A336416 -id:A336416 - cp's OEIS frontend

A376598 Points of nonzero curvature in the sequence of prime-powers inclusive (A000961).

4, 5, 7, 9, 10, 11, 12, 13, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 28, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 62, 63, 64, 65, 66, 68, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 80

Offset: 1

Gus Wiseman, Oct 05 2024

nonn

These are points at which the second differences (A376596) are nonzero.

Inclusive means 1 is a prime-power. For the exclusive version, subtract 1 from all terms.

			The prime-powers inclusive (A000961) are:
  1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23, 25, 27, 29, 31, 32, 37, 41, 43, ...
with first differences (A057820):
  1, 1, 1, 1, 2, 1, 1, 2, 2, 3, 1, 2, 4, 2, 2, 2, 2, 1, 5, 4, 2, 4, 2, 4, 6, 2, ...
with first differences (A376596):
  0, 0, 0, 1, -1, 0, 1, 0, 1, -2, 1, 2, -2, 0, 0, 0, -1, 4, -1, -2, 2, -2, 2, 2, ...
with nonzeros at (A376598):
  4, 5, 7, 9, 10, 11, 12, 13, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 28, 29, 30, ...

Gus Wiseman, Points of nonzero curvature in the prime-powers inclusive.

The first differences were A057820, see also A376340.
First differences are A376309.
These are the nonzeros of A376596 (sorted firsts A376653, exclusive A376654).
The complement is A376597.
A000961 lists prime-powers inclusive, exclusive A246655.
A001597 lists perfect-powers, complement A007916.
A023893 and A023894 count integer partitions into prime-powers, factorizations A000688.
`A064113 lists positions of adjacent equal prime gaps.
For prime-powers inclusive: A057820 (first differences), A376597 (second differences), A376597 (inflections and undulations), A376653 (sorted firsts in second differences).
For points of nonzero curvature: A333214 (prime), A376603 (composite), A376589 (non-perfect-power), A376592 (squarefree), A376595 (nonsquarefree), A376601 (non-prime-power).
Cf. A025475, A053707, A069623, A093555, A174965, A251092, A333254, A336416, A361102.

Join@@Position[Sign[Differences[Select[Range[1000], #==1||PrimePowerQ[#]&],2]],1|-1]

A376654 Sorted positions of first appearances in the second differences of consecutive prime-powers exclusive (A246655).

Original entry on oeis.org

3, 4, 9, 11, 17, 24, 44, 46, 47, 59, 67, 68, 70, 79, 117, 120, 177, 178, 198, 205, 206, 215, 243, 244, 303, 324, 326, 401, 465, 483, 604, 800, 879, 938, 1032, 1054, 1076, 1233, 1280, 1720, 1889, 1890, 1905, 1939, 1959, 1961, 2256, 2289, 2409, 2879, 3149

Offset: 1

Gus Wiseman, Oct 06 2024

nonn

			The prime-powers exclusive (A246655) are:
  2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23, 25, 27, 29, 31, 32, 37, 41, 43, 47, ...
with first differences (A057820 except first term) :
  1, 1, 1, 2, 1, 1, 2, 2, 3, 1, 2, 4, 2, 2, 2, 2, 1, 5, 4, 2, 4, 2, 4, 6, 2, 3, 3, ...
with first differences (A376596 except first term):
  0, 0, 1, -1, 0, 1, 0, 1, -2, 1, 2, -2, 0, 0, 0, -1, 4, -1, -2, 2, -2, 2, 2, -4, ...
with first appearances (A376654):
  1, 3, 4, 9, 11, 17, 24, 44, 46, 47, 59, 67, 68, 70, 79, 117, 120, 177, 178, 198, ...

