A336568 -id:A336568

A336424

Number of factorizations of n where each factor belongs to A130091 (numbers with distinct prime multiplicities).

+10
25

1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 3, 1, 1, 1, 5, 1, 3, 1, 3, 1, 1, 1, 5, 2, 1, 3, 3, 1, 1, 1, 7, 1, 1, 1, 6, 1, 1, 1, 5, 1, 1, 1, 3, 3, 1, 1, 9, 2, 3, 1, 3, 1, 5, 1, 5, 1, 1, 1, 4, 1, 1, 3, 11, 1, 1, 1, 3, 1, 1, 1, 11, 1, 1, 3, 3, 1, 1, 1, 9, 5, 1, 1, 4, 1, 1

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,4

COMMENTS

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.

LINKS

Table of n, a(n) for n=1..86.

EXAMPLE

The a(n) factorizations for n = 2, 4, 8, 60, 16, 36, 32, 48:

2 4 8 5*12 16 4*9 32 48

2*2 2*4 3*20 4*4 3*12 4*8 4*12

2*2*2 3*4*5 2*8 3*3*4 2*16 3*16

2*2*3*5 2*2*4 2*18 2*4*4 3*4*4

2*2*2*2 2*2*9 2*2*8 2*24

2*2*3*3 2*2*2*4 2*3*8

2*2*2*2*2 2*2*12

2*2*3*4

2*2*2*2*3

MATHEMATICA

facsusing[s_, n_]:=If[n<=1, {{}}, Join@@Table[Map[Prepend[#, d]&, Select[facsusing[Select[s, Divisible[n/d, #]&], n/d], Min@@#>=d&]], {d, Select[s, Divisible[n, #]&]}]];

Table[Length[facsusing[Select[Range[2, n], UnsameQ@@Last/@FactorInteger[#]&], n]], {n, 100}]

CROSSREFS

A327523 is the case when n is restricted to belong to A130091 also.

A001055 counts factorizations.

A007425 counts divisors of divisors.

A045778 counts strict factorizations.

A074206 counts ordered factorizations.

A130091 lists numbers with distinct prime multiplicities.

A181796 counts divisors with distinct prime multiplicities.

A253249 counts nonempty chains of divisors.

A281116 counts factorizations with no common divisor.

A302696 lists numbers whose prime indices are pairwise coprime.

A305149 counts stable factorizations.

A320439 counts factorizations using A289509.

A327498 gives the maximum divisor with distinct prime multiplicities.

A336500 counts divisors of n in A130091 with quotient also in A130091.

A336568 = not a product of two numbers with distinct prime multiplicities.

A336569 counts maximal chains of elements of A130091.

A337256 counts chains of divisors.

Cf. A071625, A080688, A098859, A118914, A124010, A167865, A294068, A303707, A305150, A322453, A336420, A336570, A336571.

KEYWORD

nonn

AUTHOR

Gus Wiseman, Aug 03 2020

STATUS

approved

A336414

Number of divisors of n! with distinct prime multiplicities.

+10
24

1, 1, 2, 3, 7, 10, 20, 27, 48, 86, 147, 195, 311, 390, 595, 1031, 1459, 1791, 2637, 3134, 4747, 7312, 10766, 12633, 16785, 26377, 36142, 48931, 71144, 82591, 112308, 128023, 155523, 231049, 304326, 459203, 568095, 642446, 812245, 1137063, 1441067, 1612998, 2193307, 2429362

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

COMMENTS

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

LINKS

Chai Wah Wu, Table of n, a(n) for n = 0..6245 (n = 0..94 from David A. Corneth)

FORMULA

a(n) = A181796(n!).

EXAMPLE

The first and second columns below are the a(6) = 20 counted divisors of 6! together with their prime signatures. The third column shows the A000005(6!) - a(6) = 10 remaining divisors.

