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%I #18 Jul 01 2021 03:41:52
%S 3,5,7,8,9,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,
%T 30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,
%U 53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71
%N Numbers k such that k! does not have distinct prime multiplicities.
%C The complement appears to be {0, 1, 2, 4, 6, 10}.
%C A number has distinct prime multiplicities iff its prime signature is strict.
%C From _Chai Wah Wu_, Aug 11 2020: (Start)
%C Theorem: the sequence consists of all nonnegative integers except 0, 1, 2, 4, 6, 10.
%C Proof: The cases k <= 31 follow from inspection. We show the case where k > 31.
%C 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.
%C 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.
%C 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
%C (End)
%H J. Nagura, <a href="http://dx.doi.org/10.3792/pja/1195570997">On the interval containing at least one prime number</a>, Proc. Japan Acad., 28 (1952) 177-181.
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (2,-1).
%F From _Chai Wah Wu_, Aug 11 2020: (Start)
%F a(n) = 2*a(n-1) - a(n-2) for n > 7.
%F G.f.: x*(-x^6 + x^5 - x^3 - x + 3)/(x - 1)^2. (End)
%e The sequence of indexed factorials a(n)! together with their prime signatures begins:
%e 6: (1,1)
%e 120: (3,1,1)
%e 5040: (4,2,1,1)
%e 40320: (7,2,1,1)
%e 362880: (7,4,1,1)
%e 39916800: (8,4,2,1,1)
%e 479001600: (10,5,2,1,1)
%e 6227020800: (10,5,2,1,1,1)
%e 87178291200: (11,5,2,2,1,1)
%e 1307674368000: (11,6,3,2,1,1)
%e 20922789888000: (15,6,3,2,1,1)
%e 355687428096000: (15,6,3,2,1,1,1)
%e 6402373705728000: (16,8,3,2,1,1,1)
%e 121645100408832000: (16,8,3,2,1,1,1,1)
%e 2432902008176640000: (18,8,4,2,1,1,1,1)
%t Select[Range[0,100],!UnsameQ@@Last/@FactorInteger[#!]&]
%Y A130092 is the generalization to non-factorials.
%Y A130091 lists numbers with distinct prime multiplicities.
%Y A181796 counts divisors with distinct prime multiplicities.
%Y A327498 gives the maximum divisor of n with distinct prime multiplicities.
%Y A336414 counts divisors of n! with distinct prime multiplicities.
%Y A336415 counts divisors of n! with equal prime multiplicities.
%Y A336866 counts partitions without distinct multiplicities.
%Y Cf. A098859, A118914, A124010, A336423, A336424, A336500, A336568, A336571.
%Y Factorial numbers: A000142, A007489, A022559, A027423, A048656, A048742, A071626, A325272, A325273, A325617, A336416, A336869.
%K nonn
%O 1,1
%A _Gus Wiseman_, Aug 07 2020
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