A006508 -id:A006508 - cp's OEIS frontend

A018252 The nonprime numbers: 1 together with the composite numbers, A002808.

1, 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28, 30, 32, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45, 46, 48, 49, 50, 51, 52, 54, 55, 56, 57, 58, 60, 62, 63, 64, 65, 66, 68, 69, 70, 72, 74, 75, 76, 77, 78, 80, 81, 82, 84, 85, 86, 87, 88

Offset: 1

N. J. A. Sloane

d(a(n)) != 2 (cf. A000005). - Juri-Stepan Gerasimov, Oct 17 2009
Number of prime divisors of a(n) (counted with multiplicity) != 1. - Juri-Stepan Gerasimov, Oct 30 2009
Largest nonprime < n-th composite. - Juri-Stepan Gerasimov, Oct 29 2009
The nonnegative nonprimes A141468 without zero; the natural nonprimes; the whole nonprimes; the counting nonprimes. If the nonprime numbers A141468 which are also the nonnegative integers A001477, then the nonprimes A141468 also called the nonnegative nonprimes. If the nonprime numbers A018252 which are also the natural (or whole or counting) numbers A000027, then the nonprimes A018252 also called the natural nonprimes, the whole nonprimes and the counting nonprimes. - Juri-Stepan Gerasimov, Nov 22 2009
Smallest nonprime > n-th nonnegative nonprime. - Juri-Stepan Gerasimov, Dec 04 2009
a(n) = A175944(A014284(n)) = A175944(A175965(n)). - Reinhard Zumkeller, Mar 18 2011

G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 3rd ed., Oxford Univ. Press, 1954, p. 2.

N. J. A. Sloane, List of nonprimes up to 20000: Table of n, a(n) for n = 1..17738
Eric Weisstein's World of Mathematics, Monica Set
Eric Weisstein's World of Mathematics, Suzanne Set
Index entries for "core" sequences

Cf. A000040 (complement), A002808.
Cf. A000005, A001222, A141468.
Boustrophedon transforms: A230955, A230954.
Cf. A010051, A239968.

A018252 := Difference([1..10^5], Filtered([1..10^5], IsPrime)); # Muniru A Asiru, Oct 21 2017

a018252 n = a018252_list !! (n-1)
a018252_list = filter ((== 0) . a010051) [1..]
-- Reinhard Zumkeller, Mar 31 2014

Magma
```
[n : n in [1..100] | not IsPrime(n) ];
```

with(numtheory); sort(convert(convert([ seq(i,i=1..541) ],set) minus convert([ seq(ithprime(i),i=1..100) ],set),list));
seq(`if`(not isprime(n),n,NULL),n=1..88); # Peter Luschny, Jul 29 2009
A018252 := proc(n) option remember; if n = 1 then 1; else for a from procname(n-1)+1 do if not isprime(a) then return a; end if; end do; end if; end proc: # R. J. Mathar, Oct 22 2010

nonPrime[n_Integer] := FixedPoint[n + PrimePi@# &, n + PrimePi@ n]; Array[ nonPrime, 75] (* Robert G. Wilson v, Jan 29 2015, based on the algorithm by Labos Elemer in A006508 *)
max = 90; Complement[Range[max], Prime[Range[PrimePi[max]]]] (* Harvey P. Dale, Aug 12 2011 *)
Join[{1}, Select[Range[100], CompositeQ]] (* Jean-François Alcover, Nov 07 2021 *)

isA018252(n) = !isprime(n)
A018252(n) = {local(a,b);b=n;a=1;while(a!=b,a=b;b=n+primepi(a));b} \\ Michael B. Porter, Nov 06 2009

a(n) = my(k=0); while(-n+n-=k-k=primepi(n), ); n; \\ Ruud H.G. van Tol, Jul 15 2024 (after code in A002808)

from sympy import isprime
def ok(n): return not isprime(n)
print([k for k in range(1, 89) if ok(k)]) # Michael S. Branicky, Nov 10 2022

from sympy import composite
def A018252(n): return 1 if n == 1 else composite(n-1) # Chai Wah Wu, Nov 15 2022

def A018252_list(n) :
    return [k for k in (1..n) if not k.is_prime()]
A018252_list(88)  # Peter Luschny, Feb 03 2012

