A337071 -id:A337071 - cp's OEIS frontend

A337075 Number of strict chains of divisors in A130091 (numbers with distinct prime multiplicities) starting with a proper divisor of n! and ending with 1.

1, 1, 1, 3, 14, 48, 384, 1308, 40288, 933848, 21077680, 75690016, 5471262080, 7964665440, 54595767744, 17948164982144, 3454946386353664, 5010658671663616, 723456523262697984, 950502767770273280, 165679731871366906880, 8443707247468681128448

Offset: 0

Gus Wiseman, Aug 17 2020

nonn

			The a(1) = 1 through a(4) = 14 chains (with n! prepended):
  1  2/1  6/1    24/1
          6/2/1  24/2/1
          6/3/1  24/3/1
                 24/4/1
                 24/8/1
                 24/12/1
                 24/4/2/1
                 24/8/2/1
                 24/8/4/1
                 24/12/2/1
                 24/12/3/1
                 24/12/4/1
                 24/8/4/2/1
                 24/12/4/2/1

A336571 is the generalization to not just factorial numbers.
A337104 is the version for chains containing n!.
A000005 counts divisors.
A001055 counts factorizations.
A032741 counts proper divisors.
A071625 counts distinct prime multiplicities.
A074206 counts 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.
A327498 gives the maximum divisor with distinct prime multiplicities.
A336414 counts divisors of n! with distinct prime multiplicities.
A336423 counts chains using A130091, with maximal case A336569.
A336424 counts factorizations using A130091.
A336425 counts divisible pairs of divisors of n!, both in A130091.
Cf. A002033, A098859, A124010, A294068, A327499, A327500, A327523, A337071.

chnstr[n_]:=If[n==1,1,Sum[chnstr[d],{d,Select[Most[Divisors[n]],UnsameQ@@Last/@FactorInteger[#]&]}]];
Table[chnstr[n!],{n,0,5}]

a(n) = A337104(n) whenever A337104(n) != 0.

a(n) = A336571(n!).

A337106 Number of nontrivial divisors of n!.

Original entry on oeis.org

0, 0, 0, 2, 6, 14, 28, 58, 94, 158, 268, 538, 790, 1582, 2590, 4030, 5374, 10750, 14686, 29374, 41038, 60798, 95998, 191998, 242878, 340030, 532222, 677374, 917278, 1834558, 2332798, 4665598, 5529598, 7864318, 12165118, 16422910, 19595518, 39191038, 60466174

Offset: 0

Gus Wiseman, Aug 23 2020

nonn

A divisor of n is trivial if it is 1 or n.

			The a(3) = 2 through a(5) =14 nontrivial divisions:
  6/2  24/2   120/2
  6/3  24/3   120/3
       24/4   120/4
       24/6   120/5
       24/8   120/6
       24/12  120/8
              120/10
              120/12
              120/15
              120/20
              120/24
              120/30
              120/40
              120/60

A070824 counts nontrivial divisors.
A153823 counts proper divisors of n!.
A337107 has this sequence as column k = 3.
A000005 counts divisors.
A000142 lists factorial numbers.
A001055 counts factorizations.
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.
A253249 counts chains of divisors.
A337071 counts chains of divisors starting with n!.
A337105 counts chains of divisors from n! to 1.
Cf. A022559, A124010, A251683, A325617, A336941.

Table[Length[DeleteCases[Divisors[n!],1|n!]],{n,10}]

from sympy import factorial, divisor_count
def A337106(n):
    return 0 if n <= 1 else divisor_count(factorial(n))-2 # Chai Wah Wu, Aug 24 2020

a(n) = A000005(n!) - 2 for n > 1.

a(n) = A070824(n!).

a(0) from Chai Wah Wu, Aug 24 2020

cp's OEIS Frontend

A337075 Number of strict chains of divisors in A130091 (numbers with distinct prime multiplicities) starting with a proper divisor of n! and ending with 1.

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