A277022 Primorial base representation of n is rewritten as a base-2 number with each nonzero digit k replaced by a run of k 1's (followed by one extra zero if not the rightmost run of 1's) and with each 0 kept as 0.

0, 1, 2, 5, 6, 13, 4, 9, 10, 21, 22, 45, 12, 25, 26, 53, 54, 109, 28, 57, 58, 117, 118, 237, 60, 121, 122, 245, 246, 493, 8, 17, 18, 37, 38, 77, 20, 41, 42, 85, 86, 173, 44, 89, 90, 181, 182, 365, 92, 185, 186, 373, 374, 749, 188, 377, 378, 757, 758, 1517, 24, 49, 50, 101, 102, 205, 52, 105, 106, 213, 214, 429, 108, 217, 218, 437, 438, 877, 220

Offset

0,3

Links

Antti Karttunen, Table of n, a(n) for n = 0..30030

Index entries for sequences related to binary expansion of n

Index entries for sequences related to primorial base

Formula

a(0) = 0; for n >= 1, a(n) = A000225(A276088(n))*A000079(A276084(n)) + A000079(A276088(n))*a(A276093(n)).

a(n) = A156552(A276086(n)).

Other identities. For all n >= 0:

A277021(a(n)) = n.

A005940(1+a(n)) = A276086(n).

A000035(a(n)) = A000035(n). [Preserves the parity of n.]

A000120(a(n)) = A276150(n).

A069010(a(n)) = A267263(n).

Example

9 = "111" in primorial base (A002110(0) + A002110(1) + A002110(2) = 9) is converted to three 1-bits, with separating zeros, in binary as "10101" = A007088(21), thus a(9) = 21.
91 = "3001" in primorial base (91 = 3*A002110(3) + A002110(0)) is converted to binary number "1110001" = A007088(113), thus a(91) = 113. Note how two of the zeros come from the primorial base representation and the third zero is an extra separating zero inserted after each run of 1-bits apart from the rightmost 1-run.
120 = "4000" in primorial base (120 = 4*A002110(3)) is converted to the binary number "1111000" = A007088(120), thus a(120) = 120.

Prog

(Scheme, two different implementations)
(define (A277022 n) (let loop ((n n) (z 0) (i 1) (j 0)) (if (zero? n) z (let* ((p (A000040 i)) (d (remainder n p))) (loop (quotient n p) (+ z (* (A000225 d) (A000079 j))) (+ 1 i) (+ 1 j d))))))
;; Another one based on given recurrence, utilizing memoization-macro definec:
(definec (A277022 n) (if (zero? n) n (+ (* (A000225 (A276088 n)) (A000079 (A276084 n))) (* (A000079 (A276088 n)) (A277022 (A276093 n))))))

Crossrefs

Cf. A000035, A000079, A000120, A000225, A005940, A007088, A049345, A156552, A267263, A276084, A276086, A276088, A276093, A276150.

Cf. A277018 (terms sorted into ascending order).

Cf. A277021 (a left inverse).

Differs from analogous A277012 for the first time at n=24, where a(24) = 60, while A277012(24) = 8.

Sequence in context: A243798 A057683 A277012 * A232603 A069480 A354025

Adjacent sequences: A277019 A277020 A277021 * A277023 A277024 A277025

Keyword

nonn,base

Author

Antti Karttunen, Sep 26 2016

Status

approved