A327399 -id:A327399 - cp's OEIS frontend

A259936 Number of ways to express the integer n as a product of its unitary divisors (A034444).

1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 2, 1, 1, 2, 1, 2, 2, 2, 1, 2, 1, 2, 1, 2, 1, 5, 1, 1, 2, 2, 2, 2, 1, 2, 2, 2, 1, 5, 1, 2, 2, 2, 1, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 2, 1, 5, 1, 2, 2, 1, 2, 5, 1, 2, 2, 5, 1, 2, 1, 2, 2, 2, 2, 5, 1, 2, 1, 2, 1, 5, 2, 2, 2, 2, 1, 5, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 1, 5, 1, 2, 5

Offset: 1

Geoffrey Critzer, Jul 09 2015

nonn

Equivalently, a(n) is the number of ways to express the cyclic group Z_n as a direct sum of its Hall subgroups.  A Hall subgroup of a finite group G is a subgroup whose order is coprime to its index.
a(n) is the number of ways to partition the set of distinct prime factors of n.
Also the number of singleton or pairwise coprime factorizations of n. - Gus Wiseman, Sep 24 2019

			a(60) = 5 because we have: 60 = 4*3*5 = 4*15 = 3*20 = 5*12.
For n = 36, its unitary divisors are 1, 4, 9, 36. From these we obtain 36 either as 1*36 or 4*9, thus a(36) = 2. - _Antti Karttunen_, Oct 21 2017

Alois P. Heinz, Table of n, a(n) for n = 1..20000
Wikipedia, Hall subgroup
Index entries for sequences computed from exponents in factorization of n

Cf. A000110, A001055, A001221, A034444, A089233, A258466, A281116, A285572.
Differs from A050320 for the first time at n=36.
Differs from A354870 for the first time at n=210, where a(210) = 15, while A354870(210) = 12.
Cf. A304716, A302569, A304711, A305079.
Related classes of factorizations:
- No conditions: A001055
- Strict: A045778
- Constant: A089723
- Distinct multiplicities: A255231
- Singleton or coprime: A259936
- Relatively prime: A281116
- Aperiodic: A303386
- Stable (indivisible): A305149
- Connected: A305193
- Strict relatively prime: A318721
- Uniform: A319269
- Intersecting: A319786
- Constant or distinct factors coprime: A327399
- Constant or relatively prime: A327400
- Coprime: A327517
- Not relatively prime: A327658
- Distinct factors coprime: A327695

map(combinat:-bell @ nops @ numtheory:-factorset, [$1..100]); # Robert Israel, Jul 09 2015

Table[BellB[PrimeNu[n]], {n, 1, 75}]
(* second program *)
facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Table[Length[Select[facs[n],Length[#]==1||CoprimeQ@@#&]],{n,100}] (* Gus Wiseman, Sep 24 2019 *)

a(n) = my(t=omega(n), x='x, m=contfracpnqn(matrix(2, t\2, y, z, if( y==1, -z*x^2, 1 - (z+1)*x)))); polcoeff(1/(1 - x + m[2, 1]/m[1, 1]) + O(x^(t+1)), t) \\ Charles R Greathouse IV, Jun 30 2017

a(n) = A000110(A001221(n)).

a(n > 1) = A327517(n) + 1. - Gus Wiseman, Sep 24 2019

Incorrect comment removed by Antti Karttunen, Jun 11 2022

A327400 Number of factorizations of n that are constant or whose factors are relatively prime.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 2, 2, 2, 1, 3, 1, 2, 2, 3, 1, 3, 1, 3, 2, 2, 1, 4, 2, 2, 2, 3, 1, 5, 1, 2, 2, 2, 2, 7, 1, 2, 2, 4, 1, 5, 1, 3, 3, 2, 1, 6, 2, 3, 2, 3, 1, 4, 2, 4, 2, 2, 1, 9, 1, 2, 3, 4, 2, 5, 1, 3, 2, 5, 1, 10, 1, 2, 3, 3, 2, 5, 1, 6, 3, 2, 1, 9, 2, 2, 2

Offset: 1

Gus Wiseman, Sep 22 2019

nonn

First differs from A327399 at a(24) = 4, A327399(24) = 3.

