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.

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

Views

Author

David W. Wilson

Keywords

Links

Crossrefs

Cf. A006508, A022450, A022451, A025011.

Programs

  • Mathematica
    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

Views

Author

Clark Kimberling

Keywords

References

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

Links

Crossrefs

Cf. A006508, A022450, A025010, A025011.

Programs

  • Mathematica
    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

Views

Author

Matthew Campbell, Sep 25 2015

Keywords

Comments

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

Examples

			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.

Links

Crossrefs

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.

Programs

  • Mathematica
    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 *)

Extensions

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

A377181 Rectangular array, by antidiagonals: (row 1) = r(1) = A002808 (composite numbers); (row n) = r(n) = A002808(r(n-1)) for n>=1.

Original entry on oeis.org

4, 6, 9, 8, 12, 16, 9, 15, 21, 26, 10, 16, 25, 33, 39, 12, 18, 26, 38, 49, 56, 14, 21, 28, 39, 55, 69, 78, 15, 24, 33, 42, 56, 77, 94, 106, 16, 25, 36, 49, 60, 78, 105, 125, 141, 18, 26, 38, 52, 69, 84, 106, 140, 164, 184, 20, 28, 39, 55, 74, 94, 115, 141, 183, 212, 236
Offset: 1

Views

Author

Clark Kimberling, Oct 19 2024

Keywords

Examples

			 corner:
   4     6     8     9    10    12    14    15    16    18
   9    12    15    16    18    21    24    25    26    28
  16    21    25    26    28    33    36    38    39    42
  26    33    38    39    42    49    52    55    56    60
  39    49    55    56    60    69    74    77    78    84
  56    69    77    78    84    94   100   105   106   115
  78    94   105   106   115   125   133   140   141   152

Crossrefs

Cf. A002808 (row 1), A050545 (row 2), A280327 (row 3), A006508 (column 1), A022450 (column 2), A023451 (column 3), A059981, A236356, A280327 (principal diagonal), A377173, A114577 (dispersion of the composite numbers).

Programs

  • Mathematica
    c[n_] := c[n] = Select[Range[500], CompositeQ][[n]]
r[0] = Table[c[n], {n, 1, 10}]
r[n_] := r[n] = c[r[n - 1]]
Grid[Table[r[n], {n, 0, 6}]]  (* array *)
p[n_, k_] := r[n][[k]];
Table[p[n - k + 1, k], {n, 0, 9}, {k, n + 1, 1, -1}] // Flatten  (* sequence *)

Formula

A059981(n) = number of appearances of A002808(n).
Showing 1-4 of 4 results.

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