Wah Wu, Oct 14 2023

a(n) = A014684(n), n>0. - R. J. Mathar, Aug 13 2008

Name corrected by Wolfdieter Lang, Apr 22 2015

A113523 a(n) = largest composite nonnegative integer <= n.

Original entry on oeis.org

0, 0, 0, 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

Leroy Quet, Jan 12 2006

nonn

For n > 3: a(n) = A179278(n). [From Reinhard Zumkeller, Jul 08 2010]

Cf. A007917, A014684.

nnci[n_]:=Module[{k=0},While[PrimeQ[n-k],k++];n-k]/.{1->0}; Array[nnci,80] (* Harvey P. Dale, Jul 19 2012 *)

a(1)=a(2)=a(3) = 0. For n >= 4, a(n) = A014684(n).

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

Wesley Ivan Hurt, Apr 11 2015

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

			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

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Prime Counting Function
Wikipedia, Prime-counting function

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

a256885 n = a000217 n - a000720 n  -- Reinhard Zumkeller, Apr 21 2015

[n*(n + 1)/2 - #PrimesUpTo(n): n in [1..60] ]; // Vincenzo Librandi, Apr 12 2015

with(numtheory)[pi]: A256885:=n->n*(n+1)/2-pi(n): seq(A256885(n), n=1..100);

Table[n (n + 1)/2 - PrimePi[n], {n, 1, 50}]

vector(80, n, n*(n+1)/2 - primepi(n)) \\ Michel Marcus, Apr 13 2015

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)).

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

A284383 a(1) = 1, a(2) = 2, a(3) = 3, a(4) = 2; a(n) is the largest composite number <= (a(n-a(n-1)) + a(n-a(n-2))) for n > 4.

Original entry on oeis.org

1, 2, 3, 2, 4, 4, 6, 4, 6, 6, 8, 6, 10, 6, 10, 10, 12, 8, 14, 10, 14, 10, 16, 10, 16, 16, 16, 12, 22, 12, 20, 16, 22, 14, 24, 16, 24, 16, 26, 16, 26, 26, 24, 18, 30, 22, 28, 26, 30, 20, 34, 24, 36, 20, 38, 24, 36, 24, 40, 26, 38, 26, 40, 26, 42, 26, 42, 42, 32, 28, 50, 28, 46, 34

Offset: 1

Altug Alkan, Mar 26 2017

nonn

			a(5) = 4 because a(5 - a(4)) + a(5 - a(3)) = a(3) + a(2) = 3 + 2 = 5 and largest composite number <= 5 is 4.

Altug Alkan, Table of n, a(n) for n = 1..10000
Altug Alkan, Alternative scatterplot of A284383
Altug Alkan, Illustration for A284383

Cf. A005185, A010051, A014684, A113523, A284374.

a[n_] := a[n] = If[n <= 4, n - 2 Boole[n == 4], k = 0; While[! CompositeQ@ Set[m, a[n - a[n - 1]] + a[n - a[n - 2]] - k], k++]; m]; Array[a, 74] (* Michael De Vlieger, Mar 29 2017 *)

f(n) = n-isprime(n);
a=vector(1000); a[1]=1;a[2]=2;a[3]=3; for(n=4, #a, a[n] = f(a[n-a[n-1]]+a[n-a[n-2]])); a

a(1) = 1, a(2) = 2, a(3) = 3; a(n) = a(n-a(n-1)) + a(n-a(n-2)) - A010051(a(n-a(n-1)) + a(n-a(n-2))) for n > 3.

A080786 Triangle T(n,k) = number of k-smooth numbers <= n, read by rows.

Original entry on oeis.org

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

Offset: 1

Reinhard Zumkeller, Mar 12 2003

T(n,n-1) = A014684(n) for n>1;
T(n,2) = A029837(n) for n>1; T(n,3) = A071521(n) for n>2; T(n,5) = A071520(n) for n>4.
A036234(n) = number of distinct terms in n-th row. - Reinhard Zumkeller, Sep 17 2013

			Triangle begins:
.................. 1
................ 1...2
.............. 1...2...3
............ 1...3...4...4
.......... 1...3...4...4...5
........ 1...3...5...5...6...6
...... 1...3...5...5...6...6...7
.... 1...4...6...6...7...7...8...8
.. 1...4...7...7...8...8...9...9...9.

Reinhard Zumkeller, Rows n=1..120 of triangle, flattened
Eric Weisstein's World of Mathematics, Smooth Number.

Cf. A000079, A002473, A003586, A006530, A014684, A029837, A036234, A051037, A051038, A071520, A071521, A080197, A080681, A080682, A080683.

a080786 n k = a080786_tabl !! (n-1) !! (k-1)
a080786_row n = a080786_tabl !! (n-1)
a080786_tabl = map reverse $ iterate f [1] where
   f xs@(x:_) = (x + 1) :
                (zipWith (+) xs (map (fromEnum . (lpf <=)) [x, x-1 ..]))
        where lpf = fromInteger $ a006530 $ fromIntegral (x + 1)
-- Reinhard Zumkeller, Sep 17 2013

A080786 := proc(x,y)
    local a,n ;
    a := 0 ;
    for n from 1 to x do
        if A006530(n) <= y then
            a := a+1 ;
        end if;
    end do:
    a ;
end proc: # R. J. Mathar, Aug 31 2013

