A257511 Number of 1's in factorial base representation of n (A007623).

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, 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, 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

Offset: 0

Author

Antti Karttunen, Apr 27 2015

Keywords

• nonn
• base

Links

• Antti Karttunen, Table of n, a(n) for n = 0..10080
• Index entries for sequences related to factorial base representation

Crossrefs

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

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).

Positions of records: A007489.

Cf. also A257510.

Programs

Mathematica

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}];
First@ DigitCount[#] & /@ s (* Michael De Vlieger, Apr 27 2015, after Alonso del Arte at A007623 *)
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 *)

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)))))))

Formula

a(0) = 0; for n >= 1, a(n) = A265333(n) + a(A257687(n)). - Antti Karttunen, Aug 29 2016

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

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

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

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

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

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

a(n) <= A060130(n) <= A034968(n).