A336617 -id:A336617 - cp's OEIS frontend

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

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

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!).

A336499 Irregular triangle read by rows where T(n,k) is the number of divisors of n! with distinct prime multiplicities and a total of k prime factors, counted with multiplicity.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 0, 1, 2, 1, 2, 1, 1, 3, 1, 3, 2, 0, 1, 3, 2, 5, 3, 3, 2, 1, 1, 4, 2, 7, 4, 4, 3, 2, 0, 1, 4, 2, 7, 4, 5, 7, 7, 6, 3, 2, 0, 1, 4, 2, 8, 8, 9, 10, 11, 11, 7, 8, 5, 2, 0, 1, 4, 3, 11, 8, 11, 16, 16, 15, 15, 15, 13, 9, 6, 3, 1, 1, 5, 3, 14, 10, 13, 21, 21, 20, 19, 21, 18, 13, 9, 5, 2, 0

Offset: 0

Gus Wiseman, Aug 03 2020

Row lengths are A022559(n) + 1.

			Triangle begins:
  1
  1
  1  1
  1  2  0
  1  2  1  2  1
  1  3  1  3  2  0
  1  3  2  5  3  3  2  1
  1  4  2  7  4  4  3  2  0
  1  4  2  7  4  5  7  7  6  3  2  0
  1  4  2  8  8  9 10 11 11  7  8  5  2  0
  1  4  3 11  8 11 16 16 15 15 15 13  9  6  3  1
  1  5  3 14 10 13 21 21 20 19 21 18 13  9  5  2  0
  1  5  3 14 10 14 25 23 27 24 30 28 28 25 20 16 11  5  2  0
Row n = 7 counts the following divisors:
  1  2  4  8   16  48   144  720   {}
     3  9  12  24  72   360  1008
     5     18  40  80   504
     7     20  56  112
           28
           45
           63

A000720 is column k = 1.
A022559 gives row lengths minus one.
A056172 appears to be column k = 2.
A336414 gives row sums.
A336420 is the version for superprimorials.
A336498 is the version counting all divisors.
A336865 is the generalization to non-factorials.
A336866 lists indices of rows with a final 1.
A336867 lists indices of rows with a final 0.
A336868 gives the final terms in each row.
A000110 counts divisors of superprimorials with distinct prime exponents.
A008302 counts divisors of superprimorials by number of prime factors.
A130091 lists numbers with distinct prime exponents.
A181796 counts divisors with distinct prime exponents.
A327498 gives the maximum divisor of n with distinct prime exponents.
Cf. A000005, A001222, A008278, A098859, A118914, A124010, A146291, A336422, A336500.
Factorial numbers: A000142, A002982, A027423, A048656, A048742, A054991, A071626, A336425, A336617.

Table[Length[Select[Divisors[n!],PrimeOmega[#]==k&&UnsameQ@@Last/@FactorInteger[#]&]],{n,0,6},{k,0,PrimeOmega[n!]}]

A336869 Number of divisors d of n! with distinct prime multiplicities such that the quotient n!/d also has distinct prime multiplicities.

Original entry on oeis.org

1, 1, 2, 2, 6, 4, 12, 8, 20, 28, 68, 40, 80, 0, 56, 160, 256, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Gus Wiseman, Aug 08 2020

nonn

Does this sequence converge to zero?
A number has distinct prime multiplicities iff its prime signature is strict.
From Edward Moody, Jan 18 2021: (Start)
a(n) = 0 for n >= 17.
Proof: 17 is the third Ramanujan prime (A104272). Therefore, for n>=17, there are at least three primes greater than n/2 and less than or equal to n. These primes must have exponent 1 in the prime factorization of n!, therefore, at least two of them must have exponent 1 in the prime factorization of either d or n!/d, so d and n!/d cannot both have distinct prime multiplicities. (End)

			The a(1) = 1 through a(7) = 8 divisors:
  1  1  2  1   3   1    5
     2  3  2   5   2    7
           3   24  5    45
           8   40  9    63
           12      16   80
           24      18   112
                   40   720
                   45   1008
                   80
                   144
                   360
                   720

A336419 is the version for superprimorials.
A336500 is the generalization to non-factorials.
A336616 is the maximum among these divisors.
A336617 is the minimum among these divisors.
A336939 has these row sums.
A000005 counts divisors.
A130091 lists numbers with distinct prime multiplicities.
A181796 counts divisors with distinct prime multiplicities.
A327498 gives the maximum divisor of n with distinct prime multiplicities.
A336414 counts divisors of n! with distinct prime multiplicities.
A336415 counts divisors of n! with equal prime multiplicities.
A336423 counts chains using A130091.
Cf. A098859, A118914, A124010, A327527, A336424, A336568, A336571, A336870.
Factorial numbers: A000142, A007489, A022559, A027423, A048656, A071626, A325272, A325273, A325617, A336416.

Table[Length[Select[Divisors[n!],UnsameQ@@Last/@FactorInteger[#]&&UnsameQ@@Last/@FactorInteger[n!/#]&]],{n,0,10}]

a(31)-a(80) from Edward Moody, Jan 19 2021

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!).

cp's OEIS Frontend

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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A336616 Maximum divisor of n! with distinct prime multiplicities.

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A336499 Irregular triangle read by rows where T(n,k) is the number of divisors of n! with distinct prime multiplicities and a total of k prime factors, counted with multiplicity.

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A336869 Number of divisors d of n! with distinct prime multiplicities such that the quotient n!/d also has distinct prime multiplicities.

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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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