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A059590 Numbers obtained by reinterpreting base-2 representation of n in the factorial base: a(n) = Sum_k>=0 {A030308(n,k)*A000142(k+1)}. 34
0, 1, 2, 3, 6, 7, 8, 9, 24, 25, 26, 27, 30, 31, 32, 33, 120, 121, 122, 123, 126, 127, 128, 129, 144, 145, 146, 147, 150, 151, 152, 153, 720, 721, 722, 723, 726, 727, 728, 729, 744, 745, 746, 747, 750, 751, 752, 753, 840, 841, 842, 843, 846, 847, 848, 849, 864, 865 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Numbers that are sums of distinct factorials (0! and 1! not treated as distinct).

Complement of A115945; A115944(a(n)) > 0; A115647 is a subsequence. - Reinhard Zumkeller, Feb 02 2006

A115944(a(n)) = 1. [Reinhard Zumkeller, Dec 04 2011]

From Tilman Piesk, Jun 04 2012: (Start)

The inversion vector (compare A007623) of finite permutation a(n) (compare A055089, A195663) has only zeros and ones. Interpreted as a binary number it is 2*n (or n when the inversion vector is defined without the leading 0).

The inversion set of finite permutation a(n) interpreted as a binary number (compare A211362) is A211364(n).

(End)

LINKS

Reinhard Zumkeller (terms 0..500) & Antti Karttunen, Table of n, a(n) for n = 0..8191

Index entries for sequences related to factorial base representation

Index entries for sequences related to factorial numbers

FORMULA

G.f. 1/(1-x) * sum(k>=0, (k+1)!x^2^k/(1+x^2^k)). - Ralf Stephan, Jun 24 2003

a(n) = Sum_k>=0 {A030308(n,k)*A000142(k+1)}. - Philippe Deléham, Oct 15 2011.

From Antti Karttunen, Aug 19 2016: (Start)

a(0) = 0, a(2n) = A153880(a(n)), a(2n+1) = 1+A153880(a(n)).

a(n) = A225901(A276091(n)).

a(n) = A276075(A019565(n)).

a(A275727(n)) = A276008(n).

A275736(a(n)) = n.

A276076(a(n)) = A019565(n).

A007623(a(n)) = A007088(n).

(End)

a(n) = a(n - mbs(n)) + (1 + floor(log(n) / log(2)))!. - David A. Corneth, Aug 21 2016

EXAMPLE

128 is in the sequence since 5!+3!+2!=128

a(22) = 128. a(22) = a(6) + (1 + floor(log(16) / log(2)))! = 8 + 5! = 128. Also, 22 in binary is 10110. Therefore, a(22) = 1 * 5! + 0 * 4! + 1 * 3! + 1 + 2! + 0 * 0! = 128. - David A. Corneth, Aug 21 2016

MAPLE

[seq(bin2facbase(j), j=0..64)]; bin2facbase := proc(n) local i; add((floor(n/(2^i)) mod 2)*((i+1)!), i=0..floor_log_2(n)); end;

floor_log_2 := proc(n) local nn, i; nn := n; for i from -1 to n do if(0 = nn) then RETURN(i); fi; nn := floor(nn/2); od; end;

MATHEMATICA

a[n_] :=  Reverse[id = IntegerDigits[n, 2]].Range[Length[id]]!; Table[a[n], {n, 0, 60}] (* Jean-François Alcover, Jun 19 2012, after Philippe Deléham *)

PROG

(Haskell)

import Data.List (elemIndices)

a059590 n = a059590_list !! n

a059590_list = elemIndices 1 $ map a115944 [0..]

-- Reinhard Zumkeller, Dec 04 2011

(PARI) a(n) = if(n>0, a(n-msb(n)) + (1+logint(n, 2))!, 0)

msb(n) = 2^#binary(n)>>1

{my(b = binary(n)); sum(i=1, #b, b[i]*(#b+1-i)!)} \\ David A. Corneth, Aug 21 2016

CROSSREFS

Cf. A007088, A007623, A014597, A051760, A051761, A059589.

Indices of zeros in A257684.

Cf. A275736 (left inverse).

Cf. A025494, A060112 (subsequences).

Cf. A153880, A225901.

Subsequence of A060132, A256450 and A275804.

Other sequences that are built by replacing 2^k in the binary representation with other numbers: A029931 (naturals), A089625 (primes), A022290 (Fibonacci), A197433 (Catalans), A276091 (n*n!), A275959 ((2n)!/2). Cf. also A276082 & A276083.

Sequence in context: A255527 A060132 A257262 * A276156 A144705 A028733

Adjacent sequences:  A059587 A059588 A059589 * A059591 A059592 A059593

KEYWORD

nonn

AUTHOR

Henry Bottomley, Jan 24 2001

EXTENSIONS

Name changed (to emphasize the functional nature of the sequence) with the old definition moved to the comments by Antti Karttunen, Aug 21 2016

STATUS

approved

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Last modified December 4 23:10 EST 2016. Contains 278755 sequences.