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Number of nonleading zeros in factorial base representation of n (A007623).
8

%I #18 Apr 30 2015 21:46:26

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

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

%U 0,1,0,3,2,2,1,2,1,2,1,1,0,1,0,2,1,1,0,1,0,2,1,1,0,1,0,3,2,2,1,2,1,2,1,1,0,1,0,2,1,1,0,1,0,2,1,1,0,1,0,4

%N Number of nonleading zeros in factorial base representation of n (A007623).

%C Sequence starts from n=1, because 0 is an ambiguous case.

%H Antti Karttunen, <a href="/A257510/b257510.txt">Table of n, a(n) for n = 1..10080</a>

%F a(n) = A084558(n) - A060130(n).

%F Other identities and observations:

%F For all n >= 0, a(A000142(n+1)) = n. [(n+1)! gives the position where n first appears.]

%F For all n, a(n) >= A230403(n).

%t factBaseIntDs[n_] := Module[{m, i, len, dList, currDigit}, i = 1; While[n > i!, i++]; m = n; len = i; dList = Table[0, {len}]; Do[currDigit = 0; While[m >= j!, m = m - j!; currDigit++]; dList[[len - j + 1]] = currDigit, {j, i, 1, -1}]; If[dList[[1]] == 0, dList = Drop[dList, 1]]; dList]; s = Table[FromDigits[factBaseIntDs[n]], {n, 120}]; Last@ DigitCount[#] & /@ s (* _Michael De Vlieger_, Apr 27 2015, after _Alonso del Arte_ at A007623 *)

%o (Scheme) (define (A257510 n) (let loop ((n n) (i 2) (s 0)) (cond ((zero? n) s) (else (loop (floor->exact (/ n i)) (+ 1 i) (+ s (if (zero? (modulo n i)) 1 0)))))))

%Y Cf. A227157 (numbers n such that a(n) = 0), A227187 (n for which a(n) > 0).

%Y Cf. A007623, A060130, A084558, A230403.

%Y Cf. also A257511.

%Y Cf. also A023416, A080791 (analogous sequences for base-2), A055641 (for base-10).

%K nonn,base

%O 1,6

%A _Antti Karttunen_, Apr 27 2015