A022449 -id:A022449 - cp's OEIS frontend

A073168 a(n) = A007821(n) - A022449(n).

1, 5, 9, 9, 9, 15, 16, 17, 18, 17, 25, 19, 21, 27, 31, 26, 21, 22, 21, 35, 38, 31, 35, 31, 34, 33, 37, 39, 49, 49, 33, 52, 49, 47, 39, 43, 47, 47, 48, 48, 41, 49, 48, 60, 59, 59, 49, 52, 58, 58, 63, 71, 75, 65, 65, 67, 71, 79, 75, 81, 84, 77, 65, 69, 72, 72, 67, 69, 61, 65, 65

Offset: 1

Labos Elemer, Jul 19 2002

nonn

Values of commutator of A000040 and A002808 functions i.e., of prime() and composite().

Cf. A000040, A002808, A007821, A022449.

c[n_Integer] := FixedPoint[n+PrimePi[ # ]+1&, n] Table[Prime[c[w]]-c[Prime[w]], {w, 1, 10000}];

A050435 a(n) = composite(composite(n)), where composite = A002808, composite numbers.

Original entry on oeis.org

9, 12, 15, 16, 18, 21, 24, 25, 26, 28, 32, 33, 34, 36, 38, 39, 40, 42, 45, 48, 49, 50, 51, 52, 55, 56, 57, 60, 63, 64, 65, 68, 69, 70, 72, 74, 76, 77, 78, 80, 81, 84, 86, 87, 88, 90, 91, 93, 94, 95, 98, 100, 102, 104, 105, 106, 110, 111, 112, 115, 116, 117, 118, 119

Offset: 1

Michael Lugo (mlugo(AT)thelabelguy.com), Dec 22 1999

Second-order composite numbers.

Composites (A002808) with composite (A002808) subscripts. a(n) U A022449(n) = A002808(n). Subsequence of A175251 (composites (A002808) with nonprime (A018252) subscripts), a(n) = A175251(n+1) for n >= 1. - Jaroslav Krizek, Mar 14 2010

			The 2nd composite number is 6 and the 6th composite number is 12, so a(2) = 12. a(100) = A002808(A002808(100)) = A002808(133) = 174.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
N. Fernandez, An order of primeness, F(p)
N. Fernandez, An order of primeness [cached copy, included with permission of the author]

Cf. A002808, A018252, A022449, A175251.

a050435 = a002808 . a002808
a050435_list = map a002808 a002808_list
-- Reinhard Zumkeller, Jan 12 2013

Select[ Range[ 6, 150 ], ! PrimeQ[ # ] && ! PrimeQ[ # - PrimePi[ # ] - 1 ] & ]
With[{cmps=Select[Range[200],CompositeQ]},Table[cmps[[cmps[[n]]]],{n,70}]] (* Harvey P. Dale, Feb 18 2018 *)

composite(n)=my(k=-1); while(-n + n += -k + k=primepi(n), ); n \\ M. F. Hasler
a(n)=composite(composite(n)) \\ Charles R Greathouse IV, Jun 25 2017

from sympy import composite
def a(n): return composite(composite(n))
print([a(n) for n in range(1, 65)]) # Michael S. Branicky, Sep 12 2021

Let C(n) be the n-th composite number, with C(1)=4. Then these are numbers C(C(n)).

a(n) = n + 2n/log n + O(n/log^2 n). - Charles R Greathouse IV, Jun 25 2017

More terms from Robert G. Wilson v, Dec 20 2000

A065858 m-th composite number c(m) = A002808(m), where m is the n-th prime number: a(n) = A002808(A000040(n)).

Original entry on oeis.org

6, 8, 10, 14, 20, 22, 27, 30, 35, 44, 46, 54, 58, 62, 66, 75, 82, 85, 92, 96, 99, 108, 114, 120, 129, 134, 136, 142, 144, 148, 166, 171, 178, 182, 194, 196, 204, 210, 215, 221, 230, 232, 245, 247, 252, 254, 268, 285, 289, 291, 296, 302, 304, 318, 324, 330, 338

Offset: 1

Labos Elemer, Nov 26 2001

nonn

Composites (A002808) with prime (A000040) subscripts. a(n) U A175251(n) = A002808(n). Subsequence of A022449 (composites (A002808) with noncomposite (A008578) subscripts), a(n) = A022449(n+1). - Jaroslav Krizek, Mar 14 2010

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A002808, A000040, A008578, A022449, A065856, A065857, A175251.

