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

2, 4, 6, 8, 12, 18, 20, 24, 30, 36, 48, 60, 72, 120

Offset

1,1

Comments

Numbers k for which k < A000217(tau(k)).

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.

Links

Table of n, a(n) for n=1..14.

Example

6 is a term because there is no partition of 6 into tau(6) = 4 distinct parts.

Maple

with(NumberTheory):
A385375:=proc(K)
    local k, l;
    l:=[];
    for k from 1 to K do
        if tau(k)*(tau(k)+1)/2>k then
            l:=[op(l), k];
        end if;
    end do;
    return op(l);
end proc:
A385375(1260);

Mathematica

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 *)

Crossrefs

Cf. A000005, A000217, A374793, A385374.

Sequence in context: A352107 A273457 A222603 * A177710 A032458 A284005

Adjacent sequences: A385372 A385373 A385374 * A385376 A385377 A385378

Keyword

nonn,fini,full

Author

Felix Huber, Jul 11 2025

Status

approved