Let b(0) = n + pi(n) and b(n+1) = n + pi(b(n)), with pi(n) = A000720(n); then a(n) is the limit value of b(n). - Floor van Lamoen, Oct 08 2001
a(n) = A137621(A137624(n)). - Reinhard Zumkeller, Jan 30 2008
A010051(a(n)) = 0. - Reinhard Zumkeller, Mar 31 2014
A239968(a(n)) = n. - Reinhard Zumkeller, Dec 02 2014

A236854 Self-inverse permutation of natural numbers: a(1)=1, then a(p_n)=c_{a(n)}, a(c_n)=p_{a(n)}, where p_n = n-th prime, c_n = n-th composite.

Original entry on oeis.org

1, 4, 9, 2, 16, 7, 6, 23, 3, 53, 26, 17, 14, 13, 83, 5, 12, 241, 35, 101, 59, 43, 8, 41, 431, 11, 37, 1523, 75, 149, 39, 547, 277, 191, 19, 179, 27, 3001, 31, 157, 24, 12763, 22, 379, 859, 167, 114, 3943, 1787, 1153, 67, 1063, 10, 103, 27457, 127, 919, 89, 21

Offset: 1

Antti Karttunen, Feb 01 2014, based on Katarzyna Matylla's original but misplaced definition for A135044 from Feb 11 2008

nonn

Shares with A026239 the property that the prime-positions 2, 3, 5, 7, ... contain only composite values and the composite-positions 4, 6, 8, 9, ..., etc. contain only prime values. However, instead of placing terms in those subsets in monotone order this sequence recursively permutes the order of both subsets with the emerging permutation itself, so this is a kind of "deep" variant of A026239. Alternatively, this can be viewed as yet another "entanglement permutation", where two pairs of complementary subsets of natural numbers are entangled with each other. In this case a complementary pair primes/composites (A000040/A002808) is entangled with a complementary pair composites/primes.

Maps A006508 to A007097 and vice versa.

			a(5)=c(a(3))=c(9)=16, because 5=prime(3), and the 9th composite number is c(9)=16.
Thus a(10)=prime(a(5))=prime(16)=53 (since 10 is the 5th composite), a(18)=prime(a(10))=prime(53)=241 (since 18 is the 10th composite), a(28)=prime(a(18))=prime(241)=1523.
A significant record value is a(198) = prime(a(152)) = prime(563167303) since 198=c(152); a(152)=prime(a(115)) since 152=c(115); a(115)=prime(a(84)); a(84)=prime(a(60)); a(60)=prime(a(42)); a(42)=prime(a(28)).

Chai Wah Wu, Table of n, a(n) for n = 1..735 (n = 1..150 from Alois P. Heinz)
Index entries for sequences that are permutations of the natural numbers

Differs from A135044 for the first time at n=8, where A135044(8)=13, while here a(8)=23.

Cf. A026239, A135141/A227413, A006508, A007097, A000040, A002808, A000720, A065855.

terms = 150; cc = Select[Range[4, 2 terms^2(*empirical*)], CompositeQ]; compositePi[k_?CompositeQ] := FirstPosition[cc, k][[1]]; a[1] = 1; a[p_?PrimeQ] := a[p] = cc[[a[PrimePi[p]]]]; a[k_] := a[k] = Prime[a[ compositePi[k]]]; Array[a, terms] (* Jean-François Alcover, Mar 02 2016 *)

