A007961 -id:A007961 - cp's OEIS frontend

A350674 Irregular table read by rows; the n-th row contains, in weakly decreasing order, the positive squares summing to n as obtained by the greedy algorithm.

1, 1, 1, 1, 1, 1, 4, 4, 1, 4, 1, 1, 4, 1, 1, 1, 4, 4, 9, 9, 1, 9, 1, 1, 9, 1, 1, 1, 9, 4, 9, 4, 1, 9, 4, 1, 1, 16, 16, 1, 16, 1, 1, 16, 1, 1, 1, 16, 4, 16, 4, 1, 16, 4, 1, 1, 16, 4, 1, 1, 1, 16, 4, 4, 25, 25, 1, 25, 1, 1, 25, 1, 1, 1, 25, 4, 25, 4, 1, 25, 4, 1, 1

Offset: 1

Rémy Sigrist, Jan 10 2022

The n-th row has A053610(n) terms.

			The first rows are:
     1:    [1]
     2:    [1, 1]
     3:    [1, 1, 1]
     4:    [4]
     5:    [4, 1]
     6:    [4, 1, 1]
     7:    [4, 1, 1, 1]
     8:    [4, 4]
     9:    [9]
    10:    [9, 1]
    11:    [9, 1, 1]
    12:    [9, 1, 1, 1]
    13:    [9, 4]
    14:    [9, 4, 1]
    15:    [9, 4, 1, 1]
    16:    [16]

Andrew Howroyd, Table of n, a(n) for n = 1..1733 (rows 1..500)

Cf. A007961, A048760, A053610, A350698.

row(n,e=2) = { my (g=[], r); while (n, r=sqrtnint(n,e); n-=r^e; g=concat(g,[r^e])); g }

T(n, 1) = A048760(n).

T(n, A053610(n)) = A350698(n).

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

Original entry on oeis.org

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

Henry Bottomley, Jun 07 2000

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

			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.

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

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

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

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

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

cp's OEIS Frontend

A350674 Irregular table read by rows; the n-th row contains, in weakly decreasing order, the positive squares summing to n as obtained by the greedy algorithm.

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