1: () 20: (2,1) | 6: (1,1)

2: (1) 24: (3,1) | 10: (1,1)

3: (1) 40: (3,1) | 15: (1,1)

4: (2) 45: (2,1) | 30: (1,1,1)

5: (1) 48: (4,1) | 36: (2,2)

8: (3) 72: (3,2) | 60: (2,1,1)

9: (2) 80: (4,1) | 90: (1,2,1)

12: (2,1) 144: (4,2) | 120: (3,1,1)

16: (4) 360: (3,2,1) | 180: (2,2,1)

18: (1,2) 720: (4,2,1) | 240: (4,1,1)

MATHEMATICA

Table[Length[Select[Divisors[n!], UnsameQ@@Last/@FactorInteger[#]&]], {n, 0, 15}]

PROG

(PARI) a(n) = sumdiv(n!, d, my(ex=factor(d)[, 2]); #vecsort(ex, , 8) == #ex); \\ Michel Marcus, Jul 24 2020

CROSSREFS

Perfect-powers are A001597, with complement A007916.

Numbers with distinct prime multiplicities are A130091.

Divisors with distinct prime multiplicities are counted by A181796.

The maximum divisor with distinct prime multiplicities is A327498.

Divisors of n! with equal prime multiplicities are counted by A336415.

Cf. A000005, A098859, A118914, A124010, A327527, A336424, A336500, A336568.

Factorial numbers: A000142, A007489, A022559, A027423, A048656, A048742, A071626, A325272, A325273, A325617, A336416.

KEYWORD

nonn

AUTHOR

Gus Wiseman, Jul 22 2020

EXTENSIONS

a(21)-a(41) from Alois P. Heinz, Jul 24 2020

STATUS

approved

A336500

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

+10
24

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

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.

LINKS

Table of n, a(n) for n=1..87.

EXAMPLE

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

MATHEMATICA

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

CROSSREFS

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.

KEYWORD

nonn

AUTHOR

Gus Wiseman, Aug 06 2020

STATUS

approved

A336571

Number of sets of divisors d|n, 1 < d < n, all belonging to A130091 (numbers with distinct prime multiplicities) and forming a divisibility chain.

+10
22

1, 1, 1, 2, 1, 3, 1, 4, 2, 3, 1, 5, 1, 3, 3, 8, 1, 5, 1, 5, 3, 3, 1, 14, 2, 3, 4, 5, 1, 4, 1, 16, 3, 3, 3, 17, 1, 3, 3, 14, 1, 4, 1, 5, 5, 3, 1, 36, 2, 5, 3, 5, 1, 14, 3, 14, 3, 3, 1, 16, 1, 3, 5, 32, 3, 4, 1, 5, 3, 4, 1, 35, 1, 3, 5, 5, 3, 4, 1, 36, 8, 3, 1

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,4

COMMENTS

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.

LINKS

Table of n, a(n) for n=1..83.

EXAMPLE

The a(n) sets for n = 4, 6, 12, 16, 24, 84, 36:

{} {} {} {} {} {} {}

{2} {2} {2} {2} {2} {2} {2}

{3} {3} {4} {3} {3} {3}

{4} {8} {4} {4} {4}

{2,4} {2,4} {8} {7} {9}

{2,8} {12} {12} {12}

{4,8} {2,4} {28} {18}

{2,4,8} {2,8} {2,4} {2,4}

{4,8} {2,12} {3,9}

{2,12} {2,28} {2,12}

{3,12} {3,12} {2,18}

{4,12} {4,12} {3,12}

{2,4,8} {4,28} {3,18}

{2,4,12} {7,28} {4,12}

{2,4,12} {9,18}

{2,4,28} {2,4,12}

{3,9,18}

MATHEMATICA

strchns[n_]:=If[n==1, 1, Sum[strchns[d], {d, Select[Most[Divisors[n]], UnsameQ@@Last/@FactorInteger[#]&]}]];

Table[strchns[n], {n, 100}]

CROSSREFS

A336423 is the version for chains containing n.

A336570 is the maximal version.

A000005 counts divisors.

A001055 counts factorizations.

A007425 counts divisors of divisors.

A032741 counts proper divisors.

