A055655 Efficient representation of n in "square base" where xyz means 9x+4y+z and z<4, y<9 and x<16 etc.

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

Offset: 0

Author

Henry Bottomley, Jun 07 2000

Keywords

• base
• nonn

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 16-1 = 15) through 81; 3 digits for the coefficients of 100 through 30^2, 4 digits for the coefficients of 31^2 (which may reach 32^2-1 = 1023) etc. - M. F. Hasler, Jul 25 2015

Examples

			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.
		

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

• F. Smarandache, Definitions, Solved and Unsolved Problems, Conjectures, ...

Crossrefs

Cf. A007961 for greedy representation of n in "square base".

Programs

PARI

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

Extensions

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