A236854(n)={if(isprime(n), A002808(A236854(primepi(n))), n==1&&return(1);prime(A236854(n-primepi(n)-1)))} \\ without memoization: not much slower. - M. F. Hasler, Feb 03 2014

a236854=vector(999);a236854[1]=1;A236854(n)={a236854[n]&&return(a236854[n]); a236854[n]=if(isprime(n), A002808(A236854(primepi(n))), prime(A236854(n-primepi(n)-1)))} \\ Version with memoization. - M. F. Hasler, Feb 03 2014

from sympy import primepi, prime, isprime
def a002808(n):
    m, k = n, primepi(n) + 1 + n
    while m != k: m, k = k, primepi(k) + 1 + n
    return m # this function from Chai Wah Wu
def a(n): return n if n<2 else a002808(a(primepi(n))) if isprime(n) else prime(a(n - primepi(n) - 1))
print([a(n) for n in range(1, 101)]) # Indranil Ghosh, Jun 07 2017

a(1)=1, a(p_i) = A002808(a(i)) for primes with index i, a(c_j) = A000040(a(j)) for composites with index j (where 4 has index 1, 6 has index 2, and so on).

Values double-checked by M. F. Hasler, Feb 03 2014

A022450 a(1) = 2; a(n+1) = a(n)-th composite.

Original entry on oeis.org

2, 6, 12, 21, 33, 49, 69, 94, 125, 164, 212, 270, 339, 422, 520, 636, 774, 933, 1121, 1339, 1590, 1880, 2210, 2587, 3021, 3512, 4074, 4710, 5427, 6239, 7155, 8183, 9339, 10637, 12084, 13705, 15517, 17534, 19773, 22266, 25030, 28095, 31484, 35239, 39387, 43960

Offset: 1

Clark Kimberling

nonn

C. Kimberling, Fractal sequences and interspersions, Ars Combinatoria, vol. 45 p 157 1997.

Chai Wah Wu, Table of n, a(n) for n = 1..900
C. Kimberling, Interspersions

Cf. A006508, A022451, A025010, A025011.

g[ n_Integer ] := (k = n + PrimePi[ n ] + 1; While[ k - PrimePi[ k ] - 1, k++ ]; k); NestList[ g, 2, 45 ]
With[{cmps=Select[Range[100000],CompositeQ]},NestList[cmps[[#]]&,2,50]] (* Harvey P. Dale, Jun 24 2025 *)

A025010 a(1) = 5; a(n+1) = a(n)-th nonprime, where nonprimes begin at 4.

Original entry on oeis.org

5, 10, 18, 28, 42, 60, 84, 115, 152, 198, 253, 320, 399, 494, 605, 736, 891, 1072, 1280, 1521, 1800, 2120, 2488, 2910, 3387, 3934, 4552, 5250, 6038, 6929, 7931, 9057, 10324, 11733, 13315, 15076, 17043, 19224, 21656, 24361, 27353, 30660, 34330, 38382, 42866

Offset: 1

David W. Wilson

nonn

Chai Wah Wu, Table of n, a(n) for n = 1..901

Cf. A006508, A022450, A022451, A025011.

g[ n_Integer ] := (k = n + PrimePi[ n ] + 1; While[ k - PrimePi[ k ] - 1, k++ ]; k); NestList[ g, 5, 45 ]

A025011 a(1) = 7; a(n+1) = a(n)-th composite.

Original entry on oeis.org

7, 14, 24, 36, 52, 74, 100, 133, 174, 222, 284, 356, 442, 543, 665, 805, 969, 1161, 1383, 1643, 1939, 2278, 2665, 3108, 3614, 4189, 4840, 5577, 6412, 7348, 8400, 9584, 10912, 12392, 14049, 15903, 17963, 20253, 22801, 25624, 28757, 32214, 36044, 40273, 44943

Offset: 1

David W. Wilson

nonn

Chai Wah Wu, Table of n, a(n) for n = 1..901

Cf. A006508, A025010.

g[ n_Integer ] := (k = n + PrimePi[ n ] + 1; While[ k - PrimePi[ k ] - 1, k++ ]; k); NestList[ g, 7, 45 ]
With[{cmps=Select[Range[100000],CompositeQ]},NestList[cmps[[#]]&,7,50]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jul 08 2017 *)

A135044 a(1)=1, then a(c) = p and a(p) = c, where c = T_c(r,k) and p = T_p(r,k), and where T_p contains the primes arranged in rows by the prime index chain and T_c contains the composites arranged in rows by the order of compositeness. See Formula.

