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A360382
Least integer m whose n-th power can be written as a sum of four distinct positive n-th powers.
0
10, 9, 13, 353, 144
OFFSET
1,1
FORMULA
a(n) = Minimum(m) such that m^n = a^n + b^n + c^n + d^n and 0 < a < b < c < d < m.
EXAMPLE
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.
MATHEMATICA
n = 5; SelectFirst[
Range[200], (s =
IntegerPartitions[#^n, {4, 4}, Range[1, # - 1]^n]^(1/n); (Select[
s, #[[1]] > #[[2]] > #[[3]] > #[[4]] > 0 &] != {})) &]
PROG
(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))
CROSSREFS
KEYWORD
nonn,more
AUTHOR
Zhining Yang, Feb 04 2023
STATUS
approved