A331232 Record numbers of factorizations into distinct factors > 1.

1, 2, 3, 5, 7, 9, 10, 16, 18, 25, 34, 38, 57, 59, 67, 70, 91, 100, 117, 141, 161, 193, 253, 296, 306, 426, 552, 685, 692, 960, 1060, 1067, 1216, 1220, 1589, 1591, 1912, 2029, 2157, 2524, 2886, 3249, 3616, 3875, 4953, 5147, 5285, 5810, 6023, 6112, 6623, 8129

Offset

1,2

Links

Giovanni Resta, Table of n, a(n) for n = 1..122

Jun Kyo Kim, On highly factorable numbers, Journal Of Number Theory, Vol. 72, No. 1 (1998), pp. 76-91.

Formula

a(n) = A045778(A331200(n)).

Example

Representatives for the initial records and their strict factorizations:
  ()  (6)    (12)   (24)     (48)     (60)      (96)      (120)
      (2*3)  (2*6)  (3*8)    (6*8)    (2*30)    (2*48)    (2*60)
             (3*4)  (4*6)    (2*24)   (3*20)    (3*32)    (3*40)
                    (2*12)   (3*16)   (4*15)    (4*24)    (4*30)
                    (2*3*4)  (4*12)   (5*12)    (6*16)    (5*24)
                             (2*3*8)  (6*10)    (8*12)    (6*20)
                             (2*4*6)  (2*5*6)   (2*6*8)   (8*15)
                                      (3*4*5)   (3*4*8)   (10*12)
                                      (2*3*10)  (2*3*16)  (3*5*8)
                                                (2*4*12)  (4*5*6)
                                                          (2*3*20)
                                                          (2*4*15)
                                                          (2*5*12)
                                                          (2*6*10)
                                                          (3*4*10)
                                                          (2*3*4*5)

Mathematica

nn=1000;
strfacs[n_]:=If[n<=1, {{}}, Join@@Table[Map[Prepend[#, d]&, Select[strfacs[n/d], Min@@#>d&]], {d, Rest[Divisors[n]]}]];
qv=Table[Length[strfacs[n]], {n, nn}];
Union[qv//.{foe___, x_, y_, afe___}/; x>y:>{foe, x, afe}]

Prog

(Python)
def fact(num):
....ret = []
....temp = num
....div = 2
....while temp > 1:
........while temp % div == 0:
............ret.append(div)
............temp //= div
........div += 1
....return ret
def all_partitions(lst):
....if lst:
........x = lst[0]
........for partition in all_partitions(lst[1:]):
............yield [x] + partition
............for i, _ in enumerate(partition):
................partition[i] *= x
................yield partition
................partition[i] //= x
....else:
........yield []
best = 0
terms = [0]
q = 2
while len(terms) < 100:
....total_set = set()
....factors = fact(q)
....total_set = set(tuple(sorted(x)) for x in all_partitions(factors) if len(x) == len(set(x)))
....if len(total_set) > best:
........best = len(total_set)
........terms.append(best)
........print(q, best)
....q += 2#only check evens
print(terms)
#  David Consiglio, Jr., Jan 14 2020

Crossrefs

The non-strict version is A272691.

The first appearance of a(n) in A045778 is at index A331200(n).

Factorizations are A001055 with image A045782 and complement A330976.

Strict factorizations are A045778 with image A045779 and complement A330975.

The least number with n strict factorizations is A330974(n).

The least number with A045779(n) strict factorizations is A045780(n).

Cf. A033833, A045783, A325238, A330972, A330973, A330997, A331023/A331024, A331201.

Sequence in context: A133677 A362403 A075750 * A219050 A036049 A092875

Adjacent sequences: A331229 A331230 A331231 * A331233 A331234 A331235

Keyword

nonn

Author

Gus Wiseman, Jan 12 2020

Extensions

a(26)-a(37) from David Consiglio, Jr., Jan 14 2020

a(38) and beyond from Giovanni Resta, Jan 17 2020

Status

approved