A045778 counts strict factorizations.

A071625 counts distinct prime multiplicities.

A074206 counts strict chains of divisors from n to 1.

A130091 lists numbers with distinct prime multiplicities.

A181796 counts divisors with distinct prime multiplicities.

A253249 counts chains of divisors.

A336422 counts divisible pairs of divisors, both in A130091.

A336424 counts factorizations using A130091.

A336500 counts divisors of n in A130091 with quotient also in A130091.

Cf. A001222, A002033, A005117, A098859, A118914, A124010, A294068, A305149, A327498, A327523, A336568, A336569.

KEYWORD

nonn

AUTHOR

Gus Wiseman, Jul 29 2020

STATUS

approved

A336423

Number of strict chains of divisors from n to 1 using terms of A130091 (numbers with distinct prime multiplicities).

+10
20

1, 1, 1, 2, 1, 0, 1, 4, 2, 0, 1, 5, 1, 0, 0, 8, 1, 5, 1, 5, 0, 0, 1, 14, 2, 0, 4, 5, 1, 0, 1, 16, 0, 0, 0, 0, 1, 0, 0, 14, 1, 0, 1, 5, 5, 0, 1, 36, 2, 5, 0, 5, 1, 14, 0, 14, 0, 0, 1, 0, 1, 0, 5, 32, 0, 0, 1, 5, 0, 0, 1, 35, 1, 0, 5, 5, 0, 0, 1, 36, 8, 0, 1, 0

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,4

COMMENTS

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.

LINKS

Table of n, a(n) for n=1..84.

EXAMPLE

The a(n) chains for n = 4, 8, 12, 16, 24, 32:

4/1 8/1 12/1 16/1 24/1 32/1

4/2/1 8/2/1 12/2/1 16/2/1 24/2/1 32/2/1

8/4/1 12/3/1 16/4/1 24/3/1 32/4/1

8/4/2/1 12/4/1 16/8/1 24/4/1 32/8/1

12/4/2/1 16/4/2/1 24/8/1 32/16/1

16/8/2/1 24/12/1 32/4/2/1

16/8/4/1 24/4/2/1 32/8/2/1

16/8/4/2/1 24/8/2/1 32/8/4/1

24/8/4/1 32/16/2/1

24/12/2/1 32/16/4/1

24/12/3/1 32/16/8/1

24/12/4/1 32/8/4/2/1

24/8/4/2/1 32/16/4/2/1

24/12/4/2/1 32/16/8/2/1

32/16/8/4/1

32/16/8/4/2/1

MATHEMATICA

strchns[n_]:=If[n==1, 1, If[!UnsameQ@@Last/@FactorInteger[n], 0, Sum[strchns[d], {d, Select[Most[Divisors[n]], UnsameQ@@Last/@FactorInteger[#]&]}]]];

Table[strchns[n], {n, 100}]

CROSSREFS

A336569 is the maximal case.

A336571 does not require n itself to have distinct prime multiplicities.

A000005 counts divisors.

A007425 counts divisors of divisors.

A074206 counts strict chains of divisors from n to 1.

A130091 lists numbers with distinct prime multiplicities.

A181796 counts divisors with distinct prime multiplicities.

A253249 counts nonempty strict chains of divisors.

A327498 gives the maximum divisor with distinct prime multiplicities.

A336422 counts divisible pairs of divisors, both in A130091.

A336424 counts factorizations using A130091.

A336500 counts divisors of n in A130091 with quotient also in A130091.

A337256 counts strict chains of divisors.

Cf. A001055, A002033, A005117, A032741, A067824, A071625, A118914, A124010, A167865, A336568, A336570, A336941.

KEYWORD

nonn

AUTHOR

Gus Wiseman, Jul 27 2020

STATUS

approved

A336420

Irregular triangle read by rows where T(n,k) is the number of divisors of the n-th superprimorial A006939(n) with distinct prime multiplicities and k prime factors counted with multiplicity.

