A327526 -id:A327526 - cp's OEIS frontend

A327527 Number of uniform divisors of n.

1, 2, 2, 3, 2, 4, 2, 4, 3, 4, 2, 5, 2, 4, 4, 5, 2, 5, 2, 5, 4, 4, 2, 6, 3, 4, 4, 5, 2, 8, 2, 6, 4, 4, 4, 7, 2, 4, 4, 6, 2, 8, 2, 5, 5, 4, 2, 7, 3, 5, 4, 5, 2, 6, 4, 6, 4, 4, 2, 9, 2, 4, 5, 7, 4, 8, 2, 5, 4, 8, 2, 8, 2, 4, 5, 5, 4, 8, 2, 7, 5, 4, 2, 9, 4, 4, 4, 6, 2, 9, 4, 5, 4, 4, 4, 8, 2, 5, 5, 7, 2, 8, 2, 6, 8

Offset: 1

Gus Wiseman, Sep 17 2019

nonn

A number is uniform if its prime multiplicities are all equal, meaning it is a power of a squarefree number. Uniform numbers are listed in A072774. The maximum uniform divisor of n is A327526(n).

			The uniform divisors of 40 are {1, 2, 4, 5, 8, 10}, so a(40) = 6.

Antti Karttunen, Table of n, a(n) for n = 1..65537
Gus Wiseman, Sequences counting and encoding certain classes of multisets.

See link for additional cross-references.
Cf. A000005, A000961, A005117, A006530, A007947, A071625, A112798.
Cf. A072774, A327526.
Cf. A001620, A013661, A306016, A368250, A368251.

Table[Length[Select[Divisors[n],SameQ@@Last/@FactorInteger[#]&]],{n,100}]
a[n_] := Module[{e = FactorInteger[n][[;; , 2]]}, 1 + Total[2^Accumulate[Count[e, #] & /@ Range[Max[e], 1, -1]] - 1]]; a[1] = 1; Array[a, 100] (* Amiram Eldar, Dec 19 2023 *)

isA072774(n) = { ispower(n, , &n); issquarefree(n); }; \\ From A072774
A327527(n) = sumdiv(n,d,isA072774(d)); \\ Antti Karttunen, Nov 13 2021

From Amiram Eldar, Dec 19 2023: (Start)
a(n) = A034444(n) + A368251(n).
Sum_{k=1..n} a(k) ~ (n/zeta(2)) * (log(n) + 2*gamma - 1 - 2*zeta'(2)/zeta(2) + c * zeta(2)), where gamma is Euler's constant (A001620) and c = A368250. (End)

Data section extended up to 105 terms by Antti Karttunen, Nov 13 2021

A336425 Number of ways to choose a divisor with distinct prime exponents of a divisor with distinct prime exponents of n!.

Original entry on oeis.org

1, 1, 3, 5, 24, 38, 132, 195, 570, 1588, 4193, 6086, 14561, 19232, 37142, 106479, 207291, 266871, 549726, 674330, 1465399, 3086598, 5939574, 7182133, 12324512, 28968994, 46819193, 82873443, 165205159, 196666406, 350397910, 406894074, 593725529, 1229814478, 1853300600, 4024414209, 6049714096, 6968090487, 9700557121, 16810076542, 26339337285

Offset: 0

Gus Wiseman, Aug 06 2020

nonn

			The a(4) = 24 divisors of divisors:
  1/1  2/1  3/1  4/1  8/1  12/1   24/1
       2/2  3/3  4/2  8/2  12/2   24/2
                 4/4  8/4  12/3   24/3
                      8/8  12/4   24/4
                           12/12  24/8
                                  24/12
                                  24/24

Max Alekseyev, Table of n, a(n) for n = 0..85

A336422 is the non-factorial generalization.
A130091 lists numbers with distinct prime exponents.
A181796 counts divisors with distinct prime exponents.
A327526 gives the maximum divisor of n with equal prime exponents.
A327498 gives the maximum divisor of n with distinct prime exponents.
A336414 counts divisors of n! with distinct prime exponents.
A336415 counts divisors of n! with equal prime exponents.
A336423 counts chains in A130091, with maximal version A336569.
Cf. A000005, A000110, A098859, A118914, A124010, A336424, A336500, A336568, A336570, A336571, A336865, A336866, A336869.
Factorial numbers: A000142, A022559, A027423, A048656, A048742, A071626, A325272, A325273, A325617, A327499, A336416, A336418, A336617.

strsigQ[n_]:=UnsameQ@@Last/@FactorInteger[n];
Table[Total[Cases[Divisors[n!],d_?strsigQ:>Count[Divisors[d],e_?strsigQ]]],{n,0,20}]

