A336869 -id:A336869 - 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}]

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

A336867 Numbers k such that k! does not have distinct prime multiplicities.

Original entry on oeis.org

3, 5, 7, 8, 9, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71

Offset: 1

Gus Wiseman, Aug 07 2020

nonn

The complement appears to be {0, 1, 2, 4, 6, 10}.
A number has distinct prime multiplicities iff its prime signature is strict.
From Chai Wah Wu, Aug 11 2020: (Start)
Theorem: the sequence consists of all nonnegative integers except 0, 1, 2, 4, 6, 10.
Proof: The cases k <= 31 follow from inspection. We show the case where k > 31.
Note that if p < q are successive primes, then for q <= m < 2p, the multiplicities of p and q in m! are both 1, i.e., m is a term.
Assume that p >= 29. Nagura showed that for all k >= 25 there exists a prime x such that  k < x < 1.2k. This implies that q < 1.2p and thus 2p > 1.666q, i.e. for q <= m < 1.666q, m is a term.
Again by Nagura's theorem, there exists a prime r < 1.2q. Thus intervals of the form [q, 1.666q] for q prime span all integers > 31 and the result is proved. QED
(End)

			The sequence of indexed factorials a(n)! together with their prime signatures begins:
                    6: (1,1)
                  120: (3,1,1)
                 5040: (4,2,1,1)
                40320: (7,2,1,1)
               362880: (7,4,1,1)
             39916800: (8,4,2,1,1)
            479001600: (10,5,2,1,1)
           6227020800: (10,5,2,1,1,1)
          87178291200: (11,5,2,2,1,1)
        1307674368000: (11,6,3,2,1,1)
       20922789888000: (15,6,3,2,1,1)
      355687428096000: (15,6,3,2,1,1,1)
     6402373705728000: (16,8,3,2,1,1,1)
   121645100408832000: (16,8,3,2,1,1,1,1)
  2432902008176640000: (18,8,4,2,1,1,1,1)

J. Nagura, On the interval containing at least one prime number, Proc. Japan Acad., 28 (1952) 177-181.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

A130092 is the generalization to non-factorials.
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.
A336866 counts partitions without distinct multiplicities.
Cf. A098859, A118914, A124010, A336423, A336424, A336500, A336568, A336571.
Factorial numbers: A000142, A007489, A022559, A027423, A048656, A048742, A071626, A325272, A325273, A325617, A336416, A336869.

Select[Range[0,100],!UnsameQ@@Last/@FactorInteger[#!]&]

From Chai Wah Wu, Aug 11 2020: (Start)
a(n) = 2*a(n-1) - a(n-2) for n > 7.
G.f.: x*(-x^6 + x^5 - x^3 - x + 3)/(x - 1)^2. (End)

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

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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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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A336867 Numbers k such that k! does not have distinct prime multiplicities.

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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.

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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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