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 A356233 #6 Aug 30 2022 09:41:27
%S A356233 1,1,1,2,1,2,1,3,2,1,1,4,1,1,2,5,1,4,1,2,1,1,1,7,2,1,3,2,1,4,1,7,1,1,
%T A356233 2,9,1,1,1,3,1,2,1,2,4,1,1,12,2,2,1,2,1,7,1,3,1,1,1,8,1,1,2,11,1,2,1,
%U A356233 2,1,2,1,16,1,1,4,2,2,2,1,5,5,1,1,4,1,1
%N A356233 Number of integer factorizations of n into gapless numbers (A066311).
%C A356233 A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798. We define a number to be gapless (listed by A066311) iff its prime indices cover an interval of positive integers.
%e A356233 The counted factorizations of n = 2, 4, 8, 12, 24, 36, 48:
%e A356233 (2) (4) (8) (12) (24) (36) (48)
%e A356233 (2*2) (2*4) (2*6) (3*8) (4*9) (6*8)
%e A356233 (2*2*2) (3*4) (4*6) (6*6) (2*24)
%e A356233 (2*2*3) (2*12) (2*18) (3*16)
%e A356233 (2*2*6) (3*12) (4*12)
%e A356233 (2*3*4) (2*2*9) (2*3*8)
%e A356233 (2*2*2*3) (2*3*6) (2*4*6)
%e A356233 (3*3*4) (3*4*4)
%e A356233 (2*2*3*3) (2*2*12)
%e A356233 (2*2*2*6)
%e A356233 (2*2*3*4)
%e A356233 (2*2*2*2*3)
%t A356233 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A356233 facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
%t A356233 sqq[n_]:=Max@@Differences[primeMS[n]]<=1;
%t A356233 Table[Length[Select[facs[n],And@@sqq/@#&]],{n,100}]
%Y A356233 The shortest of these factorizations is listed at A356234, length A287170.
%Y A356233 A000005 counts divisors.
%Y A356233 A001055 counts factorizations.
%Y A356233 A001221 counts distinct prime factors, sum A001414.
%Y A356233 A003963 multiplies together the prime indices.
%Y A356233 A132747 counts non-isolated divisors, complement A132881.
%Y A356233 A356069 counts gapless divisors, initial A356224 (complement A356225).
%Y A356233 A356226 lists the lengths of maximal gapless submultisets of prime indices:
%Y A356233 - length: A287170
%Y A356233 - minimum: A356227
%Y A356233 - maximum: A356228
%Y A356233 - bisected length: A356229
%Y A356233 - standard composition: A356230
%Y A356233 - Heinz number: A356231
%Y A356233 - positions of first appearances: A356232
%Y A356233 Cf. A001222, A060680-A060683, A073491-A073495, A193829, A328195, A328335-A328458.
%K A356233 nonn
%O A356233 1,4
%A A356233 _Gus Wiseman_, Aug 28 2022