Original entry on oeis.org

1, 4, 9, 2, 16, 7, 6, 13, 3, 19, 26, 17, 8, 23, 41, 5, 12, 67, 10, 29, 59, 37, 14, 83, 179, 11, 43, 331, 20, 47, 39, 109, 277, 157, 53, 431, 22, 1063, 31, 191, 15, 2221, 27, 61, 211, 71, 30, 599, 1787, 919, 241, 3001, 35, 73, 8527, 127, 1153, 79, 21, 19577, 44, 89, 283

Offset: 1

Katarzyna Matylla, Feb 11 2008

nonn

Exchanges primes with composites, primeth primes with composith composites, etc.
Exchange the k-th prime of order j with the k-th composite of order j and vice versa.
Self-inverse permutation of positive integers.
If n is the composite number A236536(r,k), then a(n) is the corresponding prime A236542(r,k) at the same position (r,k). Vice versa, if n is the prime A236542(r,k), then a(n) is the corresponding composite A236536(r,k) at the same position. - Andrew Weimholt, Jan 28 2014
The original name for this entry did not produce this sequence, but instead A236854, which differs from this permutation for the first time at n=8, where A236854(8)=23, while here a(8)=13. - Antti Karttunen, Feb 01 2014

			From _Andrew Weimholt_, Jan 29 2014: (Start)
More generally, takes the primes organized in an array according to the sieving process described in the Fernandez paper:
        Row[1](n) = 2, 7, 13, 19, 23, ...
        Row[2](n) = 3, 17, 41, 67, 83, ...
        Row[3](n) = 5, 59, 179, ...
        Row[4](n) = 11, 277, ...
        Lets call this  T_p (n, k)
Also take the composites organized in a similar manner, except we use "composite" numbered positions in our sieve:
        Row[1](n) = 4, 6, 8, 10, 14, 20, 22, ...
        Row[2](n) = 9, 12, 15, 18, 24, ...
        Row[3](n) = 16, 21, 25, ...
        Lets call this T_c (n, k)
If we now take the natural numbers and swap each number (except for 1) with the number which holds the same spot in the other array, then we get the sequence: 1, 4, 9, 2, 16, 7, 6, 13, with for example a(8) = 13 (13 holds the same position in the 'prime' table as 8 does in the 'composite' table). (End)

R. J. Mathar, Table of n, a(n) for n = 1..197
N. Fernandez, An order of primeness, F(p).
N. Fernandez, An order of primeness [cached copy, included with permission of the author]
Index to permutations of positive integers

Cf. A000040, A007097, A049076, A049078 - A049081, A058322, A058324 - A058328, A093046, A002808, A006508, A059981, A078442, A236854.

A135044 := proc(n)
    if n = 1 then
        1;
    elif isprime(n) then
        idx := -1 ;
        for r from 1 do
            for c from 1 do
                if A236542(r,c) = n then
                    idx := [r,c] ;
                end if;
                if A236542(r,c) >= n then
                    break;
                end if;
            end do:
            if type(idx,list)  then
                break;
            end if;
        end do:
        A236536(r,c) ;
    else
        idx := -1 ;
        for r from 1 do
            for c from 1 do
                if A236536(r,c) = n then
                    idx := [r,c] ;
                end if;
                if A236536(r,c) >= n then
                    break;
                end if;
            end do:
            if type(idx,list)  then
                break;
            end if;
        end do:
        A236542(r,c) ;
    end if;
end proc: # R. J. Mathar, Jan 28 2014

