A330991 -id:A330991 - cp's OEIS frontend

A330972 Sorted list containing the least number with each possible nonzero number of factorizations into factors > 1.

1, 4, 8, 12, 16, 24, 36, 48, 60, 72, 96, 120, 128, 144, 180, 192, 216, 240, 256, 288, 360, 384, 420, 432, 480, 576, 720, 768, 840, 864, 900, 960, 1024, 1080, 1152, 1260, 1440, 1680, 1728, 1800, 1920, 2048, 2160, 2304, 2520, 2592, 2880, 3072, 3360, 3456, 3600

Offset: 1

Gus Wiseman, Jan 06 2020

nonn

This is the sorted list of positions of first appearances in A001055 of each element of the range (A045782).

			Factorizations of n for n = 4, 8, 12, 16, 24, 36, 48, 60:
  4    8      12     16       24       36       48         60
  2*2  2*4    2*6    2*8      3*8      4*9      6*8        2*30
       2*2*2  3*4    4*4      4*6      6*6      2*24       3*20
              2*2*3  2*2*4    2*12     2*18     3*16       4*15
                     2*2*2*2  2*2*6    3*12     4*12       5*12
                              2*3*4    2*2*9    2*3*8      6*10
                              2*2*2*3  2*3*6    2*4*6      2*5*6
                                       3*3*4    3*4*4      3*4*5
                                       2*2*3*3  2*2*12     2*2*15
                                                2*2*2*6    2*3*10
                                                2*2*3*4    2*2*3*5
                                                2*2*2*2*3

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

All terms belong to A025487
Includes all highly factorable numbers A033833.
Factorizations are A001055, with image A045782.
The least number with A045782(n) factorizations is A045783(n).
The least number with n factorizations is A330973(n).
The strict version is A330997.
Cf. A001222, A002033, A007716, A045778, A318284, A325238, A330935, A330936, A330976, A330977, A330989, A330991, A330992.

nn=1000;
facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
nds=Length/@Array[facs,nn];
Table[Position[nds,i][[1,1]],{i,First/@Gather[nds]}]

A045782 Number of factorizations of n for some n (image of A001055).

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 9, 11, 12, 15, 16, 19, 21, 22, 26, 29, 30, 31, 36, 38, 42, 45, 47, 52, 56, 57, 64, 66, 67, 74, 77, 92, 97, 98, 101, 105, 109, 118, 135, 137, 139, 141, 162, 165, 171, 176, 181, 189, 195, 198, 203, 212, 231, 249, 250, 254, 257, 267, 269, 272, 289

Offset: 1

David W. Wilson

nonn

Also the image of A318284. - Gus Wiseman, Jan 11 2020

Florian Luca, Anirban Mukhopadhyay and Kotyada Srinivas, On the Oppenheim's "factorisatio numerorum" function, arXiv:0807.0986 [math.NT], 2008.

Factorizations are A001055 with image this sequence and complement A330976.
Strict factorizations are A045778 with image A045779 and complement A330975.
The least number with exactly a(n) factorizations is A045783(n).
The least number with exactly n factorizations is A330973(n).
Cf. A002033, A007716, A033833, A318284, A325238, A330935, A330936, A330977, A330989, A330991, A330992, A330997.

terms = 61; m0 = 10^5; dm = 10^4;
f[1, ] = 1; f[n, k_] := f[n, k] = Sum[f[n/d, d], {d, Select[Divisors[n], 1 < # <= k &]}];
Clear[seq]; seq[m_] := seq[m] = Sort[Tally[Table[f[n, n], {n, 1, m}]][[All, 1]]][[1 ;; terms]]; seq[m = m0]; seq[m += dm]; While[Print[m]; seq[m] != seq[m - dm], m += dm];
seq[m] (* Jean-François Alcover, Oct 04 2018 *)

The Luca et al. paper shows that the number of terms with a(n) <= x is x^{ O( log log log x / log log x )}. - N. J. A. Sloane, Jun 12 2009

Name edited by Gus Wiseman, Jan 11 2020

A035359 Number of partitions-into-distinct-parts of n (A000009) is a prime.

