A336865 -id:A336865 - 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

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!]}]

A336870 Irregular triangle read by rows where T(n,k) is the number of divisors d of the superprimorial A006939(n) with k prime factors (counting multiplicity), such that d and A006939(n)/d both have distinct prime multiplicities.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 4, 4, 2, 4, 4, 1, 1, 1, 1, 1, 1, 4, 4, 7, 7, 7, 7, 7, 7, 4, 4, 1, 1, 1, 1, 1, 1, 4, 4, 7, 18, 10, 10, 15, 21, 21, 15, 10, 10, 18, 7, 4, 4, 1, 1, 1, 1, 1, 1, 4, 4, 7, 18, 23, 15, 20, 37, 35, 40, 46, 32, 46, 40, 35, 37, 20, 15, 23, 18, 7, 4, 4, 1, 1, 1

Offset: 0

Gus Wiseman, Aug 08 2020

Are there any zeros (cf. A336939)?

A number has distinct prime multiplicities iff its prime signature is strict.

			Triangle begins:
  1
  1 1
  1 1 1 1
  1 1 1 4 1 1 1
  1 1 1 4 4 2 4 4 1 1 1
  1 1 1 4 4 7 7 7 7 7 7 4 4 1 1 1
Row n = 4 counts the following divisors:
  1  7  25   27   16  112   400   432  3024  10800  75600
             63   54  675  1350  1008
             75   56       1400  1200
            175  189       4725  2800

A000124 gives row lengths.
A336419 gives row sums.
A336500 is the generalization to all positive integers.
A336939 is the version for factorials.
A000005 counts divisors.
A000110 counts divisors of superprimorials with distinct prime multiplicities.
A000142 lists factorials.
A000325 counts divisors of superprimorials with equal prime multiplicities.
A006939 lists superprimorials.
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.
A336423 counts chains using A130091.
Cf. A022559, A027423, A048742, A071626, A098859, A118914, A124010, A336414, A336424, A336568, A336571, A336865, A336869.

chern[n_]:=Product[Prime[i]^(n-i+1),{i,n}];
Table[Length[Select[Divisors[chern[n]],UnsameQ@@Last/@FactorInteger[#]&&UnsameQ@@Last/@FactorInteger[chern[n]/#]&&PrimeOmega[#]==k&]],{n,0,6},{k,0,PrimeOmega[chern[n]]}]

A336939 Irregular triangle read by rows where T(n,k) is the number of divisors d of n! with k prime factors (counting multiplicity), such that both d and n!/d have distinct prime multiplicities.

Original entry on oeis.org

1, 1, 1, 1, 0, 2, 0, 1, 2, 0, 2, 1, 0, 2, 0, 0, 2, 0, 1, 2, 1, 2, 2, 1, 2, 1, 0, 2, 0, 2, 0, 2, 0, 2, 0, 0, 2, 0, 4, 2, 2, 2, 2, 4, 0, 2, 0, 0, 2, 0, 4, 0, 4, 4, 4, 4, 0, 4, 0, 2, 0, 1, 3, 2, 6, 4, 5, 7, 6, 6, 7, 5, 4, 6, 2, 3, 1, 0, 2, 0, 4, 2, 2, 4, 4, 4, 4, 4, 2, 2, 4, 0, 2, 0

Offset: 0

Gus Wiseman, Aug 08 2020

A number has distinct prime multiplicities iff its prime signature is strict.

			Triangle begins:
  1
  1
  1 1
  0 2 0
  1 2 0 2 1
  0 2 0 0 2 0
  1 2 1 2 2 1 2 1
  0 2 0 2 0 2 0 2 0
  0 2 0 4 2 2 2 2 4 0 2 0
  0 2 0 4 0 4 4 4 4 0 4 0 2 0
  1 3 2 6 4 5 7 6 6 7 5 4 6 2 3 1
Row n = 8 counts the following divisors (empty columns shown as dots):
  .  5  .  20  40   80  360   720   640  .  5760  .
     7     28  56  112  504  1008   896     8064
           45                      1440
           63                      2016

A022559 gives row lengths minus one.
A336500 is the generalization to all positive integers.
A336868 gives the first (also last) column.
A336869 gives row sums.
A336870 is the version for superprimorials.
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 factorials with distinct prime multiplicities.
A336415 counts divisors of factorials with equal prime multiplicities.
A336423 counts chains using A130091.
Cf. A006939, A098859, A118914, A124010, A336419, A336424, A336571, A336865.
Factorial numbers: A000142, A007489, A022559, A027423, A048656, A048742, A071626, A325272, A325273, A325617, A336414, A336415, A336416.

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

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Extensions

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.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A336870 Irregular triangle read by rows where T(n,k) is the number of divisors d of the superprimorial A006939(n) with k prime factors (counting multiplicity), such that d and A006939(n)/d both have distinct prime multiplicities.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A336939 Irregular triangle read by rows where T(n,k) is the number of divisors d of n! with k prime factors (counting multiplicity), such that both d and n!/d have distinct prime multiplicities.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica