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A321266 Smallest positive number for which the square cannot be written as sum of distinct squares of any subset of previous terms. 5
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified June 24 13:44 EDT 2021. Contains 345417 sequences. (Running on oeis4.)