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 1-4 of 4 results.

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

Views

Author

Labos Elemer, Nov 26 2001

Keywords

Comments

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

Links

Crossrefs

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

Programs

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

A065857 The (10^n)-th composite number.

Original entry on oeis.org

4, 18, 133, 1197, 11374, 110487, 1084605, 10708555, 106091745, 1053422339, 10475688327, 104287176419, 1039019056246, 10358018863853, 103307491450820, 1030734020030318, 10287026204717358, 102692313540015924, 1025351434864118026, 10239531292310798956, 102270102190290407386
Offset: 0

Views

Author

Labos Elemer, Nov 26 2001

Keywords

Examples

			The 100th composite number is C(100)=133, while the 100th prime is 541. In general: A000720(m) < A062298(m) < m < A002808(m) < A000040(m), for example pi(100)=25 < 75 < 100 < C(100)=133 < prime(100)=541.

References

  • A. E. Bojarincev, Asymptotic expressions for the n-th composite number. Univ. Mat. Zap. 6:21-43(1967). [in Russian]
  • J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 133, p. 45, Ellipses, Paris 2008.

Links

Crossrefs

Cf. A033844, A002808, A062298, A000720, A006988, A065855, A065856.

Programs

  • Mathematica
    Composite[n_Integer] := Block[ {k = n + PrimePi[n] + 1 }, While[ k != n + PrimePi[k] + 1, k = n + PrimePi[k] + 1]; Return[k]];
Table[Composite[10^n], {n, 0, 9}]
  • PARI
    a(n)=my(k=10^n);forcomposite(n=4,2*k+2,if(k--==0,return(n))) \\ Charles R Greathouse IV, May 30 2013

Formula

a(n) = A002808(A011557(n)).
a(n) = 10^(n + n/log n + 2n/log^2 + 4n/log^3 n + O(n/log^4 n)). See Bojarincev for an asymptotic expansion. - Charles R Greathouse IV, May 30 2013

Extensions

More terms from Robert G. Wilson v, Nov 26 2001
a(14) from Lekraj Beedassy, Jul 14 2008
a(15)-a(19) from Chai Wah Wu, Apr 16 2018
a(20) from Chai Wah Wu, Aug 23 2018

A073261 Length of FixedPointList approximating (2^n)-th composite number. See program link below.

Original entry on oeis.org

4, 4, 3, 3, 3, 4, 3, 5, 4, 4, 5, 4, 5, 6, 6, 6, 6, 5, 6, 6, 7, 6, 6, 6, 7, 7, 8, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 10, 10, 9, 10
Offset: 0

Views

Author

Labos Elemer, Jul 22 2002

Keywords

Comments

Number of iterations needed to reach the composite number using the formula in the program.

Examples

			n=30: {1073741824, 1128141853, 1130754984, 1130880243, 1130886219, 1130886489, 1130886503, 1130886504}, so a(30)=8.

Crossrefs

Cf. A002808, A073255-A073264, A065856.

Programs

  • Mathematica
    Table[ Length[ FixedPointList[ 2^n+PrimePi[ # ]+1 &, 2^n]]-1, {n, 0, 45}]

Extensions

Extended by Robert G. Wilson v, Jul 24 2002

A073719 a(n) = floor(prime(2^n)/composite(2^n)).

Original entry on oeis.org

0, 0, 0, 1, 2, 2, 3, 4, 5, 5, 6, 7, 8, 8, 9, 10, 11, 12, 12, 13, 14, 15, 15, 16, 17, 18, 18, 19, 20, 20, 21, 22, 23, 23, 24, 25, 26, 26, 27, 28, 28, 29, 30, 31, 31, 32, 33, 33, 34, 35, 36, 36, 37, 38, 38, 39, 40, 41, 41, 42, 43, 43, 44, 45, 46, 46, 47, 48, 48, 49, 50
Offset: 0

Views

Author

Labos Elemer, Aug 05 2002

Keywords

Crossrefs

Cf. A000040, A000079, A002818, A033844, A065856, A073459.

Programs

  • Mathematica
    c[x_] := FixedPoint[x + PrimePi[#] + 1 &, x]; Table[Floor[Prime[z = 2^n]/c[z]], {n, 40}] (* Jayanta Basu, Jul 08 2013 *)

Formula

a(n) = floor(A000040(A000079(n))/A002818(A000079(n))).
a(n) = floor(A033844(n)/A065856(n)). - Charles R Greathouse IV, Jul 08 2013

Extensions

Offset changed to 0, a(0) prepended and a(41)-a(70) added by Amiram Eldar, Jun 21 2024
Showing 1-4 of 4 results.

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