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.
For n = 2 we have a multiset {1, 2, 2} which can be partitioned as {{1}, {2}, {2}} or {{1, 2}, {2}} or {{1}, {2, 2}} or {{1, 2, 2}}, thus a(2) = 4.
g:= proc(n, k) option remember; uses numtheory; `if`(n>k, 0, 1)+
`if`(isprime(n), 0, add(`if`(d>k or max(factorset(n/d))>d, 0,
g(n/d, d)), d=divisors(n) minus {1, n}))
end:
a:= n-> g(mul(ithprime(i)^i, i=1..n)$2):
seq(a(n), n=0..5); # Alois P. Heinz, Jul 26 2020
chern[n_]:=Product[Prime[i]^(n-i+1),{i,n}];
facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Table[Length[facs[chern[n]]],{n,3}] (* Gus Wiseman, Aug 21 2020 *)
\\ See A318284 for count. a(n) = {if(n==0, 1, count(vector(n,i,i)))} \\ Andrew Howroyd, Aug 31 2020
For n=24, A049345(24) = "400" as 24 = 4*A002110(2) + 0*A002110(1) + 0*A002110(0). This can be partitioned in carry-free way either as "100" + "100" + "100" + "100" {6+6+6+6}, "200" + "100" + "100" {12+6+6}, "200" + "200" {12+12}, "300" + "100" {18+6}, or "400" {24}, thus a(24) = 5.
For n=0..23, A049345(n) = A007623(n), thus a(n) = A317826(n) in the same range. See the examples in the latter sequence for how the values for n=0..5 are formed.
fcnt(n, m) = {local(s); s=0; if(n == 1, s=1, fordiv(n, d, if(d > 1 & d <= m, s=s+fcnt(n/d, d)))); s};
A001055(n) = fcnt(n, n); \\ From A001055
A276086(n) = { my(i=0,m=1,pr=1,nextpr); while((n>0),i=i+1; nextpr = prime(i)*pr; if((n%nextpr),m*=(prime(i)^((n%nextpr)/pr));n-=(n%nextpr));pr=nextpr); m; };
A317836(n) = A001055(A276086(n));
\\ Slightly faster, memoized implementation: memA001055 = Map(); A001055(n) = {my(v); if(mapisdefined(memA001055,n), v = mapget(memA001055,n), v = fcnt(n, n); mapput(memA001055,n,v); (v));}; \\ Cached version. A046523(n) = { my(f=vecsort(factor(n)[, 2], , 4), p); prod(i=1, #f, (p=nextprime(p+1))^f[i]); }; \\ From A046523 A317836(n) = A001055(A046523(A276086(n)));
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