			The factorizations of 2, 4, 12, 24, 30, 36, 48, and 60 that are constant or whose factors are relatively prime:
  2   4     12      24        30      36        48          60
      2*2   3*4     3*8       5*6     4*9       3*16        3*20
            2*2*3   2*3*4     2*15    6*6       2*3*8       4*15
                    2*2*2*3   3*10    2*2*9     3*4*4       5*12
                              2*3*5   2*3*6     2*2*3*4     2*5*6
                                      3*3*4     2*2*2*2*3   3*4*5
                                      2*2*3*3               2*2*15
                                                            2*3*10
                                                            2*2*3*5

Constant factorizations are A089723.

Cf. A007359, A051424, A281116, A302569, A304709, A304711, A319269, A327399.

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

a(n) = A281116(n) + A089723(n).

A327695 Number of non-constant factorizations of n whose distinct factors are pairwise coprime.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 2, 0, 1, 1, 0, 0, 2, 0, 2, 1, 1, 0, 2, 0, 1, 0, 2, 0, 4, 0, 0, 1, 1, 1, 4, 0, 1, 1, 2, 0, 4, 0, 2, 2, 1, 0, 3, 0, 2, 1, 2, 0, 2, 1, 2, 1, 1, 0, 6, 0, 1, 2, 0, 1, 4, 0, 2, 1, 4, 0, 4, 0, 1, 2, 2, 1, 4, 0, 3, 0, 1, 0, 6, 1, 1, 1

Offset: 1

Gus Wiseman, Sep 22 2019

nonn

			The factorizations of 6, 12, 30, 48, 60, 180, and 210:
  (2*3)  (3*4)    (5*6)    (3*16)       (3*20)     (4*45)       (3*70)
         (2*2*3)  (2*15)   (3*4*4)      (4*15)     (5*36)       (5*42)
                  (3*10)   (2*2*2*2*3)  (5*12)     (9*20)       (6*35)
                  (2*3*5)               (3*4*5)    (4*5*9)      (7*30)
                                        (2*2*15)   (5*6*6)      (10*21)
                                        (2*2*3*5)  (2*2*45)     (14*15)
                                                   (3*3*20)     (2*105)
                                                   (2*2*5*9)    (5*6*7)
                                                   (3*3*4*5)    (2*3*35)
                                                   (2*2*3*3*5)  (2*5*21)
                                                                (2*7*15)
                                                                (3*5*14)
                                                                (3*7*10)
                                                                (2*3*5*7)

Factorizations that are constant or whose distinct parts are pairwise coprime are counted by A327399.
Numbers with pairwise coprime distinct prime indices are A304711.
Cf. A001055, A089723, A281116, A318721, A302569, A319269, A327407, A327517.

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

a(n) = A327399(n) - A089723(n).

A384537 Composite numbers that are equal to the concatenation of the primes and exponents in their prime factorizations in some bases.

Original entry on oeis.org

16, 27, 64, 256, 729, 1024, 3125, 4096, 4617, 16384, 19683, 29767, 65536, 255987, 262144, 395847, 531441, 631463, 823543, 1048576, 1332331, 4194304, 9765625, 14348907, 16777216, 25640947, 67108864

