A336616 -id:A336616 - cp's OEIS frontend

A336617 a(n) = n!/d where d = A336616(n) is the maximum divisor of n! with distinct prime multiplicities.

1, 1, 1, 2, 1, 3, 1, 5, 5, 5, 1, 7, 7, 77, 275, 11, 11, 143, 143, 2431, 2431, 2431, 221, 4199, 4199, 4199, 39083, 39083, 39083, 898909, 898909, 26068361, 26068361, 215441, 2141737, 2141737, 2141737, 66393847, 1009885357, 7953594143, 7953594143, 294282983291

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 maximum divisor of 13! with distinct prime multiplicities is 80870400, so a(13) = 13!/80870400 = 77.

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

A327499 is the non-factorial generalization, with quotient A327498.
A336414 counts these divisors.
A336616 is the maximum divisor d.
A336619 is the version for equal prime multiplicities.
A130091 lists numbers with distinct prime multiplicities.
A181796 counts divisors with distinct prime multiplicities.
A336415 counts divisors of n! with equal prime multiplicities.
Cf. A000005, A098859, A124010, A336424, A336500, A336568.
Factorial numbers: A000142, A007489, A022559, A027423, A048656, A048742, A071626, A325272, A325273, A325617, A327500, A336416, A336618.

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

a(n) = A327499(n!).

More terms from Jinyuan Wang, Jul 31 2020

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!).

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

A342091 a(n) is the least number k such that k! has n distinct exponents in its prime factorization.

Original entry on oeis.org

1, 2, 4, 6, 10, 15, 22, 33, 44, 55, 68, 85, 102, 119, 145, 174, 203, 232, 261, 296, 333, 370, 410, 451, 492, 533, 590, 656, 708, 767, 826, 885, 944, 1005, 1072, 1143, 1207, 1278, 1422, 1455, 1562, 1652, 1778, 1917, 2032, 2134, 2235, 2328, 2425, 2540, 2682, 2831, 2929

Offset: 0

Amiram Eldar, Feb 27 2021

nonn

After n=0, first differs from A073818 at n = 27.
a(n) is the least k such that A071625(k!) = A071626(k) = n.
Is this sequence strictly increasing?

			a(1) = 2 since 2! = 2^1 is the least factorial with a single exponent (1) in its prime factorization.
a(2) = 4 since 4! = 24 = 2^3 * 3^1 is the least factorial with 2 distinct exponents (1 and 3) in its prime factorization.
a(3) = 6 since 6! = 720 = 2^4 * 3^2 * 5^1 is the least factorial with 3 distinct exponents (1, 2 and 4) in its prime factorization.

Amiram Eldar, Table of n, a(n) for n = 0..2109 (terms below 10^7)
Paul Erdős, Miscellaneous problems in number theory, Proceedings of the Eleventh Manitoba Conference on Numerical Mathematics and Computing (Winnipeg, Man., 1981), Congr. Numer., Vol. 34 (1982), pp. 25-45.

Cf. A000142, A071625, A071626, A073818, A133924, A336616, A336619, A336867, A336868.

f[1] = 0; f[n_] := Length @ Union[FactorInteger[n!][[;; , 2]]]; seq[max_] := Module[{s = Table[0, {max}], n = 1, c = 0}, While[c < max, i = f[n] + 1; If[i <= max && s[[i]] == 0, c++; s[[i]] = n]; n++]; s]; seq[50]

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

A336617 a(n) = n!/d where d = A336616(n) is the 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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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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A342091 a(n) is the least number k such that k! has n distinct exponents in its prime factorization.

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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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