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%I #19 Sep 27 2016 06:13:12
%S 0,1,2,5,6,13,4,9,10,21,22,45,12,25,26,53,54,109,28,57,58,117,118,237,
%T 8,17,18,37,38,77,20,41,42,85,86,173,44,89,90,181,182,365,92,185,186,
%U 373,374,749,24,49,50,101,102,205,52,105,106,213,214,429,108,217,218,437,438,877,220,441,442,885,886,1773,56,113,114,229,230,461,116,233
%N Factorial 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.
%H Antti Karttunen, <a href="/A277012/b277012.txt">Table of n, a(n) for n = 0..5040</a>
%H <a href="/index/Bi#binary">Index entries for sequences related to binary expansion of n</a>
%H <a href="/index/Fa#facbase">Index entries for sequences related to factorial base representation</a>
%F a(n) = A156552(A276076(n)).
%F Other identities. For all n >= 0:
%F A277011(a(n)) = n.
%F A005940(1+a(n)) = A276076(n).
%F A000035(a(n)) = A000035(n). [Preserves the parity of n.]
%F A000120(a(n)) = A034968(n).
%F A069010(a(n)) = A060130(n).
%F A227349(a(n)) = A227153(n).
%e 9 = "111" in factorial base (3! + 2! + 1! = 9) is converted to three 1-bits with separating zeros between, in binary as "10101" = A007088(21), thus a(9) = 21.
%e 91 = "3301" in factorial base (91 = 3*4! + 3*3! + 1!) is converted to binary number "1110111001" = A007088(953), thus a(91) = 953. Between the rightmost 1-runs the other zero comes from the factorial base representation, while the other zero is an extra separating zero inserted after each run of 1-bits apart from the rightmost 1-run. The single zero between the two leftmost 1-runs is similarly used to separate the two "unary representations" of 3's.
%o (Scheme)
%o (define (A277012 n) (let loop ((n n) (z 0) (i 2) (j 0)) (if (zero? n) z (let ((d (remainder n i))) (loop (quotient n i) (+ z (* (A000225 d) (A000079 j))) (+ 1 i) (+ 1 j d))))))
%Y Cf. A000035, A000079, A000120, A000225, A007088, A007623, A034968, A060130, A069010, A156552, A227153, A227349.
%Y Cf. A277008 (terms sorted into ascending order).
%Y Cf. A277011 (a left inverse).
%Y Differs from analogous A277022 for the first time at n=24, where a(24) = 8, while A277022(24) = 60.
%K nonn,base
%O 0,3
%A _Antti Karttunen_, Sep 25 2016
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