

A055655


Efficient representation of n in "square base" where xyz means 9x+4y+z and z<4, y<9 and x<16 etc.


1



0, 1, 2, 3, 10, 11, 12, 13, 20, 21, 22, 23, 30, 31, 32, 33, 40, 41, 42, 43, 50, 51, 52, 53, 60, 61, 62, 63, 70, 71, 72, 73, 80, 81, 82, 83, 163, 170, 171, 172, 173, 180, 181, 182, 183, 263, 270, 271, 272, 273, 280, 281, 282, 283, 363, 370, 371, 372, 373, 380, 381
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OFFSET

0,3


COMMENTS

Efficient means the smallest possible a(n), cf. example. From n = 9*9+8*4+3 = 116 on, the terms (coded in base 10) become ambiguous because digits may be larger than 9, e.g., 1000 could mean 1*16 or 10*9. One possible convention to avoid ambiguity would be to reserve as many digits as might be required for the largest possible coefficient: 2 digits for the coefficients of 9 (which may reach 161 = 15) through 81; 3 digits for the coefficients of 100 through 30^2, 4 digits for the coefficients of 31^2 (which may reach 32^21 = 1023) etc.  M. F. Hasler, Jul 25 2015


REFERENCES

F. Smarandache, Definitions solved and unsolved problems, conjectures and theorems in number theory and geometry, edited by M. Perez, Xiquan Publishing House, 2000.


LINKS

Table of n, a(n) for n=0..60.
F. Smarandache, Definitions, Solved and Unsolved Problems, Conjectures, ...


EXAMPLE

a(50)=280 since 2*9+8*4+0*1=50; writing 20000 for 2*25 or 3xyz (for 3*16+x*9+y*4+z) or 5yz or 4yz or 3yz would be less efficient (larger "result" when read in base 10), and it is not possible to write 50 as 1*9+y*4+z*1 with y<9 and z<4.


PROG

(PARI) a(n, s=0)={v=[3]; until(v[#v]>=n, v=concat(v, v[#v]+((2+#v)^21)*(1+#v)^2)); for(i=1, #v1, s=s*10+t=max(ceil((nv[#vi])/(#vi+1)^2), 0); n=t*(#vi+1)^2); s*10+n} \\ M. F. Hasler, Jul 25 2015


CROSSREFS

Cf. A007961 for greedy representation of n in "square base".
Sequence in context: A225558 A301382 A288657 * A276326 A007090 A102859
Adjacent sequences: A055652 A055653 A055654 * A055656 A055657 A055658


KEYWORD

base,nonn


AUTHOR

Henry Bottomley, Jun 07 2000


EXTENSIONS

Corrected and edited by M. F. Hasler, Jul 25 2015


STATUS

approved



