A321290 Smallest positive number for which the 3rd power cannot be written as sum of 3rd powers of any subset of previous terms.

1, 2, 3, 4, 5, 7, 8, 10, 11, 13, 17, 21, 22, 28, 29, 33, 38, 41, 48, 68, 70, 96, 124, 130, 158, 179, 239, 309, 310, 351, 468, 509, 640, 843, 900, 1251, 1576, 1640, 2305, 2444, 2989, 3410, 4575, 5758, 5998, 7490, 8602, 11657, 13017, 15553, 19150, 24411, 25365

Offset

1,2

Comments

a(n)^3 forms a sum-free sequence.

Links

Bert Dobbelaere, Table of n, a(n) for n = 1..100

Wikipedia, Sum-free sequence

Example

a(10) = 13. 3rd powers of 14, 15 and 16 can be written as sums of 3rd powers of a subset of the terms {a(1)..a(10)}:
14^3 = 2^3 + 3^3 + 8^3 + 13^3,
15^3 = 4^3 + 5^3 + 7^3 + 8^3 + 10^3 + 11^3,
16^3 = 1^3 + 2^3 + 3^3 + 4^3 + 5^3 + 7^3 + 11^3 + 13^3,
17^3 cannot be written in this way, so a(11) = 17 is the next term.

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 A321290(n):
    POWER=3 ; 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

Other powers: A321266 (2), A321291 (4), A321292 (5), A321293 (6)

Sequence in context: A277992 A180968 A191847 * A274337 A107912 A190850

Adjacent sequences: A321287 A321288 A321289 * A321291 A321292 A321293

Keyword

nonn

Author

Bert Dobbelaere, Nov 02 2018

Status

approved