A321291 Smallest positive number for which the 4th power cannot be written as sum of 4th powers of any subset of previous terms.

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 16, 17, 18, 19, 20, 21, 22, 23, 24, 26, 27, 28, 32, 34, 36, 38, 40, 42, 44, 46, 48, 52, 54, 56, 64, 68, 72, 76, 80, 84, 88, 92, 96, 104, 108, 112, 128, 136, 144, 152, 160, 168, 176, 184, 192, 208, 216, 224, 256

Offset

1,2

Comments

a(n)^4 forms a sum-free sequence.

It is noteworthy that the terms of this sequence increase slower than those of similar sequences for smaller (A321266, A321290) but also larger powers (A321292, A321293).

Links

Bert Dobbelaere, Table of n, a(n) for n = 1..104

Wikipedia, Sum-free sequence

Formula

a(n) = 2 * a(n-12) for n > 25 (conjectured).

Example

The smallest number > 0 that is not in the sequence is 15, because
    15^4 = 4^4 + 6^4 + 8^4 + 9^4 + 14^4.

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 A321291(n):
....POWER=4 ; 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), A321290 (3), A321292 (5), A321293 (6).

Sequence in context: A322911 A023756 A080944 * A317294 A095736 A004829

Adjacent sequences: A321288 A321289 A321290 * A321292 A321293 A321294

Keyword

nonn

Author

Bert Dobbelaere, Nov 02 2018

Status

approved