Original entry on oeis.org

3, 4, 5, 7, 22, 70, 100, 495, 1247, 2072, 320397, 3335367, 16168775, 37472505, 52940251, 78840125, 81191852

Offset: 1

Olivier Gérard

No other terms below 10^8. - Max Alekseyev, Jul 10 2015

			From _Gus Wiseman_, Jan 13 2020: (Start)
Strict partitions of a(1) = 3 through a(4) = 7:
  (3)    (4)    (5)    (7)
  (2,1)  (3,1)  (3,2)  (4,3)
                (4,1)  (5,2)
                       (6,1)
                       (4,2,1)
(End)

Eric Weisstein's World of Mathematics, Integer Sequence Primes
Eric Weisstein's World of Mathematics, Partition Function Q
Eric Weisstein's World of Mathematics, Partition Function Q-Congruences

The non-strict version is A046063.
The version for powers of 2 instead of primes is A331022.
The version for factorizations instead of strict partitions is A330991.
The version for strict factorizations instead of strict partitions is A331201.
Cf. A000009, A051005, A056848, A265835.

n = 1; A035359 = {}; While[n < 10^7, n++; If[ PrimeQ[ PartitionsQ[n]], Print[n]; AppendTo[A035359, n]]]; A035359 (* Jean-François Alcover, Oct 12 2011 *)

More terms from Eric W. Weisstein
a(12) from Max Alekseyev, Jul 04 2009
a(13)-a(14) from Giovanni Resta, Jun 05 2015, Jun 11 2015
a(15)-a(17) from Max Alekseyev, Jul 10 2015

A330976 Numbers that are not the number of factorizations into factors > 1 of any positive integer.

Original entry on oeis.org

6, 8, 10, 13, 14, 17, 18, 20, 23, 24, 25, 27, 28, 32, 33, 34, 35, 37, 39, 40, 41, 43, 44, 46, 48, 49, 50, 51, 53, 54, 55, 58, 59, 60, 61, 62, 63, 65, 68, 69, 70, 71, 72, 73, 75, 76, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 93, 94, 95, 96, 99

Offset: 1

Gus Wiseman, Jan 07 2020

nonn

Warning: I have only confirmed the first eight terms. The rest are derived from A045782. - Gus Wiseman, Jan 07 2020

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

Complement of A045782.
The strict version is A330975.
Factorizations are A001055, with image A045782.
Strict factorizations are A045778, with image A045779.
The least number with n factorizations is A330973(n).
Cf. A033833, A045779, A045783, A317145, A330935, A330972, A330974, A330977, A330991.

nn=15;
fam[n_]:=fam[n]=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[fam[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
nds=Length/@Array[fam[#]&,2^nn];
Complement[Range[nn],nds]

A330977 Numbers whose number of factorizations into factors > 1 (A001055) is a power of 2.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 25, 26, 28, 29, 31, 33, 34, 35, 37, 38, 39, 41, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 55, 57, 58, 59, 61, 62, 63, 65, 67, 68, 69, 71, 72, 73, 74, 75, 76, 77, 79, 82, 83, 85, 86, 87

Offset: 1

Gus Wiseman, Jan 07 2020

nonn

The complement starts: 8, 16, 24, 27, 30, 32, 36, 40.

			Factorizations of n = 1, 4, 12, 72:
  ()  (4)    (12)     (72)
      (2*2)  (2*6)    (8*9)
             (3*4)    (2*36)
             (2*2*3)  (3*24)
                      (4*18)
                      (6*12)
                      (2*4*9)
                      (2*6*6)
                      (3*3*8)
                      (3*4*6)
                      (2*2*18)
                      (2*3*12)
                      (2*2*2*9)
                      (2*2*3*6)
                      (2*3*3*4)
                      (2*2*2*3*3)

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 strict integer partitions is A331022.
Factorizations are A001055, with image A045782.
The least number with exactly n factorizations is A330973(n).
The least number with exactly 2^n factorizations is A330989(n).
Numbers whose inverse prime shadow belongs to this sequence are A330990.
Numbers with a prime number of factorizations are A330991.
Cf. A002033, A033833, A045778, A045783, A317145, A326028, A330972, A330976.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Select[Range[100],IntegerQ[Log[2,Length[facs[#]]]]&]

A330989 Least positive integer with exactly 2^n factorizations into factors > 1, or 0 if no such integer exists.

