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.
%I A385375 #20 Jul 24 2025 09:38:52 %S A385375 2,4,6,8,12,18,20,24,30,36,48,60,72,120 %N A385375 Numbers k that can't be partitioned into tau(k) distinct parts. %C A385375 Numbers k for which k < A000217(tau(k)). %C A385375 To partition k into tau(k) distinct parts, k >= tau(k)*(tau(k) + 1)/2. According to A374793, k > tau(k)^2 > tau(k)*(tau(k) + 1)/2 for k > 1260. The sequence is therefore finite and contains 14 terms. %e A385375 6 is a term because there is no partition of 6 into tau(6) = 4 distinct parts. %p A385375 with(NumberTheory): %p A385375 A385375:=proc(K) %p A385375 local k,l; %p A385375 l:=[]; %p A385375 for k from 1 to K do %p A385375 if tau(k)*(tau(k)+1)/2>k then %p A385375 l:=[op(l),k]; %p A385375 end if; %p A385375 end do; %p A385375 return op(l); %p A385375 end proc: %p A385375 A385375(1260); %t A385375 s={};Do[t=DivisorSigma[0,k];If[NoneTrue[Length/@Union/@IntegerPartitions[k,{t}],#==t&],AppendTo[s,k]],{k,72}];s (* _James C. McMahon_, Jul 24 2025 *) %Y A385375 Cf. A000005, A000217, A374793, A385374. %K A385375 nonn,fini,full %O A385375 1,1 %A A385375 _Felix Huber_, Jul 11 2025