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Number of 1's in factorial base representation of n (A007623).
17

%I #21 Aug 29 2016 11:21:17

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

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

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

%N Number of 1's in factorial base representation of n (A007623).

%H Antti Karttunen, <a href="/A257511/b257511.txt">Table of n, a(n) for n = 0..10080</a>

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

%F a(0) = 0; for n >= 1, a(n) = A265333(n) + a(A257687(n)). - _Antti Karttunen_, Aug 29 2016

%F Other identities and observations. For all n >= 0:

%F a(n) = A260736(A225901(n)).

%F a(n) = A001221(A275732(n)) = A001222(A275732(n)).

%F a(n) = A007814(A275735(n)).

%F a(n) = A056169(A276076(n)).

%F a(A007489(n)) = n. [Particularly, A007489(n) gives the position where n first appears.]

%F a(n) <= A060130(n) <= A034968(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, 0, 120}];

%t First@ DigitCount[#] & /@ s (* _Michael De Vlieger_, Apr 27 2015, after _Alonso del Arte_ at A007623 *)

%t nn = 120; b = Module[{m = 1}, While[Factorial@ m < nn, m++]; MixedRadix[Reverse@ Range[2, m]]]; Table[Count[IntegerDigits[n, b], 1], {n, 0, nn}] (* _Michael De Vlieger_, Aug 29 2016, Version 10.2 *)

%o (Scheme) (define (A257511 n) (let loop ((n n) (i 2) (s 0)) (cond ((zero? n) s) (else (loop (floor->exact (/ n i)) (+ 1 i) (+ s (if (= 1 (modulo n i)) 1 0)))))))

%Y Cf. A001221, A007623, A007814, A034968, A056169, A060130, A225901, A257687, A265333, A275732, A275735, A260736, A276076.

%Y Cf. A255411 (numbers n such that a(n) = 0), A255341 (such that a(n) = 1), A255342 (such that a(n) = 2), A255343 (such that a(n) = 3).

%Y Positions of records: A007489.

%Y Cf. also A257510.

%K nonn,base

%O 0,4

%A _Antti Karttunen_, Apr 27 2015