A360382 Least integer m whose n-th power can be written as a sum of four distinct positive n-th powers.
10, 9, 13, 353, 144
Offset: 1
Examples
a(3) = 13 because 13^3 = 1^3 + 5^3 + 7^3 + 12^3 and no smaller cube may be written as the sum of 4 positive distinct cubes. Terms in this sequence and their representations are: 10^1 = 1 + 2 + 3 + 4. 9^2 = 2^2 + 4^2 + 5^2 + 6^2. 13^3 = 1^3 + 5^3 + 7^3 + 12^3. 353^4 = 30^4 + 120^4 + 272^4 + 315^4. 144^5 = 27^5 + 84^5 + 110^5 + 133^5.
Programs
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Mathematica
n = 5; SelectFirst[ Range[200], (s = IntegerPartitions[#^n, {4, 4}, Range[1, # - 1]^n]^(1/n); (Select[ s, #[[1]] > #[[2]] > #[[3]] > #[[4]] > 0 &] != {})) &] -
Python
def s(n): p=[k**n for k in range(360)] for k in range(4,360): for d in range(k-1,3,-1): if 4*p[d]>p[k]: cc=p[k]-p[d] for c in range(d-1,2,-1): if 3*p[c]>cc: bb=cc-p[c] for b in range(c-1,1,-1): if 2*p[b]>bb: aa=bb-p[b] if aa>0 and aa in p: a=round(aa**(1/n)) return(n,k,[a,b,c,d]) for n in range(1,6): print(s(n))
Formula
a(n) = Minimum(m) such that m^n = a^n + b^n + c^n + d^n and 0 < a < b < c < d < m.