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.

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A114413 Records in 3-almost prime gaps ordered by merit.

Original entry on oeis.org

4, 6, 12, 58, 83
Offset: 1

Views

Author

Jonathan Vos Post, Nov 25 2005

Keywords

Examples

			Records defined in terms of A114403 and A014612:
  n  A114403(n)  A114403(n)/log_10(A014612(n))
  =  ==========  =============================
  1      4       4/log_10(8)   = 4.42923746
  2      6       6/log_10(12)  = 5.55977045
  3      2       2/log_10(18)  = 1.59327954
  4      7       7/log_10(20)  = 5.38035251
  5      1       1/log_10(27)  = 0.698634425
  6      2       2/log_10(28)  = 1.38201907
  7      12      12/log_10(30) = 8.12390991
  ...
  19     14      14/log_10(78) = 7.3992072

Crossrefs

Cf. A014612, A065516, A111870, A111871, A114403-A114411, A114412-A114422.

Formula

a(n) = records in A114403(n)/log_10(A014612(n)) = records in (A014612(n+1) - A014612(n))/log_10(A014612(n)).

Extensions

a(4)-a(5) from Donovan Johnson, Feb 17 2010

A114416 Records in 6-almost prime gaps ordered by merit.

Original entry on oeis.org

32, 48, 56, 84, 105, 140
Offset: 1

Views

Author

Jonathan Vos Post, Nov 25 2005

Keywords

Examples

			Records defined in terms of A114406 and A046306:
n A114406(n) A114406(n)/log(A046306(n)).
1 32 32/log 64 = 17.7169498
2 48 48/log 96 = 24.2146479
3 16 16/log 144 = 7.41302726
4 56 56/log 160 = 25.4069653
5 8 8/log 216 = 3.42692589
6 16 16/log 224 = 6.80779215
7 84 84/log 240 = 35.2909853
8 12 12/log 324 = 4.77983862
...
22 105 105/log 624 = 37.5646032

Crossrefs

Cf. A046306, A065516, A111870, A111871, A114403-A114411, A114412-A114422.

Formula

a(n) = records in A114406(n)/log(A046306(n)) = records in (A046306(n+1) - A046306(n))/log(A046306(n)).

Extensions

a(6) from Donovan Johnson, Feb 17 2010

A131939 Least k such that the difference between consecutive 3-almost primes A014612(k) equals n, or 0 if no such k exists.

Original entry on oeis.org

5, 3, 13, 1, 10, 2, 4, 31, 32, 36, 12, 7, 136, 19, 302, 486, 1094, 73, 1366, 6763, 1092, 2006, 8924, 4785, 18345, 18487, 42798, 16571, 11095, 57831, 60912, 4528, 24846, 41304, 232350, 233678, 123279, 1779265, 740729, 177385, 1015228, 1772286
Offset: 1

Views

Author

Jonathan Vos Post, Oct 05 2007

Keywords

Comments

Index of smallest 3-almost prime gap equal to n. This is to 3-almost primes A014612 as A123375 is to semiprimes A001358. a(25) = 0 or a(25) > 10000. Conjecture: a(n) > 0 for all n.

Examples

			a(1) = 5 because A014612(6)-A014612(5) = 28-27 = 1.
a(2) = 3 because A014612(4)-A014612(3) = 20-18 = 2.
a(3) = 13 because 66-63 = 3.
a(4) = 1 because 12-8 = 4.
a(5) = 10 because 50-45 = 5.
a(6) = 2 because 18-12 = 6.
a(7) = 4 because 27-20 = 7.
a(8) = 31 because 138-130 = 8.
a(9) = 32 because 147-138 = 9
a(10) = 36 because 164-154 = 10.

Crossrefs

Cf. A014612, A114403, A123375.

Programs

  • Mathematica
    p3=Select[Range[2*10^6],PrimeOmega[#]==3&];s={};Do[d=0;Until[p3[[d+1]]-p3[[d]]==n,d++];AppendTo[s,d],{n,37}];s (* James C. McMahon, Mar 02 2025 *)

Formula

a(n) = MIN{k such that A114403(k) = n, or 0 if no such k exists}.
a(n) = MIN{k such that A014612(k+1) - A014612(k) = n, or 0 if no such k exists}.

Extensions

More terms from R. J. Mathar, Oct 07 2007
Previous Showing 21-23 of 23 results.

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