Least number that is the sum of four cubes of primes in n ways.
32, 10512, 289350, 1244880, 5496120, 15681708, 7025200, 101965500, 94324230, 180710530, 654503850, 821080260, 5241294000
1
It appears that these numbers exist for all positive n. However, computing them is not easy.
T. D. Noe, Plot of 13 terms
Example: 10512 = 5^3 + 11^3 + 13^3 + 19^3 = 7^3 + 7^3 + 17^3 + 17^3.
(Mma) nn = 50; t2 = Select[Sort[Flatten[Table[Prime[i]^3 + Prime[j]^3 + Prime[k]^3 + Prime[l]^3, {i, nn}, {j, i, nn}, {k, j, nn}, {l, k, nn}]]], # <= Prime[nn]^3 + 3*Prime[1]^3 &]; t3 = Tally[t2]; t4 = Table[0, {20}]; Do[cnt = i[[2]]; If[cnt <= 20 && t4[[cnt]] == 0, t4[[cnt]] = i[[1]]], {i, t3}]; t4
nonn,hard,more
T. D. Noe, May 16 2017