A249069 - OEIS

%I #12 Oct 25 2014 14:46:56

%S 0,1,1,2,3,5,1,7,9,12,2,13,3,16,5,6,18,1,19,2,21,3,25,27,30,32,35,38,

%T 40,41,43,45,48,13,14,15,16,18,6,53,20,7,57,21,8,64,24,65,27,69,28,72,

%U 10,73,11,76,12,33,80,13,34,85,14,37,89,15,41,94,17,46,96,1,97,2,99,3,103,4,48,49,50

%N a(n+1) gives the number of occurrences of the first digit of a(n) in factorial base (i.e., A099563(a(n))) so far amongst the factorial base representations of all the terms up to and including a(n), with a(0)=0.

%H Antti Karttunen, <a href="/A249069/b249069.txt">Table of n, a(n) for n = 0..10080</a>

%e    a(0) =  0 (by definition)

%e    a(1) =  1 ('1' in factorial base), as 0 has occurred once in all the preceding terms.

%e    a(2) =  1 as 1 has occurred once in all the preceding terms.

%e    a(3) =  2 ('10' in factorial base), as digit '1' has occurred two times in total in all the preceding terms.

%e    a(4) =  3 ('11' in factorial base), as '1' occurs once in each a(1) and a(2) and a(3).

%e    a(5) =  5 ('21' in factorial base), as '1' occurs once in each of a(1), a(2) and a(3) and twice at a(4).

%e    a(6) =  1 as '2' so far occurs only once at a(5)

%e    a(7) =  7 = '101'

%e    a(8) =  9 = '111'

%e    a(9) = 12 = '200'

%e   a(10) =  2 = '2'

%e   a(11) = 13 = '201'

%e   a(12) =  3 = '11'

%e   a(12) =  3 = '11'

%e   a(13) = 16 = '220'

%e   a(14) =  5 = '21'

%e   a(15) =  6 = '100'

%e   a(16) = 18 = '300'

%e   a(17) =  1 = '1'

%e   a(18) = 19 = '301'

%e   a(19) =  2 = '10'

%e   a(20) = 21 = '311'

%e   a(21) =  3 = '11'

%e   a(22) = 25 = '1001'

%e   a(23) = 27 = '1011'

%e   a(24) = 30 = '1100'

%e   a(25) = 32 = '1110'

%e   a(26) = 35 = '1121'

%e   a(27) = 38 = '1210' as the leftmost digit '1' has occurred 38 times in total in the factorial base expansions of the preceding terms a(0) - a(26).

%e etc.

%o (MIT/GNU Scheme with memoizing definec-macro from _Antti Karttunen_'s IntSeq-library)

%o (definec (A249069 n) (if (zero? n) n (vector-ref (A249069aux_digit_counts (- n 1)) (A099563 (A249069 (- n 1))))))

%o (definec (A249069aux_digit_counts n) (cond ((zero? n) (vector 1)) (else (let* ((start_n (A249069 n)) (copy-of-prevec (vector-copy (A249069aux_digit_counts (- n 1)))) (newsize (max (vector-length copy-of-prevec) (+ 1 (A246359 start_n)))) (digcounts-for-n (vector-grow copy-of-prevec newsize))) (let loop ((n start_n) (i 2)) (cond ((zero? n) digcounts-for-n) (else (vector-set! digcounts-for-n (modulo n i) (+ 1 (or (vector-ref digcounts-for-n (modulo n i)) 0))) (loop (floor->exact (/ n i)) (+ i 1)))))))))

%Y Cf. A249009 (analogous sequence in base-10).

%Y Differs from a variant A249070 for the first time at n=27, where a(27) = 38, while A249070(27) = 7.

%Y Cf. also A007623, A099563, A246359.

%K nonn,base

%O 0,4

%A _Antti Karttunen_, Oct 20 2014