For first differences we have A376340.
These are the sorted positions of first appearances in A376596 except first term.
The inclusive version is a(n) + 1 = A376653(n), except first term.
For squarefree instead of prime-power we have A376655.
A000961 lists prime-powers inclusive, exclusive A246655.
A001597 lists perfect-powers, complement A007916.
A023893 and A023894 count integer partitions into prime-powers, factorizations A000688.
For prime-powers inclusive: A057820 (first differences), A376597 (inflections and undulations), A376598 (nonzero curvature).
For second differences: A036263 (prime), A073445 (composite), A376559 (perfect-power), A376562 (non-perfect-power), A376590 (squarefree), A376593 (nonsquarefree), A376599 (non-prime-power).
Cf. A025475, A053707, A057820, A064113, A069623, A093555, A174965, A333254, A336416, A361102.

q=Differences[Select[Range[1000],PrimePowerQ[#]&],2];
Select[Range[Length[q]],!MemberQ[Take[q,#-1],q[[#]]]&]

A337074 Number of strict chains of divisors in A130091 (numbers with distinct prime multiplicities), starting with n!.

Original entry on oeis.org

1, 1, 2, 0, 28, 0, 768, 0, 0, 0, 42155360, 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, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Gus Wiseman, Aug 16 2020

nonn

Support appears to be {0, 1, 2, 4, 6, 10}.

			The a(4) = 28 chains:
  24  24/1   24/2/1   24/4/2/1   24/8/4/2/1
      24/2   24/3/1   24/8/2/1   24/12/4/2/1
      24/3   24/4/1   24/8/4/1
      24/4   24/4/2   24/8/4/2
      24/8   24/8/1   24/12/2/1
      24/12  24/8/2   24/12/3/1
             24/8/4   24/12/4/1
             24/12/1  24/12/4/2
             24/12/2
             24/12/3
             24/12/4

A336867 is the complement of the support.
A336868 is the characteristic function (image under A057427).
A336942 is half the version for superprimorials (n > 1).
A337071 does not require distinct prime multiplicities.
A337104 is the case of chains ending with 1.
A000005 counts divisors.
A000142 lists factorial numbers.
A027423 counts divisors of factorial numbers.
A067824 counts chains of divisors starting with n.
A074206 counts chains of divisors from n to 1.
A076716 counts factorizations of factorial numbers.
A130091 lists numbers with distinct prime multiplicities.
A181796 counts divisors with distinct prime multiplicities.
A253249 counts chains of divisors.
A327498 gives the maximum divisor 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, with maximal case A336569.
A336571 counts chains of divisors 1 < d < n using A130091.
Cf. A000110, A001055, A002033, A007489, A022559, A048656, A071626, A098859, A124010, A167865, A325617, A336416, A336424.

chnsc[n_]:=If[!UnsameQ@@Last/@FactorInteger[n],{},If[n==1,{{1}},Prepend[Join@@Table[Prepend[#,n]&/@chnsc[d],{d,Most[Divisors[n]]}],{n}]]];
Table[Length[chnsc[n!]],{n,0,6}]

a(n) = 2*A337104(n) = 2*A336423(n!) for n > 1.

A376561 Points of downward concavity in the sequence of perfect-powers (A001597).

Original entry on oeis.org

2, 5, 7, 13, 14, 18, 19, 21, 24, 25, 29, 30, 39, 40, 45, 51, 52, 56, 59, 66, 70, 71, 74, 87, 94, 101, 102, 108, 110, 112, 113, 119, 127, 135, 143, 144, 156, 157, 160, 161, 169, 178, 187, 196, 205, 206, 215, 224, 225, 234, 244, 263, 273, 283, 284, 293, 294, 304

Offset: 1

Gus Wiseman, Sep 30 2024

nonn

These are points at which the second differences are negative.
Perfect-powers (A001597) are numbers with a proper integer root.
Note that, for some sources, downward concavity is positive curvature.
From Robert Israel, Oct 31 2024: (Start)
The first case of two consecutive numbers in the sequence is a(4) = 13 and a(5) = 14.
The first case of three consecutive numbers is a(293) = 2735, a(294) = 2736, a(295) = 2737.
The first case of four consecutive numbers, if it exists, involves a(k) with k > 69755. (End)

