This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
nMax = 200000; t = {0}; Do[k = n; s = 0; While[s = s + k^5; s <= nMax, AppendTo[t, s]; k++], {n, nMax^(1/5)}]; t = Union[t]
nMax = 10^6; t = {0}; Do[k = n; s = 0; While[s = s + k^6; s <= nMax, AppendTo[t, s]; k++], {n, nMax^(1/6)}]; t = Union[t]
list(lim)=my(v=List(apply(n->n^6, [0..sqrtnint(lim\=1,6)])),s); for(n=2,lim, s=n*(n-1)*(2*n-1)*(3*n^4-6*n^3+3*n+1)/42; if(s>lim,break); for(k=n,lim, s+=k^6-(k-n)^6; if(s>lim,break); listput(v,s))); Set(v) \\ Charles R Greathouse IV, Apr 22 2020
nMax = 10^9; t = {0}; Do[k = n; s = 0; While[s = s + k^9; s <= nMax, AppendTo[t, s]; k++], {n, nMax^(1/9)}]; t = Union[t]
20^3 = 11^3 + 12^3 + 13^3 + 14^3, so a(20) = 2. 2856^3 = 213^3 +...+ 555^3 = 273^3 +...+ 560^3, so a(2856) = 3. See also _Donovan Johnson_'s comment in A097811. - _Antti Karttunen_, Aug 22 2019
A297199(n) = { my(s=0, k=1, c); while((c=k^3) <= n, my(u=n-c, i=k); while(u>0, i++; c = i^3; u=u-c); s += (!u); k++); (s); }; A307609(n) = A297199(n^3); \\ Antti Karttunen, Aug 22 2019
nMax = 10^7; t = {0}; Do[k = n; s = 0; While[s = s + k^7; s <= nMax, AppendTo[t, s]; k++], {n, nMax^(1/7)}]; t = Union[t]
nMax = 10^8; t = {0}; Do[k = n; s = 0; While[s = s + k^8; s <= nMax, AppendTo[t, s]; k++], {n, nMax^(1/8)}]; t = Union[t]
9 is not in the sequence since it is equal to 1^3 + 2^3.
lst:=[]; r:=26; c:=r^3; for n in [2..r] do for m in [n-1..0 by -1] do s:=((n^2+n)^2-(m^2+m)^2)/4; if s gt c then break; end if; if not s in lst then Append(~lst, s); end if; end for; end for; lst:=Sort(lst); [p: p in [2..c] | not IsPrime(p) and IsPrimePower(p) and not p in lst];