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.

User: Tomás Roca Sánchez

Tomás Roca Sánchez's wiki page.

Tomás Roca Sánchez has authored 3 sequences.

A328734 Fibonacci numbers F(k) for k in A328733.

Original entry on oeis.org

1, 2, 3, 21, 34, 610, 196418, 24157817, 39088169, 701408733, 72723460248141, 2427893228399975082453, 781774079430987230203437, 36726740705505779255899443, 150804340016807970735635273952047185, 453973694165307953197296969697410619233826
Offset: 1

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Author

Tomás Roca Sánchez, Oct 26 2019

Keywords

Examples

			The Fibonacci numbers (A000045) start 0,1,1,2,3,5,8,13,21,..., so the first 1 is a term here, but the second 1 is not. (Note that 0 has infinitely many prime divisors.)
21 = 3 * 7 and 34 = 2 * 17 both have two prime factors, and because they are contiguous in the Fibonacci sequence, 21 is included.

Links

Crossrefs

Cf. A000045, A038575, A328733 (indices).

Extensions

Edited to avoid confusion caused by the pair of 1's in the Fibonacci sequence. - N. J. A. Sloane, Nov 11 2019

A328733 List of numbers k such that Fibonacci(k) and Fibonacci(k+1) have the same number of prime factors, counted with multiplicity.

Original entry on oeis.org

1, 3, 4, 8, 9, 15, 27, 37, 38, 44, 68, 104, 116, 124, 170, 201, 202, 205, 214, 291, 302, 361, 381, 387, 403, 428, 469, 474, 502, 507, 514, 565, 584, 602, 603, 622, 628, 663, 668, 669, 675, 698, 710, 745, 763, 766, 865, 872, 873, 898, 922, 968, 1006, 1015, 1018, 1035, 1075, 1146, 1153, 1182
Offset: 1

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Author

Tomás Roca Sánchez, Oct 26 2019

Keywords

Comments

F(1) and F(2), both being 1, count as having zero prime factors each.
0 is not a term since all primes divide 0.
For the corresponding Fibonacci numbers, see A328734.

Examples

			F(8) = 21 = 3 * 7, and F(9) = 34 = 2 * 17 have 2 prime factors each, so 8 is a part of the sequence.

Links

Crossrefs

Cf. A000045, A038575, A328734.

Programs

  • PARI
    isok(k) = bigomega(fibonacci(k)) == bigomega(fibonacci(k+1)); \\ Michel Marcus, Nov 11 2019
  • Python
    # See link

Extensions

More terms from Amiram Eldar, Oct 26 2019

A328378 Number of permutations of length n that possess the maximal sum of distances between contiguous elements.

Original entry on oeis.org

1, 1, 2, 4, 2, 8, 8, 48, 72, 576, 1152, 11520, 28800, 345600, 1036800, 14515200, 50803200, 812851200, 3251404800, 58525286400, 263363788800, 5267275776000, 26336378880000, 579400335360000, 3186701844480000, 76480844267520000, 458885065605120000, 11931011705733120000
Offset: 0

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Author

Tomás Roca Sánchez, Oct 14 2019

Keywords

Comments

From Andrew Howroyd, Oct 16 2019: (Start)
No permutation with maximal sum of distances between contiguous elements can contain three contiguous elements a, b, c such that a < b < c or a > b > c. Otherwise removing b will not alter the sum and then appending b to the end of the permutation will increase it so that the original permutation could not have been maximal. In this sense all solution permutations are alternating.
For odd n consider an alternating permutation of the form p_1 p_2 ... p_n with p_1 > p2, p_2 < p_3, etc. The sum of distances is given by (p_1 + 2*p_3 + 2*p_5 + ... 2*p_{n-2} + p_n) - 2*(p_2 + p_4 + ... p_{n-1}). This is maximized by choosing the central odd p_i to be as highest possible and the even p_i to be least possible but other than that the order does not alter the sum. Similar arguments can be made for p_1 < p_2 and for the case when n is even.
The above considerations lead to a formula for this sequence with the maximum sum being given by A047838(n). (End)

Examples

			(1,3,2) is a permutation of length 3 with distance sum |1-3| + |3-2| = 2 + 1 = 3. For n = 3, the 4 permutations with maximum sum of distances are (1,3,2), (2,1,3), (2,3,1) and (3,1,2).

Links

Crossrefs

Cf. A047838 is the maximum distance for every length n, except for n = 0 and n = 1.
Cf. A070910, A096789.

Programs

  • Mathematica
    A328378[n_]:=If[n<2,1,2(Floor[n/2]-1)!^2If[Divisible[n,2],1,n-1]];Array[A328378,30,0] (* Paolo Xausa, Aug 13 2023 *)
  • PARI
    a(n)={if(n<2, n>=0, 2*(n\2-1)!^2*if(n%2, n-1, 1))} \\ Andrew Howroyd, Oct 16 2019
  • Python
    # See Github link

Formula

a(2*n) = 2*(n-1)!^2 for n > 0; a(2*n+1) = 4*n!*(n-1)! for n > 0. - Andrew Howroyd, Oct 16 2019
D-finite with recurrence: - (12*n-20)*a(n) + 4*a(n-1) + (3*n-2)*(n-3)*(n-2)*a(n-2) = 0. - Georg Fischer, Nov 25 2022
Sum_{n>=0} 1/a(n) = BesselI(0, 2)/2 + BesselI(1, 2)/4 + 2 = A070910/2 + A096789/4 + 2. - Amiram Eldar, Oct 03 2023

Extensions

Terms a(12) and beyond from Andrew Howroyd, Oct 16 2019

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