A072001 Variant of the factorial base representation of n.

1, 11, 101, 111, 201, 211, 1001, 1011, 1101, 1111, 1201, 1211, 2001, 2011, 2101, 2111, 2201, 2211, 3001, 3011, 3101, 3111, 3201, 3211, 10001, 10011, 10101, 10111, 10201, 10211, 11001, 11011, 11101, 11111, 11201, 11211, 12001, 12011, 12101, 12111

Offset

1,2

Comments

For n > 0, write n as sum(k = 0 to m, d_k*k!), where m is such that m! < n <= (m+1)!. Conditions d_0 = 1, 0 <= d_k <= k for k > 0 ensure that the representation is unique. a(n) is the concatenation of (the digits) d_m, ..., d_1, d_0.

a(n) is obtained by appending "1" to A007623(n-1), the standard factorial base representation of n-1.

References

J. S. Madachy & J. A. H. Hunter, Mathematical Diversions, pp. 73-5 VNR Co. NY

Links

Table of n, a(n) for n=1..40.

K. S. Brown, The Factorial Number System

Example

Determining a(2238): 720 = 6! < 2238 <= 7! = 5040; 2238 = 3*6! + 78; 78 = 3*4! + 6; 6 = 2*2! + 2 (to take 6 = 1*3! is not allowed since then condition d_0 = 1 cannot be met); 2 = 1*1! + 1*0!, so 2238 = 3*6! + 0*5! + 3*4! + 0*3! + 2*2! +
1*1! + 1*0! and a(2238) = 3030211.

Prog

(PARI) {for(n=1, 40, k=n-1; j=1; p=1; w=[]; while(p<=k, w=concat(p, w); j++; p=p*j); v="0"; for(i=1, length(w), d=divrem(k, w[i]); v=concat(v, d[1]); k=d[2]); print1(eval(concat(v, 1)), ", "))}

Crossrefs

Cf. A007623.

Sequence in context: A364326 A208259 A043036 * A265510 A265528 A099821

Adjacent sequences: A071998 A071999 A072000 * A072002 A072003 A072004

Keyword

nonn,base

Author

Lekraj Beedassy, Jun 18 2002

Extensions

Edited by Klaus Brockhaus, Jun 10 2003, Jun 14 2003

Status

approved