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A336867
Numbers k such that k! does not have distinct prime multiplicities.
5
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
1,1
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)
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)
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)
MATHEMATICA
Select[Range[0, 100], !UnsameQ@@Last/@FactorInteger[#!]&]
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.
Sequence in context: A080262 A025050 A196115 * A025051 A020884 A183855
KEYWORD
nonn
AUTHOR
Gus Wiseman, Aug 07 2020
STATUS
approved