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A198586 a(n) = (4^A001651(n+1) - 1)/3: numbers (4^k-1)/3 for k > 1, not multiples of 3. 3
5, 85, 341, 5461, 21845, 349525, 1398101, 22369621, 89478485, 1431655765, 5726623061, 91625968981, 366503875925, 5864062014805, 23456248059221, 375299968947541, 1501199875790165, 24019198012642645, 96076792050570581, 1537228672809129301 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Numbers coprime to 6 producing 2 odd numbers in the Collatz iteration.

Numbers appearing in A198585 (sorted and duplicates removed). These numbers occur in A002450, numbers of the form (4^k-1)/3, for k = 2, 4, 5, 7, 8, 10, ... (note that k a multiple of 3 does not appear).

A124477 \ {0,1} is a subset: for these n, 3n+1 = 2^(p-3) with p > 3 prime, whence also n !== 0 (mod 3). - M. F. Hasler, Oct 16 2018

These are exactly the odd non-multiples of 3 such 3n+1 = 2^m for some m, i.e., n = (2^m-1)/3. This is possible iff m = 2k, so we get n = (4^k-1)/3. Then n == 0 (mod 3) <=> 4^k == 1 (mod 9) <=> k == 0 (mod 3) <=> k not in A001651. This yields the FORMULA. (Multiples of 3 are excluded because the original definition implied that the terms are in the Collatz-orbit of another odd number, i.e., of the form n = (3x+1)/2^r, which is impossible for x a multiple of 3.) - M. F. Hasler, Oct 16 2018

LINKS

T. D. Noe, Table of n, a(n) for n = 1..100

Index entries for linear recurrences with constant coefficients, signature (1,64,-64).

FORMULA

a(n) = (4^A001651(n+1) - 1)/3. - M. F. Hasler, Oct 16 2018

From Colin Barker, Jan 17 2020: (Start)

G.f.: x*(5 + 80*x - 64*x^2) / ((1 - x)*(1 - 8*x)*(1 + 8*x)).

a(n) = a(n-1) + 64*a(n-2) - 64*a(n-3) for n>3.

a(n) = (-1 + (-8)^n + 3*8^n) / 3.

(End)

MATHEMATICA

e = 19; ex = Complement[Range[2, 3*e], 3*Range[e]]; (4^ex - 1)/3

(* Second program: *)

Rest@ Map[(4^# - 1)/3 &, LinearRecurrence[{1, 1, -1}, {1, 2, 4}, 21]] (* Michael De Vlieger, Oct 17 2018 *)

PROG

(PARI) is(n)=gcd(n, 6)==1&&(n=3*n+1)>>valuation(n, 2)==1 \\ M. F. Hasler, Oct 16 2018

(PARI) A198586(n)=4^(3*n\2+1)\3 \\ M. F. Hasler, Oct 16 2018

(PARI) Vec(x*(5 + 80*x - 64*x^2) / ((1 - x)*(1 - 8*x)*(1 + 8*x)) + O(x^20)) \\ Colin Barker, Jan 17 2020

(MAGMA) [4^(3*n  div 2 + 1) div 3: n in [1..25]]; // Vincenzo Librandi, Oct 20 2018

CROSSREFS

Cf. A001651, A002450, A124477, A198584.

Sequence in context: A048143 A216420 A137083 * A121290 A201797 A012743

Adjacent sequences:  A198583 A198584 A198585 * A198587 A198588 A198589

KEYWORD

nonn,easy

AUTHOR

T. D. Noe, Oct 30 2011

EXTENSIONS

Definition corrected by M. F. Hasler, Oct 16 2018

STATUS

approved

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Last modified June 20 06:56 EDT 2021. Contains 345157 sequences. (Running on oeis4.)