Original entry on oeis.org

1, 4, 12, 0, 72, 0, 480

Offset: 0

Gus Wiseman, Jan 07 2020

			The A001055(n) factorizations for n = 1, 4, 12, 72:
  ()  (4)    (12)     (72)
      (2*2)  (2*6)    (8*9)
             (3*4)    (2*36)
             (2*2*3)  (3*24)
                      (4*18)
                      (6*12)
                      (2*4*9)
                      (2*6*6)
                      (3*3*8)
                      (3*4*6)
                      (2*2*18)
                      (2*3*12)
                      (2*2*2*9)
                      (2*2*3*6)
                      (2*3*3*4)
                      (2*2*2*3*3)

All nonzero terms belong to A025487 and also A033833.
Factorizations are A001055, with image A045782.
The least number with exactly n factorizations is A330973(n).
Numbers whose number of factorizations is a power of 2 are A330977.
The least number with exactly prime(n) factorizations is A330992(n).
Cf. A002033, A045778, A045783, A318284, A330935, A330972, A330976, A330990, A330991, A331022.

A331023 Numerator: factorizations divided by strict factorizations A001055(n)/A045778(n).

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 4, 1, 1, 1, 5, 1, 4, 1, 4, 1, 1, 1, 7, 2, 1, 3, 4, 1, 1, 1, 7, 1, 1, 1, 9, 1, 1, 1, 7, 1, 1, 1, 4, 4, 1, 1, 12, 2, 4, 1, 4, 1, 7, 1, 7, 1, 1, 1, 11, 1, 1, 4, 11, 1, 1, 1, 4, 1, 1, 1, 16, 1, 1, 4, 4, 1, 1, 1, 12, 5, 1, 1, 11, 1, 1, 1, 7, 1, 11, 1, 4, 1, 1, 1, 19, 1, 4, 4, 9, 1, 1, 1, 7, 1

Offset: 1

Gus Wiseman, Jan 08 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. Factorizations and strict factorizations are counted by A001055 and A045778 respectively.

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences computed from exponents in factorization of n

Positions of 1's are A005117.
Positions of 2's appear to be A001248.
The denominators are A331024.
The rounded quotients are A331048.
The same for integer partitions is A330994.
Cf. A001055, A001222, A002033, A045778, A045779, A045780, A045782, A045783, A325755, A326028, A326622, A328966, A330972, A330977, A330991.

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

A001055(n, m=n) = if(1==n, 1, my(s=0); fordiv(n, d, if((d>1)&&(d<=m), s += A001055(n/d, d))); (s));
A045778(n, m=n) = ((n<=m) + sumdiv(n, d, if((d>1)&&(d<=m)&&(dA045778(n/d, d-1))));
A331023(n) = numerator(A001055(n)/A045778(n)); \\ Antti Karttunen, May 27 2021

a(2^n) = A330994(n).

More terms from Antti Karttunen, May 27 2021

A331024 Denominator: factorizations divided by strict factorizations A001055(n)/A045778(n).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 2, 1, 3, 1, 3, 1, 1, 1, 5, 1, 1, 2, 3, 1, 1, 1, 3, 1, 1, 1, 5, 1, 1, 1, 5, 1, 1, 1, 3, 3, 1, 1, 7, 1, 3, 1, 3, 1, 5, 1, 5, 1, 1, 1, 9, 1, 1, 3, 4, 1, 1, 1, 3, 1, 1, 1, 9, 1, 1, 3, 3, 1, 1, 1, 7, 2, 1, 1, 9, 1, 1, 1, 5, 1, 9, 1, 3, 1, 1, 1, 10, 1, 3, 3, 5, 1, 1, 1, 5, 1

Offset: 1

Gus Wiseman, Jan 08 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. Factorizations and strict factorizations are counted by A001055 and A045778 respectively.

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences computed from exponents in factorization of n

Positions of 1's include all elements of A001248 as well as A005117. The first position of a 1 that is not in A167207 is 128.
The numerators are A331023.
The rounded quotients are A331048.
The same for integer partitions is A330995.
Cf. A001055, A005117, A045778, A045779, A045780, A045782, A045783, A325755, A326028, A326622, A328966, A330972, A330977, A330991.

