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

Author

Antti Karttunen, Sep 26 2016

Keywords

• nonn
• base

Examples

			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.
		

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

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.

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).