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 1-3 of 3 results.

A165355 a(n) = 3n + 1 if n is even, or a(n) = (3n + 1)/2 if n is odd.

Original entry on oeis.org

1, 2, 7, 5, 13, 8, 19, 11, 25, 14, 31, 17, 37, 20, 43, 23, 49, 26, 55, 29, 61, 32, 67, 35, 73, 38, 79, 41, 85, 44, 91, 47, 97, 50, 103, 53, 109, 56, 115, 59, 121, 62, 127, 65, 133, 68, 139, 71, 145, 74, 151, 77, 157, 80, 163, 83, 169, 86, 175, 89, 181, 92, 187, 95, 193, 98
Offset: 0

Views

Author

Paul Curtz, Sep 16 2009

Keywords

Comments

Second trisection of A026741.
A111329(n+1) = A000041(a(n)). - Reinhard Zumkeller, Nov 19 2009
We observe that this sequence is a particular case of sequence for which there exists q: a(n+3) = (a(n+2)*a(n+1)+q)/a(n) for every n >= n0. Here q=-9 and n0=0. - Richard Choulet, Mar 01 2010
The entries are also encountered via the bilinear transform approximation to the natural log (unit circle). Specifically, evaluating 2(z-1)/(z+1) at z = 2, 3, 4, ..., A165355 entries stem from the pair (sums) seen 2 ahead of each new successive prime. For clarity, the evaluation output is 2, 3, 1, 1, 6, 5, 4, 3, 10, 7, 3, 2, 14, 9, 8, 5, 18, 11, ..., where (1+1), (4+3), (3+2), (8+5), ... generate the A165355 entries (after the first). As an aside, the same mechanism links A165355 to A140777. - Bill McEachen, Jan 08 2015
As a follow-up to the previous comment, it appears that the numerators and denominators of 2(z-1)/(z+1) are respectively given by A145979 and A060819, but with different offsets. - Michel Marcus, Jan 14 2015
Odd parts of the terms give A067745. E.g.: 1, 2/2, 7, 5, 13, 8/8 .... - Joe Slater, Nov 30 2016

Links

Crossrefs

Cf. A165351, A165367, A067745, A007814.

Programs

  • Mathematica
    f[n_] := If[ OddQ@ n, (3n +1)/2, (3n +1)]; Array[f, 66, 0] (* Robert G. Wilson v, Jan 26 2015 *)
f[n_] := (3 (-1)^(2n) + (-1)^(1 + n)) (-2 + 3n)/4; Array[f, 66] (* or *)
CoefficientList[ Series[(x^3 + 5x^2 + 2x + 1)/(x^2 - 1)^2, {x, 0, 65}], x] (* or *)
LinearRecurrence[{0, 2, 0, -1}, {1, 2, 7, 5}, 66] (* Robert G. Wilson v, Apr 13 2017 *)
  • PARI
    a(n)=n+=2*n+1; if(n%2,n,n/2) \\ Charles R Greathouse IV, Jan 13 2015

Formula

a(n) = A026741(3*n+1).
a(n)*A026741(n) = A005449(n).
a(n)*A022998(n+1) = A000567(n+1).
a(n) = A026741(n+1) + A022998(n).
a(2n) = A016921(n). a(2n+1) = A016789(n).
a(2n+1)*A026741(2n) = A045944(n).
G.f.: (1+2*x+5*x^2+x^3)/((x-1)^2 * (1+x)^2). - R. J. Mathar, Sep 26 2009
a(n) = (3+9*n)/4 + (-1)^n*(1+3*n)/4. - R. J. Mathar, Sep 26 2009
a(n) = 2*(3n+1)/(4-((2n+2) mod 4)). - Bill McEachen, Jan 09 2015
If a(2n-1) = x then a(2n) = 2x+3. - Robert G. Wilson v, Jan 26 2015
Let the reduced Collatz procedure be defined as Cr(n) = (3*n+1)/2. For odd n, a(n) = Cr(n). For even n, a(n) = Cr(4*n+1)/2. - Joe Slater, Nov 29 2016
a(n) = A067745(n+1) * 2^A007814((3n+1)/2). - Joe Slater, Nov 30 2016
a(n) = 2*a(n-2) - a(n-4). - G. C. Greubel, Apr 13 2017

