A336870 -id:A336870 - cp's OEIS frontend

A336500 Number of divisors d|n with distinct prime multiplicities such that the quotient n/d also has distinct prime multiplicities.

1, 2, 2, 3, 2, 2, 2, 4, 3, 2, 2, 4, 2, 2, 2, 5, 2, 4, 2, 4, 2, 2, 2, 6, 3, 2, 4, 4, 2, 0, 2, 6, 2, 2, 2, 6, 2, 2, 2, 6, 2, 0, 2, 4, 4, 2, 2, 8, 3, 4, 2, 4, 2, 6, 2, 6, 2, 2, 2, 4, 2, 2, 4, 7, 2, 0, 2, 4, 2, 0, 2, 8, 2, 2, 4, 4, 2, 0, 2, 8, 5, 2, 2, 4, 2, 2, 2

Offset: 1

Gus Wiseman, Aug 06 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 a(1) = 1 through a(16) = 5 divisors:
  1  1  1  1  1  2  1  1  1  2  1  1  1  2  3  1
     2  3  2  5  3  7  2  3  5 11  3 13  7  5  2
           4           4  9        4           4
                       8          12           8
                                              16

A336419 is the version for superprimorials.
A336568 gives positions of zeros.
A336869 is the restriction to factorials.
A007425 counts divisors of divisors.
A056924 counts divisors greater than their quotient.
A074206 counts chains of divisors from n to 1.
A130091 lists numbers with distinct prime exponents.
A181796 counts divisors with distinct prime multiplicities.
A336424 counts factorizations using A130091.
A336422 counts divisible pairs of divisors, both in A130091.
A327498 gives the maximum divisor with distinct prime multiplicities.
A336423 counts chains in A130091, with maximal version A336569.
A336568 gives numbers not a product of two elements of A130091.
A336571 counts divisor sets using A130091, with maximal version A336570.
Cf. A000005, A001055, A002033, A098859, A124010, A167865, A253249, A336870.

Table[Length[Select[Divisors[n],UnsameQ@@Last/@FactorInteger[#]&&UnsameQ@@Last/@FactorInteger[n/#]&]],{n,25}]

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

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

A336500 Number of divisors d|n with distinct prime multiplicities such that the quotient n/d also has distinct 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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Mathematica

Extensions

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.

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Author

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Comments

Examples

Crossrefs

Programs

Mathematica