			The perfect powers (A001597) are:
  1, 4, 8, 9, 16, 25, 27, 32, 36, 49, 64, 81, 100, 121, 125, 128, 144, 169, 196, ...
with first differences (A053289):
  3, 4, 1, 7, 9, 2, 5, 4, 13, 15, 17, 19, 21, 4, 3, 16, 25, 27, 20, 9, 18, 13, 33, ...
with first differences (A376559):
  1, -3, 6, 2, -7, 3, -1, 9, 2, 2, 2, 2, -17, -1, 13, 9, 2, -7, -11, 9, -5, 20, 2, ...
with negative positions (A376561):
  2, 5, 7, 13, 14, 18, 19, 21, 24, 25, 29, 30, 39, 40, 45, 51, 52, 56, 59, 66, 70, ...

Robert Israel, Table of n, a(n) for n = 1..10000
Gus Wiseman, Points of downward concavity in the perfect-powers.

The version for A000002 is A025505, complement A022297. See also A054354, A376604.
For first differences we have A053289, union A023055, firsts A376268, A376519.
For primes instead of perfect-powers we have A258026.
For upward concavity we have A376560 (probably the complement).
A000961 lists the prime-powers inclusive, exclusive A246655.
A001597 lists the perfect-powers.
A007916 lists the non-perfect-powers.
A112344 counts partitions into perfect-powers, factorizations A294068.
A333254 gives run-lengths of differences between consecutive primes.
Second differences: A036263 (prime), A073445 (composite), A376559 (perfect-power), A376562 (non-perfect-power), A376590 (squarefree), A376593 (nonsquarefree), A376596 (prime-power), A376599 (non-prime-power).
Cf. A025475, A045542, A052410, A054819, A064113, A069623, A174965, A216765, A251092, A336416, A336417, A375702.

N:= 10^6: # to use perfect powers <= N
P:= {seq(seq(i^m,i=2..floor(N^(1/m))), m=2 .. ilog2(N))}: nP:= nops(P):
P:= sort(convert(P,list)):
select(i -> 2*P[i] > P[i-1]+P[i+1], [$2..nP-1]); # Robert Israel, Oct 31 2024

perpowQ[n_]:=n==1||GCD@@FactorInteger[n][[All,2]]>1;
Join@@Position[Sign[Differences[Select[Range[1000],perpowQ],2]],-1]

A376588 Inflection and undulation points in the sequence of non-perfect-powers (A007916).

Original entry on oeis.org

3, 6, 7, 8, 9, 12, 13, 14, 15, 16, 17, 19, 21, 22, 25, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84

Offset: 1

Gus Wiseman, Oct 03 2024

nonn

These are points at which the second differences (A376562) are zero.

Non-perfect-powers (A007916) are numbers without a proper integer root.

			The non-perfect powers (A007916) are:
  2, 3, 5, 6, 7, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 24, 26, 28, ...
with first differences (A375706):
  1, 2, 1, 1, 3, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 2, 1, 1, ...
with first differences (A376562):
  1, -1, 0, 2, -2, 0, 0, 0, 0, 1, -1, 0, 0, 0, 0, 0, 0, 1, 0, -1, 0, 0, 1, -1, 0, ...
with zeros at (A376588):
  3, 6, 7, 8, 9, 12, 13, 14, 15, 16, 17, 19, 21, 22, 25, 28, 29, 30, 31, 32, 33, ...

Gus Wiseman, Inflection and undulation points in the non-perfect-powers.