Extensions

All comments changed to formulas by R. J. Mathar, Sep 26 2009
New name from Charles R Greathouse IV, Jan 13 2015
Name corrected by Joe Slater, Nov 29 2016

A140775 Numbers k > 1 such that p + k/p is prime for every prime p that divides k.

Original entry on oeis.org

2, 6, 10, 22, 30, 34, 42, 58, 70, 78, 82, 102, 118, 130, 142, 190, 202, 210, 214, 274, 298, 310, 322, 330, 358, 382, 394, 442, 454, 462, 478, 510, 538, 562, 582, 610, 622, 658, 694, 714, 730, 742, 790, 838, 862, 922, 930, 970, 1002, 1038, 1042, 1110, 1138, 1198
Offset: 1

Views

Author

Leroy Quet, May 29 2008

Keywords

Comments

All terms of this sequence are even and squarefree.
The only term == 2 (mod 3) is 2. - Robert Israel, Jan 09 2024

Examples

			The primes dividing 70 are 2, 5, 7. Now, 2 + 70/2 = 37; 5 + 70/5 = 19; 7 + 70/7 = 17. Since 37, 19 and 17 are each prime, then 70 is included in this sequence.

Links

Crossrefs

Cf. A140776, A140777, A294925.

Programs

  • Maple
    filter:= t -> andmap(p -> isprime(p+t/p), numtheory:-factorset(t)):
select(filter, [seq(i,i=2..2000,4)]); # Robert Israel, Jan 09 2024
  • Mathematica
    fQ[n_] := Block[{p = First@ Transpose@ FactorInteger@ n}, Union@ PrimeQ[p + n/p] == {True}]; Select[ Range[2, 1221], fQ@# &] (* Robert G. Wilson v, May 30 2008 *)
pnpQ[n_]:=AllTrue[#+n/#&/@Transpose[FactorInteger[n]][[1]],PrimeQ]; Select[ Range[2,1200],pnpQ] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Apr 15 2016 *)

Extensions

More terms from Robert G. Wilson v, May 30 2008
Definition edited by Robert Israel, Jan 09 2024

A140776 A number n is included if (p + n/p) is prime, where p is the largest prime that divides n.

Original entry on oeis.org

2, 6, 10, 12, 22, 24, 28, 30, 34, 40, 42, 48, 52, 58, 60, 66, 70, 76, 78, 82, 84, 88, 90, 102, 112, 118, 120, 130, 132, 138, 142, 148, 160, 168, 172, 180, 184, 186, 190, 192, 198, 202, 204, 208, 210, 214, 220, 222, 228, 232, 234, 238, 240, 246, 252, 268, 270, 274
Offset: 1

Views

Author

Leroy Quet, May 29 2008

Keywords

Comments

All terms are even. - Robert Israel, Feb 04 2018

Examples

			The largest prime dividing 48 is 3. Since 3 + 48/3 = 19 is prime, then 48 is included in this sequence.

Links

Crossrefs

Cf. A140775, A140777.

Programs

  • Maple
    filter:= proc(n) local p;
  p:= max(numtheory:-factorset(n));
  isprime(p+n/p)
end proc:
select(filter, 2*[$1..1000]); # Robert Israel, Feb 04 2018
  • Mathematica
    fQ[n_] := Block[{p = Last@ First@ Transpose@ FactorInteger@n}, PrimeQ[p + n/p] == True]; Select[ Range[2, 279], fQ@# &] (* Robert G. Wilson v, May 30 2008 *)

Extensions

More terms from Robert G. Wilson v, May 30 2008
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.