cp's OEIS Frontend

A385375 Numbers k that can't be partitioned into tau(k) distinct parts.

%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