+10
19

1, 1, 1, 1, 2, 1, 1, 1, 3, 2, 5, 2, 1, 1, 1, 4, 3, 11, 7, 7, 10, 5, 2, 1, 1, 1, 5, 4, 19, 14, 18, 37, 25, 23, 15, 23, 10, 5, 2, 1, 1, 1, 6, 5, 29, 23, 33, 87, 70, 78, 74, 129, 84, 81, 49, 39, 47, 23, 10, 5, 2, 1, 1, 1, 7, 6, 41, 34, 52, 165, 144, 183, 196, 424, 317, 376, 325, 299, 431, 304, 261, 172, 129, 81, 103, 47, 23, 10, 5, 2, 1, 1

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,5

COMMENTS

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 n-th superprimorial or Chernoff number is A006939(n) = Product_{i = 1..n} prime(i)^(n - i + 1).

T(n,k) is also the number of length-n vectors 0 <= v_i <= i summing to k whose nonzero values are all distinct.

LINKS

Table of n, a(n) for n=0..91.

EXAMPLE

Triangle begins:

1

1 1

1 2 1 1

1 3 2 5 2 1 1

1 4 3 11 7 7 10 5 2 1 1

1 5 4 19 14 18 37 25 23 15 23 10 5 2 1 1

The divisors counted in row n = 4 are:

1 2 4 8 16 48 144 432 2160 10800 75600

3 9 12 24 72 360 720 3024

5 25 18 40 80 400 1008

7 20 54 108 504 1200

27 56 112 540 2800

28 135 200 600

45 189 675 756

50 1350

63 1400

75 4725

175

MATHEMATICA

chern[n_]:=Product[Prime[i]^(n-i+1), {i, n}];

Table[Length[Select[Divisors[chern[n]], PrimeOmega[#]==k&&UnsameQ@@Last/@FactorInteger[#]&]], {n, 0, 5}, {k, 0, n*(n+1)/2}]

CROSSREFS

A000110 gives row sums.

A000124 gives row lengths.

A000142 counts divisors of superprimorials.

A006939 lists superprimorials or Chernoff numbers.

A008278 is the version counting only distinct prime factors.

A008302 counts divisors of superprimorials by bigomega.

A022915 counts permutations of prime indices of superprimorials.

A076954 can be used instead of A006939.

A130091 lists numbers with distinct prime multiplicities.

A146291 counts divisors by bigomega.

A181796 counts divisors with distinct prime multiplicities.

A181818 gives products of superprimorials.

A317829 counts factorizations of superprimorials.

A336417 counts perfect-power divisors of superprimorials.

A336498 counts divisors of factorials by bigomega.

A336499 uses factorials instead superprimorials.

Cf. A000005, A001222, A008278, A027423, A071625, A124010, A327498, A336419, A336421, A336426, A336500, A336568.

KEYWORD

nonn,tabf

AUTHOR

Gus Wiseman, Jul 25 2020

STATUS

approved

A336419

Number of divisors d of the n-th superprimorial A006939(n) with distinct prime exponents such that the quotient A006939(n)/d also has distinct prime exponents.

+10
15

1, 2, 4, 10, 24, 64, 184, 536, 1608, 5104, 16448, 55136, 187136, 658624, 2339648, 8618208, 31884640, 121733120, 468209408, 1849540416, 7342849216

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

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

The n-th superprimorial or Chernoff number is A006939(n) = Product_{i = 1..n} prime(i)^(n - i + 1).

LINKS

Table of n, a(n) for n=0..20.

EXAMPLE

The a(0) = 1 through a(3) = 10 divisors:

1 2 12 360

-----------------

1 1 1 1

2 3 5

4 8

12 9

18

20

40

45

72

360

MATHEMATICA

chern[n_]:=Product[Prime[i]^(n-i+1), {i, n}];

Table[Length[Select[Divisors[chern[n]], UnsameQ@@Last/@FactorInteger[#]&&UnsameQ@@Last/@FactorInteger[chern[n]/#]&]], {n, 0, 6}]

PROG

(PARI) recurse(n, k, b, d)={if(k>n, 1, sum(i=0, k, if((i==0||!bittest(b, i)) && (i==k||!bittest(d, k-i)), self()(n, k+1, bitor(b, 1<<i), bitor(d, 1<<(k-i))))))}

a(n)={recurse(n, 1, 1, 1)} \\ Andrew Howroyd, Aug 30 2020

CROSSREFS

A000110 shifted once to the left dominates this sequence.