Offset: 1

Jianing Song, Jun 02 2025

Someone called James Davis found that 13532385396179 = 13 * 53^2 * 3853 * 96179, showing that a composite number can be equal the concatenation of the primes and exponents in its canonical prime factorization. In general, if a composite number is equal the concatenation in base b of the primes and exponents in its prime factorization, then let's call it a Davis number to base b.
Conjecture: a composite number can be a Davis number to at most one base.
Let (d_1,...,d_r) be the ordered tuple of prime factors and exponents > 1 in the prime factorization of n (e.g., 4617 = 3^5 * 19 -> (3,5,19), 13532385396179 = 13 * 53^2 * 3853 * 96179 -> (13,53,2,3853,96179)), then n is a Davis number to base b if and only if n = d_1*b^{s_1} + ... + d_{r-1}*b^{s_{r-1}} + d_r, where s_i = (Sum_{j=i+1..r} floor(log_b(d_j))) + r-i. In particular, we must have b dividing n - d_r.
Suppose that p^e is a Davis number to some base b, with e >= 2. We have p^e = p*b^(floor(log_b(e))+1) + e in base b, hence e is divisible by p. If b <= e, then we have p^e <= p*b^(log_b(e)+1) + e <= p*e^2 + e, which is impossible, and so we must have b > e. Conversely, when e is divisible by p and p^e > 4, p^e is a Davis number to base (p^e-e)/p > e.
No term can be squarefree: for primes p_1 < ... < p_r, the concatenation of p_1, ..., p_r in base b is p_1 * b^(Sum_{i=2..r} (floor(log_b(p_i))+1)) + ... >= p_1*...*p_r + ... > p_1*...*p_r.
Here are some examples that are near-miss of being Davis numbers to base 10. Each is equal to the concatenation of the factors and exponents in its generalized factorization (we call n = (q_1)^(e_1) * ... * (q_k)^(e_k) a generalized factorization of n, where 1 < q_1 < ... < q_k, (q_1,...,q_k) are pairwise coprime but are not necessarily primes, and exponents 1 are omitted; the number of such factorizations is A327399(n)):
  2592 = 2^5 * 9^2;
  34425 = 3^4 * 425;
  312325 = 31^2 * 325;
  492205 = 49^2 * 205;
  36233196159122085048010973936921313644799483579440006455257 = 3^6 * 2331961591220850480109739369 * 21313644799483579440006455257. (Note that in the last four examples, we can add as many trailing zeros as we want).

			In base 6: 24 = 2^4 (in decimal: 16 = 2^4);
In base 8: 33 = 3^3 (in decimal: 27 = 3^3);
In base 29: 26 = 2^6 (in decimal: 64 = 2^6);
In base 124: 28 = 2^8 (in decimal: 256 = 2^8);
In base 241: 36 = 3^6 (in decimal: 729 = 3^6);
In base 507: 2A = 2^A (in decimal: 1024 = 2^10);
In base 624: 55 = 5^5 (in decimal: 3125 = 5^5);
In base 2042: 2C = 2^C (in decimal: 4096 = 2^12);
In base 11: 3518 = 3^5 * 18 (in decimal: 4617 = 3^5 * 19).
See A384540 for more nontrivial examples.

Cf. A080670, A195264, A230625, A384540, A327399.

F(n,b) = my(f=factor(n), d=[]); for(i=1, #f~, d=concat(d, digits(f[i,1],b)); if(f[i,2]>1, d=concat(d, digits(f[i,2],b)))); fromdigits(d,b)
isA384537(n) = {
if(issquarefree(n), return(0)); my(f=factor(n), dr);
if(#f~ == 1, return(n > 4 && f[1,2] % f[1,1] == 0));
dr = if(f[#f~,2] == 1, f[#f~,1], f[#f~,2]);
fordiv(n - dr, b, if(b>=2 && F(n,b)==n, return(b))); return(0);
} \\ returns the (smallest) base to which n is a Davis number whenever possible

cp's OEIS Frontend

A259936 Number of ways to express the integer n as a product of its unitary divisors (A034444).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

Extensions

A327400 Number of factorizations of n that are constant or whose factors are relatively prime.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

A327695 Number of non-constant factorizations of n whose distinct factors are pairwise coprime.

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

Formula

A384537 Composite numbers that are equal to the concatenation of the primes and exponents in their prime factorizations in some bases.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

PARI