A255411 - OEIS

%I #47 Apr 27 2020 02:33:05

%S 0,4,12,16,18,22,48,52,60,64,66,70,72,76,84,88,90,94,96,100,108,112,

%T 114,118,240,244,252,256,258,262,288,292,300,304,306,310,312,316,324,

%U 328,330,334,336,340,348,352,354,358,360,364,372,376,378,382,408,412,420,424,426,430,432,436,444

%N Shift factorial base representation of n one digit left (with 0 added to right), increment all nonzero digits by one, then convert back to decimal; Numbers with no digit 1 in their factorial base representation.

%C Nonnegative integers such that the number of ones (A257511) in their factorial base representation (A007623) is zero.

%C Nonnegative integers such that the least missing nonzero digit (A257079) in their factorial base representation is one.

%C a(n) can be also directly computed from n by "shifting left" its factorial base representation (that is, by appending one zero to the right, see A153880) and then incrementing all nonzero digits by one, and then converting the resulting (still valid) factorial base number back to decimal. See the examples.

%C The sequences A227130 and A227132 are closed under a(n), in other words, permutation listed as the a(n)-th entry in tables A060117 & A060118 has the same parity as the n-th entry in those same tables.

%H Antti Karttunen, <a href="/A255411/b255411.txt">Table of n, a(n) for n = 0..40319</a>

%e Factorial base representation (A007623) of 1 is "1", shifting it left yields "10", and when we increment all nonzero digits by one, we get "20", which is the factorial base representation of 4 (as 4 = 2*2! + 0*1!), thus a(1) = 4.

%e F.b.r. of 2 is "10", shifting it left yields "100", and "200" is f.b.r. of 12, thus a(2) = 12.

%e F.b.r. of 43 is "1301", shifting it left and incrementing all nonzeros by one yields "24020", which is f.b.r of 340, thus a(43) = 340.

%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, 500}]; {0}~Join~Flatten@ Position[s, x_ /; DigitCount[x][[1]] == 0](* _Michael De Vlieger_, Apr 27 2015, after _Alonso del Arte_ at A007623 *)

%t Select[Range[0, 444], ! MemberQ[IntegerDigits[#, MixedRadix[Reverse@ Range@ 12]], 1] &] (* _Michael De Vlieger_, May 30 2016, Version 10.2 *)

%t r = MixedRadix[Reverse@Range[2, 12]]; Table[FromDigits[Map[If[# == 0, 0, # + 1] &, IntegerDigits[n, r]]~Join~{0}, r], {n, 0, 60}] (* _Michael De Vlieger_, Aug 14 2016, Version 10.2 *)

%o (Scheme, with _Antti Karttunen_'s IntSeq-library)

%o (define (A255411 n) (let loop ((n n) (z 0) (i 2) (f 2)) (cond ((zero? n) z) (else (let ((d (remainder n i))) (loop (quotient n i) (+ z (* f (+ d (if (zero? d) 0 1)))) (+ 1 i) (* f (+ i 1))))))))

%o (define A255411 (MATCHING-POS 0 0 isA255411?)) ;; Slow one.

%o (define (isA255411? n) (let loop ((n n) (i 2)) (cond ((zero? n) #t) ((= 1 (modulo n i)) #f) (else (loop (floor->exact (/ n i)) (+ 1 i))))))

%o ;; Alternative definitions:

%o (define A255411 (ZERO-POS 0 0 A257511))

%o (define A255411 (ZERO-POS 0 0 (COMPOSE -1+ A257079)))

%o (define A255411 (FIXED-POINTS 0 0 A257080))

%o (define A255411 (ZERO-POS 0 0 A257081))

%o (Python)

%o from sympy import factorial as f

%o def a007623(n, p=2): return n if n<p else a007623((n//p), p+1)*10 + n%p

%o def a(n):

%o     x=(str(a007623(n)) + '0')

%o     y="".join(str(int(i) + 1) if int(i)>0 else '0' for i in x)[::-1]

%o     return 0 if n==0 else sum(int(y[i])*f(i + 1) for i in range(len(y)))

%o print([a(n) for n in range(101)]) # _Indranil Ghosh_, Jun 20 2017

%Y Complement: A256450.

%Y Positions of ones in A257079, fixed points of A257080, positions of zeros in A257511, A257081 and A257261.

%Y Cf. A007623, A227157, A153880, A231720.

%Y Cf. A255341, A255342, A255343.

%Y Cf. A257262, A257263.

%Y Cf. also A227130/A227132, A060117/A060118 and also arrays A257503 & A257505.

%K nonn,base

%O 0,2

%A _Antti Karttunen_, Apr 16 2015