Terms a(21) onward from Max Alekseyev, Nov 07 2024

A336498 Irregular triangle read by rows where T(n,k) is the number of divisors of n! with k prime factors, counted with multiplicity.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 2, 2, 2, 1, 1, 3, 4, 4, 3, 1, 1, 3, 5, 6, 6, 5, 3, 1, 1, 4, 8, 11, 12, 11, 8, 4, 1, 1, 4, 8, 11, 12, 12, 12, 12, 11, 8, 4, 1, 1, 4, 8, 12, 16, 19, 20, 20, 19, 16, 12, 8, 4, 1, 1, 4, 9, 15, 21, 26, 29, 30, 30, 29, 26, 21, 15, 9, 4, 1

Offset: 0

Gus Wiseman, Aug 03 2020

Row n is row n! of A146291. Row lengths are A022559(n) + 1.

			Triangle begins:
  1
  1
  1  1
  1  2  1
  1  2  2  2  1
  1  3  4  4  3  1
  1  3  5  6  6  5  3  1
  1  4  8 11 12 11  8  4  1
  1  4  8 11 12 12 12 12 11  8  4  1
  1  4  8 12 16 19 20 20 19 16 12  8  4  1
Row n = 6 counts the following divisors:
  1  2   4   8  16   48  144  720
     3   6  12  24   72  240
     5   9  18  36   80  360
        10  20  40  120
        15  30  60  180
            45  90
Row n = 7 counts the following divisors:
  1  2   4    8   16   48   144   720  5040
     3   6   12   24   72   240  1008
     5   9   18   36   80   336  1680
     7  10   20   40  112   360  2520
        14   28   56  120   504
        15   30   60  168   560
        21   42   84  180   840
        35   45   90  252  1260
             63  126  280
             70  140  420
            105  210  630
                 315

A000720 is column k = 1.
A008302 is the version for superprimorials.
A022559 gives row lengths minus one.
A027423 gives row sums.
A146291 is the generalization to non-factorials.
A336499 is the restriction to divisors in A130091.
A000142 lists factorial numbers.
A336415 counts uniform divisors of n!.
Cf. A000005, A001222, A118914, A124010, A181796, A327526, A336420.
Factorial numbers: A002982, A007489, A048656, A054991, A071626, A325272, A325617, A336414, A336415, A336416, A336418.

Table[Length[Select[Divisors[n!],PrimeOmega[#]==k&]],{n,0,10},{k,0,PrimeOmega[n!]}]

A336616 Maximum divisor of n! with distinct prime multiplicities.

Original entry on oeis.org

1, 1, 2, 3, 24, 40, 720, 1008, 8064, 72576, 3628800, 5702400, 68428800, 80870400, 317011968, 118879488000, 1902071808000, 2487324672000, 44771844096000, 50039119872000, 1000782397440000, 21016430346240000, 5085976143790080000, 6156707963535360000

Offset: 0

Gus Wiseman, Jul 29 2020

nonn

A number's prime signature (row n of A124010) is the sequence of positive exponents in its prime factorization, so a number has distinct prime multiplicities iff all the exponents in its prime signature are distinct.

			The sequence of terms together with their prime signatures begins:
             1: ()
             1: ()
             2: (1)
             3: (1)
            24: (3,1)
            40: (3,1)
           720: (4,2,1)
          1008: (4,2,1)
          8064: (7,2,1)
         72576: (7,4,1)
       3628800: (8,4,2,1)
       5702400: (8,4,2,1)
      68428800: (10,5,2,1)
      80870400: (10,5,2,1)
     317011968: (11,5,2,1)
  118879488000: (11,6,3,2,1)

David A. Corneth, Table of n, a(n) for n = 0..589
Gus Wiseman, Sequences counting and encoding certain classes of multisets

A327498 is the version not restricted to factorials, with quotient A327499.
A336414 counts these divisors.
A336617 is the quotient n!/a(n).
A336618 is the version for equal prime multiplicities.
A130091 lists numbers with distinct prime multiplicities.
A181796 counts divisors with distinct prime multiplicities.
A327526 gives the maximum divisor of n with equal prime multiplicities.
A336415 counts divisors of n! with equal prime multiplicities.
Cf. A000005, A000110, A098859, A118914, A124010, A336424, A336500, A336568.
Factorial numbers: A000142, A007489, A022559, A027423, A048656, A048742, A071626, A325272, A325273, A325617, A336416.

Table[Max@@Select[Divisors[n!],UnsameQ@@Last/@If[#==1,{},FactorInteger[#]]&],{n,0,15}]

a(n) = { if(n < 2, return(1)); my(pr = primes(primepi(n)), res = pr[#pr]); for(i = 1, #pr, pr[i] = [pr[i], val(n, pr[i])] ); forstep(i = #pr, 2, -1, if(pr[i][2] < pr[i-1][2], res*=pr[i-1][1]^pr[i-1][2] ) ); res }
val(n, p) = my(r=0); while(n, r+=n\=p); r \\ David A. Corneth, Aug 25 2020

a(n) = A327498(n!).