Composite[n_Integer] := Block[{k = n + PrimePi@n + 1}, While[k != n + PrimePi@k + 1, k++ ]; k]; Compositeness[n_] := Block[{c = 1, k = n}, While[ !(PrimeQ@k || k == 1), k = k - 1 - PrimePi@k; c++ ]; c]; Primeness[n_] := Block[{c = 1, k = n}, While[ PrimeQ@k, k = PrimePi@k; c++ ]; c];
ckj[k_, j_] := Select[ Table[Composite@n, {n, 10000}], Compositeness@# == j &][[k]]; pkj[k_, j_] := Select[ Table[Prime@n, {n, 3000}], Primeness@# == j &][[k]]; f[0]=0; f[1] = 1;
f[n_] := If[ PrimeQ@ n, pn = Primeness@n; ckj[ Position[ Select[ Table[ Prime@ i, {i, 150}], Primeness@ # == pn &], n][[1, 1]], pn], cn = Compositeness@n; pkj[ Position[ Select[ Table[ Composite@ i, {i, 500}], Compositeness@ # == cn &], n][[1, 1]], cn]]; Array[f, 64] (* Robert G. Wilson v *)

a(1)=1, a(A236536(r,k))=A236542(r,k), a(A236542(r,k))=A236536(r,k)

Edited, corrected and extended by Robert G. Wilson v, Feb 18 2008

Name corrected by Andrew Weimholt, Jan 29 2014

A236536 Array T(n,k) read along antidiagonals: the composites of order of compositeness n in row n.

Original entry on oeis.org

4, 6, 9, 8, 12, 16, 10, 15, 21, 26, 14, 18, 25, 33, 39, 20, 24, 28, 38, 49, 56, 22, 32, 36, 42, 55, 69, 78, 27, 34, 48, 52, 60, 77, 94, 106, 30, 40, 50, 68, 74, 84, 105, 125, 141, 35, 45, 57, 70, 93, 100, 115, 140, 164, 184, 44, 51, 64, 80, 95, 124, 133, 152, 183, 212, 236, 46, 63, 72, 88, 110, 126, 162, 174, 198, 235, 270, 299

Offset: 1

R. J. Mathar, Jan 28 2014

Row n contains the composites A002808(j) for which A059981(j) = n.
The 1st row contains the composites with a nonprime index, A002808(1)=4, A002808(2)=6, A002808(3)=8, A002808(5)=10, A002808(7)=14,...
The 2nd row contains the composites with an index in the 1st row.
Recursively the followup rows contain the composites that need a higher number of applications of A002808 to reach a nonprime.

			The array starts:
  4,  6,  8, 10, 14, 20, 22, 27, 30, 35,...
  9, 12, 15, 18, 24, 32, 34, 40, 45, 51,...
 16, 21, 25, 28, 36, 48, 50, 57, 64, 72,...
 26, 33, 38, 42, 52, 68, 70, 80, 88, 98,...
 39, 49, 55, 60, 74, 93, 95,110,119,130,...
 56, 69, 77, 84,100,124,126,145,156,170,...
 78, 94,105,115,133,162,165,188,203,218,...
106,125,140,152,174,209,213,242,259,278,...
141,164,183,198,222,266,272,305,326,348,...

Cf. A006508 (column 1), A022449 (row 1), A135044, A236542, A002808.

A236536 := proc(n,k)
    option remember ;
    if n = 1 then
        A022449(k) ;
    else
        A002808(procname(n-1,k)) ;
    end if:
end proc:
for d from 2 to 10 do
     for k from d-1 to  by -1 do
        printf("%3d,",A236536(d-k,k)) ;
     end do:
end do:

Composite[n_] := FixedPoint[n + PrimePi[#] + 1&, n + PrimePi[n] + 1];
T[n_, k_] := T[n, k] = If[n == 1, Composite[If[k == 1, 1, Prime[k - 1]]], Composite[T[n - 1, k]]];
Table[T[n - k + 1, k], {n, 1, 12}, {k, n, 1, -1}] // Flatten (* Jean-François Alcover, Sep 16 2023 *)

T(1,k) = A022449(k).

T(n,k) = A002808( T(n-1,k) ), n>1 .

A022451 a(1) = 3; a(n+1) = a(n)-th composite.

