A221905 -id:A221905 - cp's OEIS frontend

A370658 Numbers k such that (3^k + 3*k)/3 is prime.

Original entry on oeis.org

1, 2, 4, 16, 46, 118, 160, 3740, 3778, 9704, 10624, 21460

Offset: 1

Hugo Pfoertner, Feb 27 2024

If it exists, a(13) > 75000.

Cf. A104743, A221905.

A381705 Length of iteration sequence of shortest unimodal Collatz (3x+1)/2 sequence that begins with exactly n increases and ends with continuous decreases until reaching 1.

Original entry on oeis.org

3, 6, 13, 32, 87, 250, 737, 2196, 6571, 19694, 59061, 177160, 531455, 1594338, 4782985, 14348924, 43046739, 129140182, 387420509, 1162261488, 3486784423, 10460353226, 31381059633, 94143178852, 282429536507, 847288609470, 2541865828357, 7625597485016, 22876792454991

Offset: 1

David Dewan, Mar 04 2025

A unimodal Collatz sequence has one peak because it starts with only odd numbers (which increase) followed by only even numbers (which decrease). It uses the rule odd x -> (3x+1)/2.
A sequence of length a(n) starts with exactly n odd numbers and ends with 3^(n-1) even numbers and the final 1 for a total length of n + 3^(n-1) + 1.
The peak of a given sequence is 2^(3^(n-1)). See A023365.

			For n=2, the shortest unimodal sequence has length a(2) = 6 terms and one such sequence is
  3 -> 5 ->  8  -> 4 -> 2 -> 1
    \-----/     \----------/
  2 increases, then decreases

David Dewan, Unimodal Collatz Sequences
Index entries for linear recurrences with constant coefficients, signature (5,-7,3).

Cf. A014682, A100702, A287319, A221905.

LinearRecurrence[{5,-7,3},{3,6,13},29] (* James C. McMahon, Apr 02 2025 *)

a(n) = n + 3^(n-1) + 1.
From Stefano Spezia, Mar 07 2025: (Start)
G.f.: x*(3 - 9*x + 4*x^2)/((1 - x)^2*(1 - 3*x)).
E.g.f.: (exp(3*x) + 3*exp(x)*(1 + x) - 4)/3. (End)

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A370658 Numbers k such that (3^k + 3*k)/3 is prime.

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A381705 Length of iteration sequence of shortest unimodal Collatz (3x+1)/2 sequence that begins with exactly n increases and ends with continuous decreases until reaching 1.

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