A336618 Maximum divisor of n! with equal prime multiplicities.

Original entry on oeis.org

1, 1, 2, 6, 8, 30, 36, 210, 210, 1296, 1296, 2310, 7776, 30030, 44100, 46656, 46656, 510510, 1679616, 9699690, 9699690, 10077696, 10077696, 223092870, 223092870, 729000000, 901800900, 13060694016, 13060694016, 13060694016, 78364164096, 200560490130

Offset: 0

Gus Wiseman, Jul 30 2020

nonn

A number has equal prime multiplicities iff it is a power of a squarefree number. We call such numbers uniform, so a(n) is the maximum uniform divisor of n!.

			The sequence of terms together with their prime signatures begins:
       1: ()
       1: ()
       2: (1)
       6: (1,1)
       8: (3)
      30: (1,1,1)
      36: (2,2)
     210: (1,1,1,1)
     210: (1,1,1,1)
    1296: (4,4)
    1296: (4,4)
    2310: (1,1,1,1,1)
    7776: (5,5)
   30030: (1,1,1,1,1,1)
   44100: (2,2,2,2)

Amiram Eldar, Table of n, a(n) for n = 0..1000
Gus Wiseman, Sequences counting and encoding certain classes of multisets.

A327526 is the non-factorial generalization, with quotient A327528.
A336415 counts these divisors.
A336616 is the version for distinct prime multiplicities.
A336619 is the quotient n!/a(n).
A047966 counts uniform partitions.
A071625 counts distinct prime multiplicities.
A072774 lists uniform numbers.
A130091 lists numbers with distinct prime multiplicities.
A181796 counts divisors with distinct prime multiplicities.
A319269 counts uniform factorizations.
A327524 counts factorizations of uniform numbers into uniform numbers.
A327527 counts uniform divisors.
Cf. A000005, A001222, A001597, A007916, A098859, A124010, A182853, A327498.
Factorial numbers: A000142, A007489, A022559, A027423, A048656, A071626, A108731, A325272, A325273, A325617, A336414, A336416.

Table[Max@@Select[Divisors[n!],SameQ@@Last/@FactorInteger[#]&],{n,0,15}]

a(n) = A327526(n!).

A327528 Quotient of n over the maximum uniform divisor of n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 3, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 2, 3, 1, 1, 3, 1, 2, 1, 2, 1, 2, 1, 4, 1, 1, 1, 2, 1, 1, 3, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 3, 2, 1, 1, 1, 5, 1, 1, 1, 2, 1, 1, 1

Offset: 1

Gus Wiseman, Sep 17 2019

nonn

Requires A071625(n) steps to reach 1, the only fixed point.

A number is uniform if its prime multiplicities are all equal, meaning it is a power of a squarefree number. Uniform numbers are listed in A072774. The maximum uniform divisor of n is A327526(n).

			The uniform divisors of 40 are {1, 2, 4, 5, 8, 10}, so a(40) = 40/10 = 4.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Gus Wiseman, Sequences counting and encoding certain classes of multisets.

See link for additional cross-references.

Cf. A000005, A000961, A005117, A006530, A007947, A071625, A072774, A112798, A327526.

Table[n/Max[Select[Divisors[n],SameQ@@Last/@FactorInteger[#]&]],{n,100}]
a[n_] := Module[{f = FactorInteger[n], p, e}, p = f[[;; , 1]]; e = f[[;; , 2]]; n / Max@ Table[(Times @@ p[[Position[e, ?(# >= k &)] // Flatten]])^k, {k, Union[e]}]]; Array[a, 100] (* _Amiram Eldar, Dec 19 2023 *)

a(n) = n / A327526(n). - Amiram Eldar, Dec 19 2023

A336619 a(n) = n!/d where d is the maximum divisor of n! with equal prime exponents.

Original entry on oeis.org

1, 1, 1, 1, 3, 4, 20, 24, 192, 280, 2800, 17280, 61600, 207360, 1976832, 28028000, 448448000, 696729600, 3811808000, 12541132800, 250822656000, 5069704640000, 111533502080000, 115880067072000, 2781121609728000, 21277380032004096, 447206762741760000

Offset: 0

Gus Wiseman, Jul 30 2020

nonn

A number has equal prime exponents iff it is a power of a squarefree number. We call such numbers uniform, so a(n) is n! divided by the maximum uniform divisor of n!.

After the first three terms, is this sequence strictly increasing?

