A259614 Numbers congruent to {17,29} mod 36.

17, 29, 53, 65, 89, 101, 125, 137, 161, 173, 197, 209, 233, 245, 269, 281, 305, 317, 341, 353, 377, 389, 413, 425, 449, 461, 485, 497, 521, 533, 557, 569, 593, 605, 629, 641, 665, 677, 701, 713, 737, 749, 773, 785, 809, 821, 845, 857, 881, 893, 917, 929, 953

Offset: 1

Author

Bob Selcoe, Jun 30 2015

Keywords

• nonn
• easy

Comments

Subsequence of A087445.

Let terms in this sequence be T:

Collatz sequences (C) that contain no T must terminate at 1.

Define C containing at least one T as C(T), and let T(i) {i=1..z} be T in order of appearance in C(T).

All T(i) i>=2 have odd preimages congruent to either {1,5} mod 12 or {11,19} mod 24. Preimages of the second type (P2) are congruent to B mod 2^m (m>=4), where B is a set of numbers with a predictable recurrence pattern (a bit cumbersome to describe here) starting with A259663(n,2), i.e., {11, 19, 3, 35, 99, 483, ...}. All P2 lead to T(i) == A002450((m-2)/2) mod 2^(m-1) when m is even, and T(i) == A072197((m-3)/2) mod 2^(m-1) when m is odd. So, for example, T(i) == 1 mod 8 when P2 == 11 mod 16; T(i) == 13 mod 16 when P2 == 19 mod 32; T(i) == 5 mod 32 when P2 == 3 mod 64; T(i) == 53 mod 64 when P2 == 35 mod 128; etc.

If the Collatz conjecture is true (i.e., all C terminate at 1), then all C(T) contain T(z) after which all subsequent odd terms decrease and are congruent to {1,5} mod 12 that are not congruent to {17,29} mod 36. The first few T(z) are {17, 53, 341, 1109, 1205, ...}. So, for example, the trajectory of odd terms in C with initial term 950 is [475, 713, 535, 803, 1205, 113, 85, 1], where T(1) = 713 and T(2) = T(z) = 1205. In this example, P2 = 803 because 803 == 11 mod 24.

Links

• G. C. Greubel, Table of n, a(n) for n = 1..2000
• Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Crossrefs

Cf. A002450, A072197, A087445, A259663.

Programs

Magma

[ n : n in [1..1000] | n mod 36 in [17, 29] ] // Vincenzo Librandi Jul 01 2015

Mathematica

Select[Range[1000], MemberQ[{17, 29}, Mod[#, 36]] &] (* Vincenzo Librandi, Jul 01 2015 *)

Formula

G.f.: x*(17+12*x+7*x^2) / ( (1+x)*(x-1)^2 ). - R. J. Mathar, Jul 26 2015

E.g.f.: 7 + (18*x - 4)*exp(x) - 3*exp(-x). - David Lovler, Sep 10 2022