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A153880 Shift factorial base representation left by one digit.

%I A153880 #31 Apr 27 2020 02:32:50
%S A153880 0,2,6,8,12,14,24,26,30,32,36,38,48,50,54,56,60,62,72,74,78,80,84,86,
%T A153880 120,122,126,128,132,134,144,146,150,152,156,158,168,170,174,176,180,
%U A153880 182,192,194,198,200,204,206,240,242,246,248,252,254,264,266,270,272
%N A153880 Shift factorial base representation left by one digit.
%C A153880 Equally, append 0 to the end of the factorial base representation of n (= A007623(n)), then convert back to decimal.
%C A153880 Involution A225901 maps each term of this sequence to a unique term of A255411, and vice versa.
%H A153880 Antti Karttunen, <a href="/A153880/b153880.txt">Table of n, a(n) for n = 0..40319</a>
%H A153880 <a href="/index/Fa#facbase">Index entries for sequences related to factorial base representation</a>
%F A153880 Other identities. For all n >= 0:
%F A153880 A266193(a(n)) = n.
%e A153880 Factorial base representation of 5 is A007623(5) = "21". Shifting this once left (that is, appending 0 to the end) yields "210", which is factorial base representation for 14. Thus a(5) = 14.
%t A153880 Table[Function[b, FromDigits[IntegerDigits[n, b]~Join~{0}, b]]@ MixedRadix[Reverse@ Range@ 12], {n, 0, 57}] (* _Michael De Vlieger_, May 30 2016, Version 10.2 *)
%o A153880 (Scheme)
%o A153880 (define (A153880 n) (let loop ((n n) (z 0) (i 2) (f 2)) (cond ((zero? n) z) (else (loop (floor->exact (/ n i)) (+ (* f (modulo n i)) z) (+ 1 i) (* f (+ i 1)))))))
%o A153880 (Python)
%o A153880 from sympy import factorial as f
%o A153880 def a007623(n, p=2): return n if n<p else a007623((n//p), p+1)*10 + n%p
%o A153880 def a(n):
%o A153880     x = (str(a007623(n)) + '0')[::-1]
%o A153880     return 0 if n==0 else sum(int(x[i])*f(i + 1) for i in range(len(x)))
%o A153880 print([a(n) for n in range(101)]) # _Indranil Ghosh_, Jun 20 2017
%Y A153880 Indices of zeros in A260736.
%Y A153880 Cf. A153883 (terms divided by 2).
%Y A153880 Cf. A266193 (a left inverse).
%Y A153880 Cf. A273670 (complement).
%Y A153880 Cf. also A007623, A225901, A255411.
%K A153880 base,nonn
%O A153880 0,2
%A A153880 _Antti Karttunen_, Jan 03 2009