			The sequence of terms together with their prime signatures begins:
           1: ()
           1: ()
           1: ()
           1: ()
           3: (1)
           4: (2)
          20: (2,1)
          24: (3,1)
         192: (6,1)
         280: (3,1,1)
        2800: (4,2,1)
       17280: (7,3,1)
       61600: (5,2,1,1)
      207360: (9,4,1)
     1976832: (9,3,1,1)
    28028000: (5,3,2,1,1)
   448448000: (9,3,2,1,1)
   696729600: (14,5,2,1)
  3811808000: (8,3,2,1,1,1)

Amiram Eldar, Table of n, a(n) for n = 0..500
Gus Wiseman, Sequences counting and encoding certain classes of multisets.

A327528 is the non-factorial generalization, with quotient A327526.
A336415 counts these divisors.
A336617 is the version for distinct prime exponents.
A336618 is the quotient n!/a(n).
A047966 counts uniform partitions.
A071625 counts distinct prime exponents.
A072774 gives Heinz numbers of uniform partitions, with nonprime terms A182853.
A130091 lists numbers with distinct prime exponents.
A181796 counts divisors with distinct prime exponents.
A319269 counts uniform factorizations.
A327524 counts factorizations of uniform numbers into uniform numbers.
A327527 counts uniform divisors.
Cf. A000005, A001222, A001597, A007916, A098859, A118914, A124010, A327498, A327499, A336616.
Factorial numbers: A000142, A007489, A022559, A027423, A048656, A071626, A108731, A325272, A325273, A325617, A336414, A336416.

Table[n!/Max@@Select[Divisors[n!],SameQ@@Last/@FactorInteger[#]&],{n,0,15}]

a(n) = n!/A336618(n) = n!/A327526(n!).

A375931 The product of the prime powers in the prime factorization of n that have an exponent that is equal to the maximum exponent in this factorization.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 4, 13, 14, 15, 16, 17, 9, 19, 4, 21, 22, 23, 8, 25, 26, 27, 4, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 8, 41, 42, 43, 4, 9, 46, 47, 16, 49, 25, 51, 4, 53, 27, 55, 8, 57, 58, 59, 4, 61, 62, 9, 64, 65, 66, 67, 4, 69, 70, 71

Offset: 1

Amiram Eldar, Sep 03 2024

Differs from A327526 at n = 12, 20, 28, 40, 44, 45, ... .
Each positive number appears in this sequence either once or infinitely many times:
1. If m is squarefree then the only solution to a(x) = m is x = m.
2. If m = s^k is a power of a squarefree number s with k >= 2, then x = m * i is a solution to a(x) = m for all numbers i that are k-free numbers (i.e., having exponents in their prime factorizations that are all less than k) that are coprime to m.

			180 = 2^2 * 3^2 * 5, and the maximum exponent in the prime factorization of 180 is 2, which is the exponent of its prime factors 2 and 3. Therefore a(180) = 2^2 * 3^2 = (2*3)^2 = 36.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A001221 (omega), A001222 (Omega), A005117, A051903, A060818, A072774, A104141, A261969, A327526, A356862, A362611, A375932.

a[n_] := Module[{f = FactorInteger[n], p, e, i, m}, p = f[[;; , 1]]; e = f[[;; , 2]]; m = Max[e]; i = Position[e, m] // Flatten; (Times @@ p[[i]])^m]; Array[a, 100]

a(n) = {my(f = factor(n), p = f[,1], e = f[,2], m); if(n == 1, 1, m = vecmax(e); prod(i = 1, #p, if(e[i] == m, p[i], 1))^m);}

If n = Product_{i} p_i^e_i (where p_i are distinct primes) then a(n) = Product_{i} p_i^(e_i * [e_i = max_{j} e_j]), where [] is the Iverson bracket.
a(n) = A261969(n)^A051903(n).
a(n) = n / A375932(n).
a(n) = n if and only if n is a power of a squarefree number (A072774).
A051903(a(n)) = A051903(n).
omega(a(n)) = A362611(n).
omega(a(n)) = 1 if and only if n is in A356862.
Omega(a(n)) = A051903(n) * A362611(n).
a(n!) = A060818(n) for n != 3.
Sum_{k=1..n} a(k) ~ c * n^2, where c = 3/Pi^2 = 0.303963... (A104141).

cp's OEIS Frontend

A327527 Number of uniform divisors of n.

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A336425 Number of ways to choose a divisor with distinct prime exponents of a divisor with distinct prime exponents of n!.

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A336498 Irregular triangle read by rows where T(n,k) is the number of divisors of n! with k prime factors, counted with multiplicity.

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A336616 Maximum divisor of n! with distinct prime multiplicities.

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A336618 Maximum divisor of n! with equal prime multiplicities.

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A327528 Quotient of n over the maximum uniform divisor of n.

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A336619 a(n) = n!/d where d is the maximum divisor of n! with equal prime exponents.

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A375931 The product of the prime powers in the prime factorization of n that have an exponent that is equal to the maximum exponent in this factorization.

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