P,C:= selectremove(isprime,[seq(i,i=2..10^3)]):
seq(C[P[i]],i=1..100); # Robert Israel, Mar 09 2025

Composite[n_] := FixedPoint[n + PrimePi[#] + 1 & , n + PrimePi[n] + 1];
a[n_] := Composite[Prime[n]];
Array[a, 100] (* Jean-François Alcover, Jan 26 2018, after Robert G. Wilson v *)

A059981 Order of compositeness for the n-th composite number.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 2, 3, 2, 1, 3, 1, 2, 3, 4, 1, 3, 1, 2, 4, 2, 1, 3, 4, 5, 2, 4, 1, 2, 1, 3, 5, 3, 2, 4, 1, 5, 6, 3, 1, 5, 1, 2, 3, 2, 1, 4, 6, 4, 3, 5, 1, 2, 6, 7, 4, 2, 1, 6, 1, 2, 3, 4, 3, 2, 1, 5, 7, 5, 1, 4, 1, 6, 2, 3, 7, 8, 1, 5, 3, 2, 1, 7, 2, 3, 4

Offset: 1

Robert G. Wilson v, Mar 06 2001

Let c(k) = k-th composite number, let S(c) = S(c(k)) = k, the subscript of c; a(n) = order of compositeness of c(n) = 1+m where m is largest number such that S(S(..S(c(n))...)) with m S's is a composite.

Number of steps in the composite index chain for the n-th composite. - Daniel Forgues, Sep 28 2012

			16 is 9th composite number, so S(16)=9, 9 is 4th composite, so S(S(16))=4, 4 is first composite number, so S(S(S(16)))=1, not a composite number. Thus a(9)=3.
4 is the first composite number, so S(4)=1, not a composite number. Thus a(1)=1.

T. D. Noe, Table of n, a(n) for n = 1..1000

Cf. A049076, A022449 (composites with compositeness 1).

Composite[ n_Integer ] := (k = n + PrimePi[ n ] + 1; While[ k != n + PrimePi[ k ] + 1, k++ ]; k); CompositePi[ n_Integer ] := (n - 1 - PrimePi[ n ]); Attributes[ Composite ] = Attributes[ CompositePi ] = Listable; Table[ c = 1; k = CompositePi[ Composite[ n ] ]; While[ ! (PrimeQ[ k ] || k == 1), k = CompositePi[ k ]; c++ ]; c, {n, 100} ]

A073169 a(n)=A002808(n)-n, difference between n-th composite and n.

Original entry on oeis.org

3, 4, 5, 5, 5, 6, 7, 7, 7, 8, 9, 9, 9, 10, 10, 10, 10, 10, 11, 12, 12, 12, 12, 12, 13, 13, 13, 14, 15, 15, 15, 16, 16, 16, 16, 16, 17, 17, 17, 17, 17, 18, 19, 19, 19, 19, 19, 20, 20, 20, 21, 22, 22, 22, 22, 22, 23, 23, 23, 24, 24, 24, 24, 24, 25, 25, 25, 25, 25, 25, 25, 26, 26

Offset: 1

Labos Elemer, Jul 19 2002

nonn

a(n) = the number of numbers of set {1, prime} (A008578(n)) less than n-th composite numbers (A002808(n)). a(n) = inverse (frequency distribution) sequence of A162177(n), i.e. number of terms of sequence A162177(n) less than n for n >= 1. a(n) = A002808(n) + A162177(n) - A158611(n+1) for n >= 1. a(n) = A002808(n) + A162177(n) - A008578(n) for n >= 1. [From Jaroslav Krizek, Jul 23 2009]

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A007821, A022449, A000040, A002808, A073168.

c[n_Integer] := FixedPoint[n+PrimePi[ # ]+1&, n] Table[c[w]-w, {w, 1, 128}]
With[{c=Select[Range[100],CompositeQ]},#[[1]]-#[[2]]&/@Thread[ {c,Range[ Length[ c]]}]] (* Harvey P. Dale, Feb 03 2015 *)

a(n)=1+A073425(n). [From R. J. Mathar, Jul 31 2009]

Correction for change of offset in A158611 and A008578 in Aug 2009 Jaroslav Krizek, Jan 27 2010

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 .

A175251 Composites (A002808) with nonprime (A018252) subscripts.