The version for A000002 is empty.
For first differences we had A375706, ones A375740, complement A375714.
Positions of zeros in A376562, complement A376589.
Runs of non-perfect-powers:
- length: A375702 = A053289(n+1) - 1
- first: A375703 (same as A216765 with 2 exceptions)
- last: A375704 (same as A045542 with 8 removed)
- sum: A375705
A000961 lists prime-powers inclusive, exclusive A246655.
A007916 lists non-perfect-powers, complement A001597.
A305631 counts integer partitions into non-perfect-powers, factorizations A322452.
A333254 gives run-lengths of differences between consecutive primes.
For non-perfect-powers: A375706 (first differences), A376562 (second differences), A376589 (nonzero curvature).
For second differences: A064113 (prime), A376602 (composite), {} (perfect-power), A376591 (squarefree), A376594 (nonsquarefree), A376597 (prime-power inclusive), A376600 (non-prime-power inclusive).
Cf. A025475, A052410, A053707, A069623, A073445, A093555, A174965, A294068, A336416, A361102, A376599, A376590.

radQ[n_]:=n>1&&GCD@@Last/@FactorInteger[n]==1;
Join@@Position[Differences[Select[Range[100],radQ],2],0]

A376589 Points of nonzero curvature in the sequence of non-perfect-powers (A007916).

Original entry on oeis.org

1, 2, 4, 5, 10, 11, 18, 20, 23, 24, 26, 27, 38, 39, 52, 53, 68, 69, 86, 87, 106, 107, 109, 110, 111, 112, 126, 127, 150, 151, 176, 177, 195, 196, 203, 204, 220, 221, 232, 233, 264, 265, 298, 299, 316, 317, 333, 334, 371, 372, 411, 412, 453, 454, 480, 481, 496

Offset: 1

Gus Wiseman, Oct 03 2024

nonn

These are points at which the second differences (A376562) are nonzero.

Non-perfect-powers (A007916) are numbers without a proper integer root.

			The non-perfect powers (A007916) are:
  2, 3, 5, 6, 7, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 24, 26, 28, ...
with first differences (A375706):
  1, 2, 1, 1, 3, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 2, 1, 1, ...
with first differences (A376562):
  1, -1, 0, 2, -2, 0, 0, 0, 0, 1, -1, 0, 0, 0, 0, 0, 0, 1, 0, -1, 0, 0, 1, -1, 0, ...
with nonzeros at (A376589):
  1, 2, 4, 5, 10, 11, 18, 20, 23, 24, 26, 27, 38, 39, 52, 53, 68, 69, 86, 87, ...

Gus Wiseman, Points of nonzero curvature in the non-perfect-powers.

For first differences we had A375706, ones A375740, complement A375714.
These are the positions of nonzeros in A376562, complement A376588.
Runs of non-perfect-powers:
- length: A375702 = A053289(n+1) - 1
- first: A375703 (same as A216765 with 2 exceptions)
- last: A375704 (same as A045542 with 8 removed)
- sum: A375705
A000961 lists prime-powers inclusive, exclusive A246655.
A007916 lists non-perfect-powers, complement A001597.
A305631 counts integer partitions into non-perfect-powers, factorizations A322452.
For non-perfect-powers: A375706 (first differences), A376562 (second differences), A376588 (inflection and undulation points).
For second differences: A064113 (prime), A376602 (composite), A376591 (squarefree), A376594 (nonsquarefree), A376597 (prime-power), A376600 (non-prime-power).
Cf. A025475, A052410, A053707, A069623, A073445, A093555, A174965, A182853, A294068, A333254, A336416, A376599.

radQ[n_]:=n>1&&GCD@@Last/@FactorInteger[n]==1;
Join@@Position[Sign[Differences[Select[Range[1000],radQ],2]],1|-1]

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!).

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

Original entry on oeis.org

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

cp's OEIS Frontend

A376598 Points of nonzero curvature in the sequence of prime-powers inclusive (A000961).

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A376654 Sorted positions of first appearances in the second differences of consecutive prime-powers exclusive (A246655).

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A337074 Number of strict chains of divisors in A130091 (numbers with distinct prime multiplicities), starting with n!.

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Formula

A376561 Points of downward concavity in the sequence of perfect-powers (A001597).

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Maple

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A376588 Inflection and undulation points in the sequence of non-perfect-powers (A007916).

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A376589 Points of nonzero curvature in the sequence of non-perfect-powers (A007916).

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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.