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

1,1

Links

Table of n, a(n) for n=1..5.

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

Cf. A007666, A039664, A003294, A134341.

Sequence in context: A154533 A167608 A063596 * A284518 A369603 A309660

Adjacent sequences: A360379 A360380 A360381 * A360383 A360384 A360385

Keyword

nonn,more

Author

Zhining Yang, Feb 04 2023

Status

approved