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

A001055(n, m=n) = if(1==n, 1, my(s=0); fordiv(n, d, if((d>1)&&(d<=m), s += A001055(n/d, d))); (s));
A045778(n, m=n) = ((n<=m) + sumdiv(n, d, if((d>1)&&(d<=m)&&(dA045778(n/d, d-1))));
A331024(n) = denominator(A001055(n)/A045778(n)); \\ Antti Karttunen, May 27 2021

a(2^n) = A330995(n).

More terms from Antti Karttunen, May 27 2021

A331050 Positive integers whose number of factorizations into factors > 1 (A001055) is odd.

Original entry on oeis.org

1, 2, 3, 5, 7, 8, 11, 13, 16, 17, 19, 23, 24, 27, 29, 30, 31, 32, 36, 37, 40, 41, 42, 43, 47, 53, 54, 56, 59, 60, 61, 64, 66, 67, 70, 71, 73, 78, 79, 81, 83, 84, 88, 89, 90, 96, 97, 100, 101, 102, 103, 104, 105, 107, 109, 110, 113, 114, 120, 125, 126, 127, 128

Offset: 1

Gus Wiseman, Jan 10 2020

nonn

First differs from A319239 in lacking 256.

Complement of A331051.
The version for powers of two (instead of odds) is A330977.
The version for primes (instead of odds) 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[100],OddQ[Length[facs[#]]]&]

A330992 Least positive integer with exactly prime(n) factorizations into factors > 1, or 0 if no such integer exists.

Original entry on oeis.org

4, 8, 16, 24, 60, 0, 0, 96, 0, 144, 216, 0, 0, 0, 288, 0, 0, 0, 768, 0, 0, 0, 0, 0, 864, 8192, 0, 0, 1080, 0, 0, 0, 1800, 3072, 0, 0, 0, 0, 0, 0, 0, 2304, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3456, 0, 3600, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 24576

Offset: 1

Gus Wiseman, Jan 07 2020

nonn

			Factorizations of the initial positive terms are:
  4    8      16       24       60       96
  2*2  2*4    2*8      3*8      2*30     2*48
       2*2*2  4*4      4*6      3*20     3*32
              2*2*4    2*12     4*15     4*24
              2*2*2*2  2*2*6    5*12     6*16
                       2*3*4    6*10     8*12
                       2*2*2*3  2*5*6    2*6*8
                                3*4*5    3*4*8
                                2*2*15   4*4*6
                                2*3*10   2*2*24
                                2*2*3*5  2*3*16
                                         2*4*12
                                         2*2*3*8
                                         2*2*4*6
                                         2*3*4*4
                                         2*2*2*12
                                         2*2*2*2*6
                                         2*2*2*3*4
                                         2*2*2*2*2*3

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

All positive terms belong to A025487 and also A033833.
Factorizations are A001055, with image A045782, with complement A330976.
Numbers whose number of partitions is prime are A046063.
Numbers whose number of strict partitions is prime are A035359.
Numbers whose number of set partitions is prime are A051130.
Numbers with a prime number of factorizations are A330991.
The least number with exactly 2^n factorizations is A330989(n).
Cf. A001222, A045783, A325238, A330972, A330973, A330976, A330993, A330998.

More terms from Jinyuan Wang, Jul 07 2021

cp's OEIS Frontend

A330972 Sorted list containing the least number with each possible nonzero number of factorizations into factors > 1.

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A045782 Number of factorizations of n for some n (image of A001055).

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A035359 Number of partitions-into-distinct-parts of n (A000009) is a prime.

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A330976 Numbers that are not the number of factorizations into factors > 1 of any positive integer.

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A330977 Numbers whose number of factorizations into factors > 1 (A001055) is a power of 2.

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A330989 Least positive integer with exactly 2^n factorizations into factors > 1, or 0 if no such integer exists.

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A331023 Numerator: factorizations divided by strict factorizations A001055(n)/A045778(n).

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A331024 Denominator: factorizations divided by strict factorizations A001055(n)/A045778(n).

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