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.

A350098 a(n) is the lower end of a record gap A349995(n) between consecutive odd squarefree semiprimes (A046388).

Original entry on oeis.org

15, 21, 95, 267, 2369, 6559, 8817, 13705, 15261, 21583, 35981, 66921, 113009, 340891, 783757, 872219, 3058853, 3586843, 5835191, 12345473, 108994623, 248706917, 268749691, 679956119, 709239621, 3648864859, 3790337723, 4171420481, 33955869693, 34279038379, 34840796369
Offset: 1

Views

Author

Hugo Pfoertner, Dec 26 2021

Keywords

Examples

			See A349995.
		

Crossrefs

Starting at a(3)=95 the terms coincide with the known terms of A114057.

Formula

a(n) = A350099(n) - A349995(n).

A350099 a(n) is the upper end of a record gap A349995(n) between consecutive odd squarefree semiprimes (A046388).

Original entry on oeis.org

21, 33, 111, 287, 2391, 6583, 8843, 13733, 15293, 21619, 36019, 66961, 113053, 340941, 783809, 872279, 3058917, 3586913, 5835265, 12345557, 108994713, 248707009, 268749791, 679956221, 709239737, 3648864977, 3790337843, 4171420613, 33955869829, 34279038517, 34840796509
Offset: 1

Views

Author

Hugo Pfoertner, Dec 26 2021

Keywords

Examples

			See A349995.
		

Crossrefs

Formula

a(n) = A350098(n) + A349995(n).

A114057 Start of record gap in odd semiprimes A046315.

Original entry on oeis.org

9, 25, 39, 95, 267, 2369, 6559, 8817, 13705, 15261, 21583, 35981, 66921, 113009, 340891, 783757, 872219, 3058853, 3586843, 5835191, 12345473, 108994623, 248706917, 268749691, 679956119, 709239621, 3648864859, 3790337723, 4171420481, 33955869693, 34279038379
Offset: 1

Views

Author

Jonathan Vos Post, Feb 02 2006

Keywords

Comments

3 of the first 5 values of record gaps in odd semiprimes are also record merits = (A046315(k+1)-A046315(k))/log_10(A046315(k)), namely: (15 - 9) / log_10(9) = 6.28770982; (111 - 95) / log_10(95) = 8.09010923; (287 - 267) / log_10(267) = 8.24228608. It is easy to prove that there are gaps of arbitrary length in even semiprimes (A100484); can we prove that there are gaps of arbitrary length in odd semiprimes (A046315) and in semiprimes (A001358)?
The record gaps have lengths 6, 8, 10, 16, 20, 22, 24, 26, 28, 32, 36, 38, 40, 44, 50, 52, 60, 64, 70, 74. - T. D. Noe, Feb 03 2006

Examples

			a(1) = A046315(2)-A046315(1) = 15 - 9 = 6.
a(2) = A046315(5)-A046315(4) = 33 - 25 = 8.
a(3) = A046315(8)-A046315(7) = 49 - 39 = 10.
a(4) = A046315(20)-A046315(19) = 111 - 95 = 16.
a(5) = A046315(55)-A046315(54) = 287 - 267 = 20.
		

Crossrefs

Starting at a(4)=95 the known terms of this sequence coincide with A350098.

Programs

  • Mathematica
    f[n_] := Block[{k = n + 2}, While[ Plus @@ Last /@ FactorInteger@k != 2, k += 2]; k]; lst = {}; d = 0; a = b = 9; Do[{a, b} = {b, f[a]}; If[b - a > d, d = b - a; AppendTo[lst, a]], {n, 10^8}]; lst (* Robert G. Wilson v, Feb 03 2006 *)

Formula

{a(n)} = {A046315(k) such that A046315(k+1)-A046315(k) is a record}.

Extensions

More terms from Robert G. Wilson v and T. D. Noe, Feb 03 2006
a(23)-a(28) from Donovan Johnson, Mar 14 2010
a(29)-a(31) from Donovan Johnson, Oct 20 2012

A350096 a(n) is the larger of 2 consecutive primes bounding an interval containing a record number A350097(n) of odd squarefree semiprimes (A046388).

Original entry on oeis.org

17, 37, 97, 211, 223, 907, 1151, 1361, 10007, 15727, 19661, 44351, 156007, 370373, 396833, 492227, 1357333, 1671907, 3826157, 17836561, 20831533, 47465443, 107534789, 122164969, 434865671, 436273291, 2300942869, 4302407713, 10726905041, 25056082543, 42652618807
Offset: 1

Views

Author

Hugo Pfoertner, Dec 25 2021

Keywords

Examples

			See A350095.
		

Crossrefs

A350097 gives the corresponding counts.

Formula

a(n) = nextprime(A350095(n)).

Extensions

a(29)-a(31) from Martin Ehrenstein, Dec 28 2021
a(32) from Lucas A. Brown, Mar 21 2024
Showing 1-4 of 4 results.