Original entry on oeis.org

3, 8, 15, 25, 38, 55, 77, 105, 140, 183, 235, 298, 372, 462, 566, 692, 838, 1007, 1205, 1432, 1698, 2002, 2352, 2755, 3210, 3731, 4322, 4990, 5747, 6601, 7562, 8638, 9854, 11211, 12731, 14422, 16315, 18425, 20765, 23372, 26258, 29460, 32998, 36912, 41229

Offset: 1

Clark Kimberling

nonn

C. Kimberling, Fractal sequences and interspersions, Ars Combinatoria, vol. 45 p 157 1997.

Chai Wah Wu, Table of n, a(n) for n = 1..900
C. Kimberling, Interspersions

Cf. A006508, A022450, A025010, A025011.

g[ n_Integer ] := (k = n + PrimePi[ n ] + 1; While[ k - PrimePi[ k ] - 1, k++ ]; k); NestList[ g, 3, 45 ]
With[{comps=Complement[Range[80000],Prime[Range[PrimePi[80000]]]]}, NestList[comps[[#+1]]&,3,50]] (* Harvey P. Dale, Mar 17 2012 *)

A059407 a(n+1) = a(n)-th composite number, with a(1) = 11.

Original entry on oeis.org

11, 20, 32, 48, 68, 93, 124, 162, 209, 266, 334, 415, 513, 628, 764, 922, 1108, 1325, 1574, 1858, 2186, 2562, 2992, 3480, 4038, 4670, 5379, 6184, 7094, 8115, 9263, 10552, 11991, 13600, 15400, 17403, 19629, 22107, 24856, 27902, 31275, 35008

Offset: 1

Robert G. Wilson v, Jan 29 2001

nonn

Chai Wah Wu, Table of n, a(n) for n = 1..900

Cf. A006508, A059408.

g[ n_Integer ] := (k = n + PrimePi[ n ] + 1; While[ k - PrimePi[ k ] - 1, k++ ]; k); NestList[ g, 11, 45 ]
Module[{c=Select[Range[500000],CompositeQ]},NestList[c[[#]]&,11,50]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Sep 12 2019 *)

lista(nn) = {print1(a = 11, ", "); nb = 0; forcomposite(c=1, nn, nb++; if (nb==a, print1(c, ", "); a = c););} \\ Michel Marcus, May 14 2018

A059408 a(n+1) = a(n)-th composite and a(1) = 13.

Original entry on oeis.org

13, 22, 34, 50, 70, 95, 126, 165, 213, 272, 341, 424, 524, 640, 778, 938, 1127, 1345, 1596, 1886, 2217, 2596, 3031, 3523, 4086, 4724, 5445, 6259, 7176, 8205, 9364, 10666, 12118, 13744, 15560, 17583, 19827, 22328, 25099, 28171, 31569, 35334

Offset: 1

Robert G. Wilson v, Jan 29 2001

nonn

Chai Wah Wu, Table of n, a(n) for n = 1..900

Cf. A006508, A059407.

g[ n_Integer ] := (k = n + PrimePi[ n ] + 1; While[ k - PrimePi[ k ] - 1, k++ ]; k); NestList[ g, 13, 45 ]

lista(nn) = {print1(a = 13, ", "); nb = 0; forcomposite(c=1, nn, nb++; if (nb==a, print1(c, ", "); a = c););} \\ Michel Marcus, May 14 2018

cp's OEIS Frontend

A018252 The nonprime numbers: 1 together with the composite numbers, A002808.

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A236854 Self-inverse permutation of natural numbers: a(1)=1, then a(p_n)=c_{a(n)}, a(c_n)=p_{a(n)}, where p_n = n-th prime, c_n = n-th composite.

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A022450 a(1) = 2; a(n+1) = a(n)-th composite.

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A025010 a(1) = 5; a(n+1) = a(n)-th nonprime, where nonprimes begin at 4.

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A025011 a(1) = 7; a(n+1) = a(n)-th composite.

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A135044 a(1)=1, then a(c) = p and a(p) = c, where c = T_c(r,k) and p = T_p(r,k), and where T_p contains the primes arranged in rows by the prime index chain and T_c contains the composites arranged in rows by the order of compositeness. See Formula.

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A236536 Array T(n,k) read along antidiagonals: the composites of order of compositeness n in row n.

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