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A336867
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Numbers k such that k! does not have distinct prime multiplicities.
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5
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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
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1,1
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COMMENTS
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The complement appears to be {0, 1, 2, 4, 6, 10}.
A number has distinct prime multiplicities iff its prime signature is strict.
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)
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LINKS
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FORMULA
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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)
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EXAMPLE
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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)
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MATHEMATICA
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Select[Range[0, 100], !UnsameQ@@Last/@FactorInteger[#!]&]
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CROSSREFS
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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.
Factorial numbers: A000142, A007489, A022559, A027423, A048656, A048742, A071626, A325272, A325273, A325617, A336416, A336869.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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