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

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

Author

Gus Wiseman, Aug 07 2020

Keywords

• nonn

Comments

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)

Examples

			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)
		

Links

• 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).

Crossrefs

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.

Programs

Mathematica

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

Formula

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)