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 A331232 #20 May 22 2025 10:21:48
%S A331232 1,2,3,5,7,9,10,16,18,25,34,38,57,59,67,70,91,100,117,141,161,193,253,
%T A331232 296,306,426,552,685,692,960,1060,1067,1216,1220,1589,1591,1912,2029,
%U A331232 2157,2524,2886,3249,3616,3875,4953,5147,5285,5810,6023,6112,6623,8129
%N A331232 Record numbers of factorizations into distinct factors > 1.
%H A331232 Giovanni Resta, <a href="/A331232/b331232.txt">Table of n, a(n) for n = 1..122</a>
%H A331232 Jun Kyo Kim, <a href="https://doi.org/10.1006/jnth.1998.2238">On highly factorable numbers</a>, Journal Of Number Theory, Vol. 72, No. 1 (1998), pp. 76-91.
%F A331232 a(n) = A045778(A331200(n)).
%e A331232 Representatives for the initial records and their strict factorizations:
%e A331232 () (6) (12) (24) (48) (60) (96) (120)
%e A331232 (2*3) (2*6) (3*8) (6*8) (2*30) (2*48) (2*60)
%e A331232 (3*4) (4*6) (2*24) (3*20) (3*32) (3*40)
%e A331232 (2*12) (3*16) (4*15) (4*24) (4*30)
%e A331232 (2*3*4) (4*12) (5*12) (6*16) (5*24)
%e A331232 (2*3*8) (6*10) (8*12) (6*20)
%e A331232 (2*4*6) (2*5*6) (2*6*8) (8*15)
%e A331232 (3*4*5) (3*4*8) (10*12)
%e A331232 (2*3*10) (2*3*16) (3*5*8)
%e A331232 (2*4*12) (4*5*6)
%e A331232 (2*3*20)
%e A331232 (2*4*15)
%e A331232 (2*5*12)
%e A331232 (2*6*10)
%e A331232 (3*4*10)
%e A331232 (2*3*4*5)
%t A331232 nn=1000;
%t A331232 strfacs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[strfacs[n/d],Min@@#>d&]],{d,Rest[Divisors[n]]}]];
%t A331232 qv=Table[Length[strfacs[n]],{n,nn}];
%t A331232 Union[qv//.{foe___,x_,y_,afe___}/;x>y:>{foe,x,afe}]
%o A331232 (Python)
%o A331232 def fact(num):
%o A331232 ret = []
%o A331232 temp = num
%o A331232 div = 2
%o A331232 while temp > 1:
%o A331232 while temp % div == 0:
%o A331232 ret.append(div)
%o A331232 temp //= div
%o A331232 div += 1
%o A331232 return ret
%o A331232 def all_partitions(lst):
%o A331232 if lst:
%o A331232 x = lst[0]
%o A331232 for partition in all_partitions(lst[1:]):
%o A331232 yield [x] + partition
%o A331232 for i, _ in enumerate(partition):
%o A331232 partition[i] *= x
%o A331232 yield partition
%o A331232 partition[i] //= x
%o A331232 else:
%o A331232 yield []
%o A331232 best = 0
%o A331232 terms = [0]
%o A331232 q = 2
%o A331232 while len(terms) < 100:
%o A331232 total_set = set()
%o A331232 factors = fact(q)
%o A331232 total_set = set(tuple(sorted(x)) for x in all_partitions(factors) if len(x) == len(set(x)))
%o A331232 if len(total_set) > best:
%o A331232 best = len(total_set)
%o A331232 terms.append(best)
%o A331232 print(q,best)
%o A331232 q += 2#only check evens
%o A331232 print(terms)
%o A331232 # _David Consiglio, Jr._, Jan 14 2020
%Y A331232 The non-strict version is A272691.
%Y A331232 The first appearance of a(n) in A045778 is at index A331200(n).
%Y A331232 Factorizations are A001055 with image A045782 and complement A330976.
%Y A331232 Strict factorizations are A045778 with image A045779 and complement A330975.
%Y A331232 The least number with n strict factorizations is A330974(n).
%Y A331232 The least number with A045779(n) strict factorizations is A045780(n).
%Y A331232 Cf. A033833, A045783, A325238, A330972, A330973, A330997, A331023/A331024, A331201.
%K A331232 nonn
%O A331232 1,2
%A A331232 _Gus Wiseman_, Jan 12 2020
%E A331232 a(26)-a(37) from _David Consiglio, Jr._, Jan 14 2020
%E A331232 a(38) and beyond from _Giovanni Resta_, Jan 17 2020