cp's OEIS Frontend

A336869 Number of divisors d of n! with distinct prime multiplicities such that the quotient n!/d also has distinct prime multiplicities.

%I A336869 #18 Jan 19 2021 11:08:29
%S A336869 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,
%T A336869 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,
%U A336869 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0
%N A336869 Number of divisors d of n! with distinct prime multiplicities such that the quotient n!/d also has distinct prime multiplicities.
%C A336869 Does this sequence converge to zero?
%C A336869 A number has distinct prime multiplicities iff its prime signature is strict.
%C A336869 From _Edward Moody_, Jan 18 2021: (Start)
%C A336869 a(n) = 0 for n >= 17.
%C A336869 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)
%e A336869 The a(1) = 1 through a(7) = 8 divisors:
%e A336869   1  1  2  1   3   1    5
%e A336869      2  3  2   5   2    7
%e A336869            3   24  5    45
%e A336869            8   40  9    63
%e A336869            12      16   80
%e A336869            24      18   112
%e A336869                    40   720
%e A336869                    45   1008
%e A336869                    80
%e A336869                    144
%e A336869                    360
%e A336869                    720
%t A336869 Table[Length[Select[Divisors[n!],UnsameQ@@Last/@FactorInteger[#]&&UnsameQ@@Last/@FactorInteger[n!/#]&]],{n,0,10}]
%Y A336869 A336419 is the version for superprimorials.
%Y A336869 A336500 is the generalization to non-factorials.
%Y A336869 A336616 is the maximum among these divisors.
%Y A336869 A336617 is the minimum among these divisors.
%Y A336869 A336939 has these row sums.
%Y A336869 A000005 counts divisors.
%Y A336869 A130091 lists numbers with distinct prime multiplicities.
%Y A336869 A181796 counts divisors with distinct prime multiplicities.
%Y A336869 A327498 gives the maximum divisor of n with distinct prime multiplicities.
%Y A336869 A336414 counts divisors of n! with distinct prime multiplicities.
%Y A336869 A336415 counts divisors of n! with equal prime multiplicities.
%Y A336869 A336423 counts chains using A130091.
%Y A336869 Cf. A098859, A118914, A124010, A327527, A336424, A336568, A336571, A336870.
%Y A336869 Factorial numbers: A000142, A007489, A022559, A027423, A048656, A071626, A325272, A325273, A325617, A336416.
%K A336869 nonn
%O A336869 0,3
%A A336869 _Gus Wiseman_, Aug 08 2020
%E A336869 a(31)-a(80) from _Edward Moody_, Jan 19 2021