A045783 -id:A045783 - cp's OEIS frontend

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

Gus Wiseman, Jan 12 2020

nonn

			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)

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.

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.

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}]

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

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

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

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

A330993 Numbers k such that a multiset whose multiplicities are the prime indices of k has a prime number of multiset partitions.

Original entry on oeis.org

3, 4, 5, 7, 8, 10, 11, 12, 13, 21, 22, 25, 33, 38, 41, 45, 46, 49, 50, 55, 57, 58, 63

Offset: 1

Gus Wiseman, Jan 07 2020

This multiset (row k of A305936) is generally not the same as the multiset of prime indices of k. For example, the prime indices of 12 are {1,1,2}, while a multiset whose multiplicities are {1,1,2} is {1,1,2,3}.

Also numbers whose inverse prime shadow has a prime number of factorizations. A prime index of k is a number m such that prime(m) divides k. The multiset of prime indices of k is row k of A112798. The inverse prime shadow of k is the least number whose prime exponents are the prime indices of k.

			The multiset partitions for n = 1..6:
  {11}    {12}    {111}      {1111}        {123}      {1112}
  {1}{1}  {1}{2}  {1}{11}    {1}{111}      {1}{23}    {1}{112}
                  {1}{1}{1}  {11}{11}      {2}{13}    {11}{12}
                             {1}{1}{11}    {3}{12}    {2}{111}
                             {1}{1}{1}{1}  {1}{2}{3}  {1}{1}{12}
                                                      {1}{2}{11}
                                                      {1}{1}{1}{2}
The factorizations for n = 1..8:
  4    6    8      16       30     24       32         60
  2*2  2*3  2*4    2*8      5*6    3*8      4*8        2*30
            2*2*2  4*4      2*15   4*6      2*16       3*20
                   2*2*4    3*10   2*12     2*2*8      4*15
                   2*2*2*2  2*3*5  2*2*6    2*4*4      5*12
                                   2*3*4    2*2*2*4    6*10
                                   2*2*2*3  2*2*2*2*2  2*5*6
                                                       3*4*5
                                                       2*2*15
                                                       2*3*10
                                                       2*2*3*5

R. E. Canfield, P. Erdős and C. Pomerance, On a Problem of Oppenheim concerning "Factorisatio Numerorum", J. Number Theory 17 (1983), 1-28.

The same for powers of 2 (instead of primes) is A330990.
Factorizations are A001055, with image A045782, with complement A330976.
Numbers whose number of integer partitions is prime are A046063.
Numbers whose number of strict integer partitions is prime are A035359.
Numbers whose number of set partitions is prime are A051130.
Numbers whose number of factorizations is a power of 2 are A330977.
The least number with prime(n) factorizations is A330992(n).
Factorizations of a number's inverse prime shadow are A318284.
Numbers with a prime number of factorizations are A330991.
Cf. A033833, A045783, A056239, A181819, A181821, A305936, A318286, A325755, A330972, A330973, A330998.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
unsh[n_]:=Times@@MapIndexed[Prime[#2[[1]]]^#1&,Reverse[Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]];
Select[Range[30],PrimeQ[Length[facs[unsh[#]]]]&]

A001055(A181821(a(n))) belongs to A000040.

A331048 Nearest integer to A001055(n)/A045778(n), where A001055 is factorizations and A045778 is strict factorizations.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1

Offset: 1

Gus Wiseman, Jan 10 2020

A factorization of n is a finite, nondecreasing sequence of positive integers > 1 with product n. It is strict if the factors are all different.

The exact quotient is A331023/A331024.
The same for integer partitions is A330996 ~ A330994/A330995.
Cf. A001055, A001222, A002033, A005117, A045778, A045779, A045780, A045782, A045783, A330972, A330977, A330991.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Table[Round[Length[facs[n]]/Length[Select[facs[n],UnsameQ@@#&]]],{n,100}]

A331051 Numbers whose number of factorizations into factors > 1 (A001055) is even.

Original entry on oeis.org

4, 6, 9, 10, 12, 14, 15, 18, 20, 21, 22, 25, 26, 28, 33, 34, 35, 38, 39, 44, 45, 46, 48, 49, 50, 51, 52, 55, 57, 58, 62, 63, 65, 68, 69, 72, 74, 75, 76, 77, 80, 82, 85, 86, 87, 91, 92, 93, 94, 95, 98, 99, 106, 108, 111, 112, 115, 116, 117, 118, 119, 121, 122

Offset: 1

Gus Wiseman, Jan 10 2020

nonn

First differs from A319240 in having 256.

