A336867 - OEIS

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

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OFFSET

1,1

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)

LINKS

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

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).

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)

EXAMPLE

The sequence of indexed factorials a(n)! together with their prime signatures begins:

6: (1,1)