Original entry on oeis.org

4, 9, 12, 15, 16, 18, 21, 24, 25, 26, 28, 32, 33, 34, 36, 38, 39, 40, 42, 45, 48, 49, 50, 51, 52, 55, 56, 57, 60, 63, 64, 65, 68, 69, 70, 72, 74, 76, 77, 78, 80, 81, 84, 86, 87, 88, 90, 91, 93, 94, 95, 98, 100

Offset: 1

Jaroslav Krizek, Mar 13 2010

nonn

a(n) = composite(nonprime(n)) = A002808(A018252(n)). a(n) U A065858(n) = A002808(n), a(n+1) U A022449(n) = A002808(n) for n >= 1. a(1) = 4, a(n) = A050435(n-1) = composites (A002808) with composite (A002808) subscripts for n >=2.

			a(5) = 16 because a(5) = c(b(5)) = c(9) = 16, c = composite, b = nonprime.

A260621 Let b(k, n) = number obtained when the map x->A002808(x) is applied k times to n; a(n) is the smallest k such that b(k, n) + 1 is prime.

Original entry on oeis.org

1, 1, 12, 2, 1, 1, 3, 11, 1, 1, 7, 9, 1, 2, 10, 4, 2, 1, 1, 6, 8, 3, 3, 1, 9, 3, 1, 1, 18, 3, 1, 5, 7, 2, 2, 1, 4, 8, 2, 14, 1, 1, 6, 17, 2, 6, 1, 4, 6, 1, 1, 2, 2, 3, 7, 1, 13, 6, 1, 4, 16, 5, 16, 1, 5, 31, 35, 3, 5, 2, 1, 2, 3, 1, 1, 2, 6, 1, 1, 12, 5, 1, 2

Offset: 1

Matthew Campbell, Sep 25 2015

nonn

a(n) is also the smallest value of k at which b(k, n+1) - b(k, n) > 1.

			When n = 3, writing Composite(x) for A002808(x):
1. Composite(3) = 8. 8 + 1 = 9 = 3^2. 9 is not prime.
2. Composite(8) = 15. 15 + 1 = 16 = 2^4. 16 is not prime.
3. Composite(15) = 25. 25 + 1 = 26 = 2*13. 26 is not prime.
4. Composite(25) = 38. 38 + 1 = 39 = 3*13. 39 is not prime.
5. Composite(38) = 55. 55 + 1 = 56 = 2^3*7. 56 is not prime.
6. Composite(55) = 77. 77 + 1 = 78 = 2*3*13. 78 is not prime.
7. Composite(77) = 105. 105 + 1 = 106 = 2*53. 106 is not prime.
8. Composite(105) = 140. 140 + 1 = 141 = 3*47. 141 is not prime.
9. Composite(140) = 183. 183 + 1 = 184 = 2^3*23. 184 is not prime.
10. Composite(183) = 235. 235 + 1 = 236 = 2^2*59. 236 is not prime.
11. Composite(235) = 298. 298 + 1 = 299 = 13*23. 299 is not prime.
12. Composite(298) = 372. 372 + 1 = 373. 373 is prime.
--------------------------------------------------------------
Since the composite function was applied 12 times, a(3)=12.

Jon E. Schoenfield, Table of n, a(n) for n = 1..10000

Primes and nonprimes: A000040, A002808, A008578, A018252.
a(1) = p, a(n+1) = a(n)-th composite number: A006508, A022450, A022451, A025010, A025011, A059407, A059408.
Composites with order n > 1: A050435, A050436, A050438, A050439, A050440.
Composites with order n = b, n >= 1: A022449.
Composites with prime subscripts: A065858.
Composites without prime subscripts: A175251.
Order of compositeness: A059981, A236536.
Prime(n)-1: A006093.

c = Select[Range[10^5], CompositeQ]; Table[k = 1; While[! PrimeQ[Nest[c[[#]] &, n, k] + 1], k++]; k, {n, 120}] (* Michael De Vlieger, Jul 15 2016 *)

Terms from a(12) onward from Jon E. Schoenfield, Sep 27 2015

cp's OEIS Frontend

A073168 a(n) = A007821(n) - A022449(n).

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A050435 a(n) = composite(composite(n)), where composite = A002808, composite numbers.

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A065858 m-th composite number c(m) = A002808(m), where m is the n-th prime number: a(n) = A002808(A000040(n)).

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A059981 Order of compositeness for the n-th composite number.

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A073169 a(n)=A002808(n)-n, difference between n-th composite and n.

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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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A175251 Composites (A002808) with nonprime (A018252) subscripts.

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A260621 Let b(k, n) = number obtained when the map x->A002808(x) is applied k times to n; a(n) is the smallest k such that b(k, n) + 1 is prime.

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