

A072001


Variant of the factorial base representation of n.


0



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
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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(n1), the standard factorial base representation of n1.


REFERENCES

J. S. Madachy & J. A. H. Hunter, Mathematical Diversions, pp. 735 VNR Co. NY


LINKS



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=n1; 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



KEYWORD

nonn,base


AUTHOR



EXTENSIONS



STATUS

approved



