A321266 Smallest positive number for which the square cannot be written as sum of distinct squares of any subset of previous terms.

1, 2, 3, 4, 6, 8, 12, 16, 17, 24, 32, 34, 48, 64, 68, 96, 128, 136, 192, 256, 272, 384, 512, 544, 768, 1024, 1088, 1536, 2048, 2176, 3072, 4096, 4352, 6144, 8192, 8704, 12288, 16384, 17408, 24576, 32768, 34816, 49152, 65536, 69632, 98304, 131072, 139264

Offset

1,2

Comments

a(n)^2 = A226076(n) forms a sum-free sequence.

Links

Bert Dobbelaere, Table of n, a(n) for n = 1..89

Wikipedia, Sum-free sequence

Formula

a(n) = 2 * a(n-3) for n > 9 (conjectured).

Example

0^2 = 0 (sum of squares of the empty set).
1^2 cannot be written as sum of squares of the empty set, so a(1)=1.
Suppose we determined all terms up to a(7)=12:
13^2 = 12^2 + 4^2 + 3^2,
14^2 = 12^2 + 6^2 + 4^2,
15^2 = 12^2 + 8^2 + 4^2 + 1^2.
16^2 cannot be written as sum of squares of distinct smaller terms, hence a(8)=16.

Prog

(Python)
def findSum(nopt, tgt, a, smax, pwr):
....if nopt==0:
........return [] if tgt==0 else None
....if tgt<0 or tgt>smax[nopt-1]:
........return None
....rv=findSum(nopt-1, tgt - a[nopt-1]**pwr, a, smax, pwr)
....if rv!=None:
........rv.append(a[nopt-1])
....else:
........rv=findSum(nopt-1, tgt, a, smax, pwr)
....return rv
def A321266(n):
....POWER=2 ; x=0 ; a=[] ; smax=[] ; sumpwr=0
....while len(a)<n:
........while True:
............x+=1
............lst=findSum(len(a), x**POWER, a, smax, POWER)
............if lst==None:
................break
............rhs = " + ".join(["%d^%d"%(i, POWER) for i in lst])
............print("    %d^%d = %s"%(x, POWER, rhs))
........a.append(x) ; sumpwr+=x**POWER
........print("a(%d) = %d"%(len(a), x))
........smax.append(sumpwr)
....return a[-1]

Crossrefs

Square root of A226076.

Other powers: A321290 (3), A321291 (4), A321292 (5), A321293 (6).

Sequence in context: A171966 A034893 A187448 * A018662 A240557 A326712

Adjacent sequences: A321263 A321264 A321265 * A321267 A321268 A321269

Keyword

nonn

Author

Bert Dobbelaere, Nov 01 2018

Status

approved