A025011 -id:A025011 - cp's OEIS frontend

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

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 ]

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 *)

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

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

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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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