

A220655


For n with a unique factorial base representation n = du*u! + ... + d2*2! + d1*1! (each di in range 0..i, cf. A007623), a(n) = (du+1)*u! + ... + (d2+1)*2! + (d1+1)*1!; a(n) = n + A007489(A084558(n)).


5



2, 5, 6, 7, 8, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99
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OFFSET

1,1


COMMENTS

Term a(n) can be obtained by adding one to each digit of factorial base representation of n (A007623(n)) and then reinterpreting it as a kind of pseudofactorial base representation, ignoring the fact that now some of the digits might be over the maximum allowed in that position. Please see the example section.  Antti Karttunen, Nov 29 2013


LINKS



FORMULA

a(n) = n + A007489(A084558(n)). [The above formula reduces to this, which proves that the original Dec 17 2012 description and the new main description produce the same sequence. Essentially, we are adding to n a factorial base repunit '1...111' with as many fact.base digits as n has.]  Antti Karttunen, Nov 29 2013


EXAMPLE

1 has a factorial base representation A007623(1) = '1', as 1 = 1*1!. Incrementing the digit 1 with 1, we get 2*1! = 2, thus a(1) = 2. (Note that although '2' is not a valid factorial base representation, it doesn't matter here.)
2 has a factorial base representation '10', as 2 = 1*2! + 0*1!. Incrementing the digits by one, we get 2*2! + 1*1! = 5, thus a(2) = 5.
3 has a factorial base representation '11', as 3 = 1*2! + 1*1!. Incrementing the digits by one, we get 2*2! + 2*1! = 6, thus a(3) = 6.


MATHEMATICA

Block[{nn = 66, m = 1}, While[Factorial@ m < nn, m++]; m = MixedRadix[Reverse@ Range[2, m]]; Array[FromDigits[1 + IntegerDigits[#, m], m] &, nn]] (* Michael De Vlieger, Jan 20 2020 *)


PROG

(Scheme)
;; Standalone iterative implementation (Nov 29 2013):
(define (A220655 n) (let loop ((n n) (z 0) (i 2) (f 1)) (cond ((zero? n) z) (else (loop (quotient n i) (+ (* f (+ 1 (remainder n i))) z) (+ 1 i) (* f i))))))
;; Alternative implementation:


CROSSREFS



KEYWORD

nonn


AUTHOR



EXTENSIONS



STATUS

approved



