A293445 A multiplicative encoding (base-2 compressed) for the exponents of 3 obtained when using Shevelev's algorithm for computing A053446.

2, 2, 3, 12, 36, 3, 12, 24, 6, 48, 12, 20736, 82944, 12, 18, 864, 248832, 6, 20, 19906560, 59719680, 80, 8640, 720, 25920, 34560, 5, 80, 103195607040, 240, 480, 622080, 137594142720, 138240, 20, 59440669655040, 138240, 20, 14929920, 29859840, 240, 59719680, 8640, 720, 414720, 8640, 540, 447897600, 960, 46080, 34560, 59719680, 295814814232058265600, 5, 80

Offset

1,1

Links

Antti Karttunen, Table of n, a(n) for n = 1..1458

Formula

A048675(a(n)) = A053446(n).

Example

A001651(5) = 7 as 7 is the fifth number not divisible by 3. According to the algorithm described in the comment of A053446 we have in the form of a "finite continued fraction"
    1 + 14
    ------ + 7
       3^1
    ---------- + 14
          3^1
    ----------------- + 7
            3^2
    ---------------------- = 1
               3^2
Cumulatively multiplying (with A019565) together the prime-numbers corresponding to 1-bits in the binary expansions of the exponents of 3 in the denominators (that are 1, 1, 2, 2, in binary 1, 1, 10, 10, with 1's in bit-positions 0 and 1), yields prime(0+1) * prime(0+1) * prime(1+1) * prime(1+1) = 2^2 * 3^2 = 36, thus a(5) = 36.
(Adapted from Vladimir Shevelev's explanation in A053446.)
Another example: A001651(19) = 28 as 28 is the 19th number not divisible by 3. (1 + 28) is not a multiple of 3, so we start with (1 + 2*28) = 1+56 = 57 and proceed as:
    1 + 56
    ------ + 56                     [that is, (57/3) + 56 = 75]
       3^1
    ---------- + 56                 [that is, (75/3) + 56 = 81]
          3^1
    -----------------  = 1          [that is, (81/81) = 1]
            3^4
So we obtained exponents 1, 1, 4 (in binary "1", "1" and "100") where the 1-bits are in positions 0, 0 and 2. We form a product prime(0+1) * prime(0+1) * prime(2+1) = 2*2*5, thus a(19) = 20.

Prog

(Scheme)
(define (A293445 n) (define (next_one k m) (if (zero? (modulo (+ k m) 3)) (+ k m) (+ k m m))) (let* ((u (A001651 n)) (x_init (next_one 1 u))) (let loop ((x x_init) (z (A019565 (A007949 x_init)))) (let ((r (A038502 x))) (if (= 1 r) z (let ((x_next (next_one r u))) (loop x_next (* z (A019565 (A007949 x_next))))))))))
(define (A001651 n) (let ((x (- n 1))) (if (even? x) (+ 1 (* 3 (/ x 2))) (- (* 3 (/ (+ x 1) 2)) 1))))
(define (A038500 n) (A000244 (A007949 n)))
(define (A038502 n) (/ n (A038500 n)))

Crossrefs

Cf. A001651, A007949, A019565, A038502, A053446, A293220.

Cf. A293446 (restricted growth transform of this sequence).

Cf. also A292265.

Sequence in context: A375218 A335942 A240133 * A126339 A268725 A153929

Adjacent sequences: A293442 A293443 A293444 * A293446 A293447 A293448

Keyword

nonn

Author

Antti Karttunen, Oct 09 2017

Status

approved