Complement of A331050.
The version for powers of two (instead of evens) is A330977.
The version for primes (instead of evens) is A330991.
Cf. A001055, A001221, A001222, A002033, A005117, A045778, A045782, A045783, A330972, A330973.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Select[Range[300],EvenQ[Length[facs[#]]]&]

A272691 Number of factorizations of the highly factorable numbers A033833.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 9, 12, 16, 19, 21, 29, 30, 31, 38, 47, 52, 57, 64, 77, 98, 105, 109, 118, 171, 212, 289, 382, 392, 467, 484, 662, 719, 737, 783, 843, 907, 1097, 1261, 1386, 1397, 1713, 1768, 2116, 2179, 2343, 3079, 3444, 3681, 3930, 5288, 5413, 5447

Offset: 1

N. J. A. Sloane, Jun 02 2016, following a suggestion from George Beck

nonn

These are defined as record numbers of factorizations (A001055). - Gus Wiseman, Jan 13 2020

			From _Gus Wiseman_, Jan 13 2020: (Start)
The a(1) = 1 through a(8) = 12 factorizations of highly factorable numbers:
  ()  (4)    (8)      (12)     (16)       (24)       (36)       (48)
      (2*2)  (2*4)    (2*6)    (2*8)      (3*8)      (4*9)      (6*8)
             (2*2*2)  (3*4)    (4*4)      (4*6)      (6*6)      (2*24)
                      (2*2*3)  (2*2*4)    (2*12)     (2*18)     (3*16)
                               (2*2*2*2)  (2*2*6)    (3*12)     (4*12)
                                          (2*3*4)    (2*2*9)    (2*3*8)
                                          (2*2*2*3)  (2*3*6)    (2*4*6)
                                                     (3*3*4)    (3*4*4)
                                                     (2*2*3*3)  (2*2*12)
                                                                (2*2*2*6)
                                                                (2*2*3*4)
                                                                (2*2*2*2*3)
(End)

Amiram Eldar, Table of n, a(n) for n = 1..235 (terms 1..118 from E. R. Canfield et al.)
R. E. Canfield, P. Erdős and C. Pomerance, On a Problem of Oppenheim concerning "Factorisatio Numerorum", J. Number Theory 17 (1983), 1-28.
Jun Kyo Kim, On highly factorable numbers, Journal Of Number Theory, Vol. 72, No. 1 (1998), pp. 76-91.

The strict version is A331232.
Factorizations are A001055, with image A045782 and complement A330976.
Highly factorable numbers are A033833, with strict version A331200.
Cf. A033834, A045778, A045783, A330972, A330973, A330998.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Table[Length[facs[n]],{n,100}]//.{foe___,x_,y_,afe___}/;x>=y:>{foe,x,afe} (* Gus Wiseman, Jan 13 2020 *)

a(n) = A001055(A033833(n)).

a(n) = A033834(n) + 1. - Amiram Eldar, Jun 07 2019

A331231 Numbers k such that the number of factorizations of k into distinct factors > 1 is even.

Original entry on oeis.org

6, 8, 10, 14, 15, 16, 21, 22, 26, 27, 33, 34, 35, 38, 39, 46, 51, 55, 57, 58, 62, 64, 65, 69, 74, 77, 81, 82, 85, 86, 87, 91, 93, 94, 95, 96, 106, 111, 115, 118, 119, 120, 122, 123, 125, 129, 133, 134, 141, 142, 143, 144, 145, 146, 155, 158, 159, 160, 161, 166

Offset: 1

Gus Wiseman, Jan 12 2020

nonn

First differs from A319238 in having 300.

The version for integer partitions is A001560.
The version for strict integer partitions is A090864.
The version for set partitions appears to be A016789.
The non-strict version is A331051.
The version for primes (instead of evens) is A331201.
The odd version is A331230.
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).
Cf. A001318, A045780, A045783, A318286, A328966, A330973, A330991, A330997, A331022, A331023/A331024, A331050, A331200, A331232.

strfacs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[strfacs[n/d],Min@@#>d&]],{d,Rest[Divisors[n]]}]];
Select[Range[100],EvenQ[Length[strfacs[#]]]&]

A331049 Number of factorizations of A055932(n), the least representative of the n'th distinct unsorted prime signature, into factors > 1.