A006939 lists superprimorials or Chernoff numbers.

A022915 counts permutations of prime indices of superprimorials.

A130091 lists numbers with distinct prime exponents.

A181796 counts divisors with distinct prime exponents.

A181818 gives products of superprimorials.

A317829 counts factorizations of superprimorials.

A336417 counts perfect-power divisors of superprimorials.

Cf. A000005, A000178, A008278, A071625, A076954, A118914, A124010, A327498, A327527, A336420, A336421, A336426, A336500, A336568.

KEYWORD

nonn,more

AUTHOR

Gus Wiseman, Jul 25 2020

EXTENSIONS

a(10)-a(20) from Andrew Howroyd, Aug 31 2020

STATUS

approved

A336426

Numbers that cannot be written as a product of superprimorials {2, 12, 360, 75600, ...}.

+10
14

3, 5, 6, 7, 9, 10, 11, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 25, 26, 27, 28, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

The n-th superprimorial is A006939(n) = Product_{i = 1..n} prime(i)^(n - i + 1).

LINKS

Table of n, a(n) for n=1..66.

EXAMPLE

We have 288 = 2*12*12 so 288 is not in the sequence.

MATHEMATICA

chern[n_]:=Product[Prime[i]^(n-i+1), {i, n}];

facsusing[s_, n_]:=If[n<=1, {{}}, Join@@Table[(Prepend[#, d]&)/@Select[facsusing[Select[s, Divisible[n/d, #]&], n/d], Min@@#>=d&], {d, Select[s, Divisible[n, #]&]}]];

Select[Range[100], facsusing[Array[chern, 30], #]=={}&]

CROSSREFS

A181818 is the complement.

A336497 is the version for superfactorials.

A001055 counts factorizations.

A006939 lists superprimorials or Chernoff numbers.

A022915 counts permutations of prime indices of superprimorials.

A317829 counts factorizations of superprimorials.

A336417 counts perfect-power divisors of superprimorials.

Cf. A000325, A005117, A076954, A124010, A294068, A336419, A336420, A336421, A336496, A336500, A336568.

KEYWORD

nonn

AUTHOR

Gus Wiseman, Jul 26 2020

STATUS

approved

A336422

Number of ways to choose a divisor of a divisor of n, both having distinct prime exponents.

+10
13

1, 3, 3, 6, 3, 5, 3, 10, 6, 5, 3, 13, 3, 5, 5, 15, 3, 13, 3, 13, 5, 5, 3, 24, 6, 5, 10, 13, 3, 7, 3, 21, 5, 5, 5, 21, 3, 5, 5, 24, 3, 7, 3, 13, 13, 5, 3, 38, 6, 13, 5, 13, 3, 24, 5, 24, 5, 5, 3, 20, 3, 5, 13, 28, 5, 7, 3, 13, 5, 7, 3, 42, 3, 5, 13, 13, 5, 7, 3

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

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

LINKS

Table of n, a(n) for n=1..79.

EXAMPLE

The a(n) ways for n = 1, 2, 4, 6, 8, 12, 30, 210:

1/1/1 2/1/1 4/1/1 6/1/1 8/1/1 12/1/1 30/1/1 210/1/1

2/2/1 4/2/1 6/2/1 8/2/1 12/2/1 30/2/1 210/2/1

2/2/2 4/2/2 6/2/2 8/2/2 12/2/2 30/2/2 210/2/2

4/4/1 6/3/1 8/4/1 12/3/1 30/3/1 210/3/1

4/4/2 6/3/3 8/4/2 12/3/3 30/3/3 210/3/3

4/4/4 8/4/4 12/4/1 30/5/1 210/5/1

8/8/1 12/4/2 30/5/5 210/5/5

8/8/2 12/4/4 210/7/1

8/8/4 12/12/1 210/7/7

8/8/8 12/12/2

12/12/3

12/12/4

12/12/12

MATHEMATICA

strdivs[n_]:=Select[Divisors[n], UnsameQ@@Last/@FactorInteger[#]&];

Table[Sum[Length[strdivs[d]], {d, strdivs[n]}], {n, 30}]

CROSSREFS

A336421 is the case of superprimorials.

