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%I #29 Sep 01 2016 10:54:15
%S 0,1,2,3,10,11,12,13,20,21,22,23,30,31,32,33,40,41,100,101,102,103,
%T 110,111,112,113,120,121,122,123,130,131,132,133,140,141,200,201,202,
%U 203,210,211,212,213,220,221,222,223,230,231,232,233,240,241,300,301,302,303,310,311,312,313,320,321,322,323,330,331,332,333,340,341,400
%N Numbers expressed in greedy A001563-base.
%C Terms A001563(1) = 1, A001563(2) = 4, A001563(3) = 18, ... give the base values for the digit positions from 1 onward. Digit places are filled by always trying to find the largest possible term of A001563 that still fits into the sum.
%C A130744(8) = 3225600 = 10*A001563(8) is the first number which yields an ambiguous representation when expressed in decimal, because in this base it is actually "A0000000" (where digit "A" stands for ten).
%H Antti Karttunen, <a href="/A276326/b276326.txt">Table of n, a(n) for n = 0..4320</a>
%e To recover n from a(n) the digits in positions i = 1, 2, 3, ... (starting indexing from the least significant digit at right) are multiplied by A001563(i) and added together:
%e ----------------
%e n a(n)
%e ----------------
%e 0 0
%e 1 1
%e 2 2
%e 3 3
%e 4 10
%e 5 11
%e 6 12
%e 7 13
%e 8 20
%e 9 21
%e 10 22
%e 11 23
%e 12 30
%e 13 31
%e 14 32
%e 15 33
%e 16 40
%e 17 41 (as 4*A001563(2) + 1*A001563(1) = 17)
%e 18 100 (as 1*A001563(3) + 0*A001563(2) + 0*A001563(1) = 18)
%e and:
%e 3225599 99111111 (as 3225599 = 9*b(8) + 9*b(7) + b(6) + b(5) + b(4) + b(3) + b(2) + b(1)), where b(n) = A001563(n).
%t f[n_] := Block[{a = {{0, n}}}, Do[AppendTo[a, {First@ #, Last@ #} &@ QuotientRemainder[a[[-1, -1]], (# #!) &[# - i]]], {i, 0, # - 1}] &@ NestWhile[# + 1 &, 0, (# #!) &[# + 1] <= n &]; Rest[a][[All, 1]]]; Table[FromDigits@ f@ n, {n, 72}] (* _Michael De Vlieger_, Aug 31 2016 *)
%o (Scheme)
%o (define (A276326 n) (let loop ((n n) (s 0)) (if (zero? n) s (let ((dig (A276333 n))) (if (> dig 9) (error "A276326: ambiguous representation of n, digit > 9 would be needed: " n dig) (loop (A276335 n) (+ s (* dig (expt 10 (- (A258198 n) 1))))))))))
%Y Cf. A001563, A130744, A258198, A276335.
%Y Cf. A276327 (the least significant nonzero digit).
%Y Cf. A276328 (the sum of digits).
%Y Cf. A276333 (the most significant digit).
%Y Cf. A276336 (a largest digit).
%Y Cf. A276337 (number of nonzero digits).
%Y Cf. A033312 (repunits).
%Y Cf. A276091 (no digits larger than one).
%Y Differs from A007090 for the first time at n=16 and from A055655 at n=18.
%Y Cf. also A007623, A007961, A000433, A014418, A265747.
%K nonn,base
%O 0,3
%A _Antti Karttunen_, Aug 30 2016
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