Original entry on oeis.org

1, 1, 2, 2, 3, 4, 5, 4, 7, 5, 7, 9, 12, 7, 11, 11, 16, 11, 19, 16, 21, 15, 29, 11, 12, 26, 30, 15, 31, 38, 22, 21, 47, 26, 29, 52, 45, 36, 57, 26, 64, 19, 30, 52, 77, 52, 36, 57, 98, 21, 67, 38, 74, 97, 66, 105, 47, 42, 36, 109, 118, 98, 92, 109, 52, 171, 30

Offset: 1

Gus Wiseman, Jan 10 2020

nonn

A factorization of n is a finite, nondecreasing sequence of positive integers > 1 with product n. Factorizations are counted by A001055.

The unsorted prime signature of A055932(n) is given by row n of A124829.

			The a(1) = 1 through a(11) = 7 factorizations:
  {}  2  4    6    8      12     16       18     24       30     32
         2*2  2*3  2*4    2*6    2*8      2*9    3*8      5*6    4*8
                   2*2*2  3*4    4*4      3*6    4*6      2*15   2*16
                          2*2*3  2*2*4    2*3*3  2*12     3*10   2*2*8
                                 2*2*2*2         2*2*6    2*3*5  2*4*4
                                                 2*3*4           2*2*2*4
                                                 2*2*2*3         2*2*2*2*2

The sorted-signature version is A050322.
This sequence has range A045782.
Factorizations are A001055.
Cf. A025487, A033833, A045778, A045783, A070175, A181821, A325238, A330972, A330973, A330976, A330989, A330990, A330998, A331050.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Length@*facs/@First/@GatherBy[Range[1500],If[#==1,{},Last/@FactorInteger[#]]&]

a(n) = A001055(A055932(n)).

A331198 Numbers n with exactly three times as many factorizations (A001055) as strict factorizations (A045778).

Original entry on oeis.org

128, 2187, 10368, 34992, 78125, 80000, 307328, 823543, 1250000, 1366875, 1874048, 3655808, 5250987, 6328125, 10690688, 13176688, 16681088, 19487171, 32019867, 35819648, 62462907, 62748517, 66706983, 90531968, 118210688, 182660427, 187578125, 239892608, 285012027

Offset: 1

Gus Wiseman, Jan 12 2020

nonn

Contains p^7 for all primes p.

			The 15 factorizations and 5 strict factorizations of 2187:
  (2187)           (2187)
  (27*81)          (27*81)
  (3*729)          (3*729)
  (9*243)          (9*243)
  (3*9*81)         (3*9*81)
  (9*9*27)
  (3*27*27)
  (3*3*243)
  (3*9*9*9)
  (3*3*3*81)
  (3*3*9*27)
  (3*3*3*9*9)
  (3*3*3*3*27)
  (3*3*3*3*3*9)
  (3*3*3*3*3*3*3)

Factorizations are A001055.
Strict factorizations are A045778.
Taking "twice" instead of "three times" gives A001248.
Cf. A045779, A045780, A045782, A045783, A330972, A331023/A331024, A331048, A331050.

facsm[n_]:=facsm[n]=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facsm[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Select[Range[100000],3==Length[facsm[#]]/Length[Select[facsm[#],UnsameQ@@#&]]&]

a(7)-(10) from Alois P. Heinz, Jan 17 2020

a(11)-a(29) from Giovanni Resta, Jan 20 2020

cp's OEIS Frontend

A331232 Record numbers of factorizations into distinct factors > 1.

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A330993 Numbers k such that a multiset whose multiplicities are the prime indices of k has a prime number of multiset partitions.

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A331048 Nearest integer to A001055(n)/A045778(n), where A001055 is factorizations and A045778 is strict factorizations.

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A331051 Numbers whose number of factorizations into factors > 1 (A001055) is even.

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A272691 Number of factorizations of the highly factorable numbers A033833.

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A331231 Numbers k such that the number of factorizations of k into distinct factors > 1 is even.

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A331049 Number of factorizations of A055932(n), the least representative of the n'th distinct unsorted prime signature, into factors > 1.

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A331198 Numbers n with exactly three times as many factorizations (A001055) as strict factorizations (A045778).

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