P[n_] := FactorInteger[n][[-1, 1]]; P[1]=1; T[n_, k_] := (For[j=0; m=1, m <= n, m++, If[P[m] <= k, j++]]; j); Table[T[n, k], {n, 1, 15}, {k, 1, n}] // Flatten (* Jean-François Alcover, Nov 22 2015 *)

from itertools import count, islice
from sympy import prevprime, integer_log
def A080786_T(n,k):
    if k==1: return 1
    def g(x,m): return x.bit_length() if m==2 else sum(g(x//(m**i),prevprime(m))for i in range(integer_log(x,m)[0]+1))
    return g(n,prevprime(k+1))
def A080786_gen(): # generator of terms
    return (A080786_T(n,k) for n in count(1) for k in range(1,n+1))
A080786_list = list(islice(A080786_gen(),100)) # Chai Wah Wu, Oct 22 2024

A135681 a(n)=n if n=1 or if n=prime. Otherwise, n=4 if n is even and n=1 if n is odd.

Original entry on oeis.org

1, 2, 3, 4, 5, 4, 7, 4, 1, 4, 11, 4, 13, 4, 1, 4, 17, 4, 19, 4, 1, 4, 23, 4, 1, 4, 1, 4, 29, 4, 31, 4, 1, 4, 1, 4, 37, 4, 1, 4, 41, 4, 43, 4, 1, 4, 47, 4, 1, 4, 1, 4, 53, 4, 1, 4, 1, 4, 59, 4, 61, 4, 1, 4, 1, 4, 67, 4, 1, 4, 71, 4, 73

Offset: 1

Mohammad K. Azarian, Dec 01 2007

nonn

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A135679; A135580; A014681; A014682; A014683; A014684; A014685; A014686; A014687; A014688; A014689; A014690.

a[n_] := If[PrimeQ[n] || n == 1, n, If[EvenQ[n], 4, 1] ]; Table[a[n], {n,1,25}] (* G. C. Greubel, Oct 26 2016 *)

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

Cino Hilliard, Jan 15 2006

This is the complement of sequence A014683.

Möbius transform of A380449(n). - Wesley Ivan Hurt, Jun 21 2025

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

Array[# + Boole[! PrimeQ@ #] &, 72] (* Michael De Vlieger, Nov 05 2020 *)

a(n) = if (!isprime(n), n+1, n); \\ Michel Marcus, Nov 06 2020

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)

Offset 1 from Michel Marcus, Nov 06 2020

A135682 a(n)=n if n=1 or if n=prime. Otherwise, n=4 if n is even and n=7 if n is odd.

Original entry on oeis.org

1, 2, 3, 4, 5, 4, 7, 4, 7, 4, 11, 4, 13, 4, 7, 4, 17, 4, 19, 4, 7, 4, 23, 4, 7, 4, 7, 4, 29, 4, 31, 4, 7, 4, 7, 4, 37, 4, 7, 4, 41, 4, 43, 4, 7, 4, 47, 4, 7, 4, 7, 4, 53, 4, 7, 4, 7, 4, 59, 4, 61, 4, 7, 4, 7, 4, 67, 4, 7, 4, 71, 4, 73

Offset: 1

Mohammad K. Azarian, Dec 01 2007

nonn

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A000027; A000040; A135679; A135680; A014681; A014682; A014683; A014684; A014685; A014686; A014687; A014688; A014689; A014690; A135681.

a[n_] := If[PrimeQ[n] || n == 1, n, If[EvenQ[n], 4, 7] ]; Table[a[n], {n,1,25}] (* G. C. Greubel, Oct 26 2016 *)

A135684 a(n)=11 if n is a prime number. Otherwise, a(n)=n.

Original entry on oeis.org

1, 11, 11, 4, 11, 6, 11, 8, 9, 10, 11, 12, 11, 14, 15, 16, 11, 18, 11, 20, 21, 22, 11, 24, 25, 26, 27, 28, 11, 30, 11, 32, 33, 34, 35, 36, 11, 38, 39, 40, 11, 42, 11, 44, 45, 46, 11, 48, 49, 50, 51, 52, 11, 54, 55, 56, 57, 58, 11

Offset: 1

Mohammad K. Azarian, Dec 01 2007

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A000027; A000040; A135679; A135680; A014681; A014682; A014683; A014684; A014685; A014686; A014687; A014688; A014689; A014690; A135681; A135682; A135683.

[IsPrime(n) select 11 else n: n in [1..70]]; // Vincenzo Librandi, Feb 22 2013

Table[If[PrimeQ[n], 11, n], {n, 70}] (* Vincenzo Librandi, Feb 22 2013 *)

cp's OEIS Frontend

A179278 Largest nonprime integer <= n.

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A113638 In the sequence of nonnegative integers subtract 1 from each prime number.

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A113523 a(n) = largest composite nonnegative integer <= n.

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

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A284383 a(1) = 1, a(2) = 2, a(3) = 3, a(4) = 2; a(n) is the largest composite number <= (a(n-a(n-1)) + a(n-a(n-2))) for n > 4.

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A080786 Triangle T(n,k) = number of k-smooth numbers <= n, read by rows.

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A135681 a(n)=n if n=1 or if n=prime. Otherwise, n=4 if n is even and n=1 if n is odd.

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