A007425 counts divisors of divisors.

A130091 lists numbers with distinct prime exponents.

A181796 counts divisors with distinct prime exponents.

A327498 gives the maximum divisor with distinct prime exponents.

A336500 counts divisors with quotient also having distinct prime exponents.

A336568 = not a product of two numbers with distinct prime exponents.

Cf. A000005, A001055, A005117, A032741, A071625, A118914, A124010, A336419, A336420.

KEYWORD

nonn

AUTHOR

Gus Wiseman, Jul 26 2020

STATUS

approved

A336569

Number of maximal strict chains of divisors from n to 1 using elements of A130091 (numbers with distinct prime multiplicities).

+10
13

1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 2, 1, 0, 0, 1, 1, 2, 1, 2, 0, 0, 1, 3, 1, 0, 1, 2, 1, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0, 3, 1, 0, 1, 2, 2, 0, 1, 4, 1, 2, 0, 2, 1, 3, 0, 3, 0, 0, 1, 0, 1, 0, 2, 1, 0, 0, 1, 2, 0, 0, 1, 5, 1, 0, 2, 2, 0, 0, 1, 4, 1, 0, 1, 0, 0, 0, 0

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,12

COMMENTS

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.

LINKS

Table of n, a(n) for n=1..87.

EXAMPLE

The a(n) chains for n = 12, 72, 144, 192 (ones not shown):

12/3 72/18/2 144/72/18/2 192/96/48/24/12/3

12/4/2 72/18/9/3 144/72/18/9/3 192/64/32/16/8/4/2

72/24/12/3 144/48/24/12/3 192/96/32/16/8/4/2

72/24/8/4/2 144/72/24/12/3 192/96/48/16/8/4/2

72/24/12/4/2 144/48/16/8/4/2 192/96/48/24/8/4/2

144/48/24/8/4/2 192/96/48/24/12/4/2

144/72/24/8/4/2

144/48/24/12/4/2

144/72/24/12/4/2

MATHEMATICA

strsigQ[n_]:=UnsameQ@@Last/@FactorInteger[n];

fasmax[y_]:=Complement[y, Union@@(Most[Subsets[#]]&/@y)];

strchs[n_]:=If[n==1, {{}}, If[!strsigQ[n], {}, Join@@Table[Prepend[#, d]&/@strchs[d], {d, Select[Most[Divisors[n]], strsigQ]}]]];

Table[Length[fasmax[strchs[n]]], {n, 100}]

CROSSREFS

A336423 is the non-maximal version.

A336570 is the version for chains not necessarily containing n.

A000005 counts divisors.

A001055 counts factorizations.

A001222 counts prime factors with multiplicity.

A007425 counts divisors of divisors.

A032741 counts proper divisors.

A045778 counts strict factorizations.

A071625 counts distinct prime multiplicities.

A074206 counts strict chains of divisors from n to 1.

A130091 lists numbers with distinct prime multiplicities.

A181796 counts divisors with distinct prime multiplicities.

A253249 counts chains of divisors.

A336422 counts divisible pairs of divisors, both in A130091.

A336424 counts factorizations using A130091.

A336571 counts divisor sets of elements of A130091.

Cf. A002033, A005117, A098859, A118914, A124010, A305149, A327498, A327523, A336414, A336425, A336500, A336568.

KEYWORD

nonn

AUTHOR

Gus Wiseman, Jul 29 2020

STATUS

approved