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)).
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
Offset: 1
Keywords
Examples
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.
Links
- Antti Karttunen, Table of n, a(n) for n = 1..5039
Crossrefs
Programs
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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 *) -
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: (define (A220655 n) (+ n (A007489 (A084558 n))))
Formula
a(n) = A220656(n)-1 = A003422(A084558(n)+1) + A000142(A084558(n)) + A212598(n) - 1. [The original definition]
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
Extensions
Name changed by Antti Karttunen, Nov 29 2013
Comments