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One-based position of the rightmost one in the factorial base representation (A007623) of n, 0 if no one is present.
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%I #18 Feb 07 2024 01:17:34

%S 0,1,2,1,0,1,3,1,2,1,3,1,0,1,2,1,0,1,0,1,2,1,0,1,4,1,2,1,4,1,3,1,2,1,

%T 3,1,4,1,2,1,4,1,4,1,2,1,4,1,0,1,2,1,0,1,3,1,2,1,3,1,0,1,2,1,0,1,0,1,

%U 2,1,0,1,0,1,2,1,0,1,3,1,2,1,3,1,0,1,2,1,0,1,0,1,2,1,0,1,0,1,2,1,0,1,3,1,2,1,3,1,0,1,2,1,0,1,0,1,2,1,0,1,5

%N One-based position of the rightmost one in the factorial base representation (A007623) of n, 0 if no one is present.

%H Antti Karttunen, <a href="/A257261/b257261.txt">Table of n, a(n) for n = 0..10081</a>

%H <a href="/index/Fa#facbase">Index entries for sequences related to factorial base representation</a>.

%F Other identities:

%F For all n >= 1, a(n!) = n.

%e For n = 0, with factorial base representation (A007623) "0", there are no ones present at all, thus a(0) = 0.

%e For n = 1, with representation "1", the rightmost one occurs at digit-position 1 (when the least significant digit has index 1, etc.), thus a(1) = 1.

%e For n = 6, with representation "100", the rightmost one occurs at position 3, thus a(6) = 3.

%e For n = 11, with representation "121", the rightmost one occurs at digit-position 1 (when the least significant digit has index 1, etc.), thus a(11) = 1.

%t a[n_] := Module[{k = n, m = 2, r, s = {}, p}, While[{k, r} = QuotientRemainder[k, m]; k != 0|| r != 0, AppendTo[s, r]; m++]; If[MissingQ[(p = FirstPosition[s, 1])], 0, p[[1]]]]; Array[a, 100, 0] (* _Amiram Eldar_, Feb 07 2024 *)

%o (Scheme) (define (A257261 n) (let loop ((n n) (i 2)) (cond ((zero? n) 0) ((= 1 (modulo n i)) (- i 1)) (else (loop (floor->exact (/ n i)) (+ 1 i))))))

%Y Cf. A007623, A257260.

%Y Cf. A000142 (positions of records, where each n first occurs as a value), A255411 (positions of zeros), A000012 (odd bisection).

%K nonn,base

%O 0,3

%A _Antti Karttunen_, Apr 29 2015