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Numbers whose number of divisors is not a 3-smooth number.
3

%I #12 May 21 2022 14:52:11

%S 16,48,64,80,81,112,144,162,176,192,208,240,272,304,320,324,336,368,

%T 400,405,432,448,464,496,512,528,560,567,576,592,624,625,648,656,688,

%U 704,720,729,752,784,810,816,832,848,880,891,912,944,960,976,1008,1024,1040

%N Numbers whose number of divisors is not a 3-smooth number.

%C Number whose prime factorization includes an exponent e such that e+1 is in A059485.

%C The asymptotic density of this sequence is 1 - Product_{p prime} ((1 - 1/p) * Sum_{k in A003586} 1/p^(k-1)) = 0.0512963858... (Hilberdink, 2022).

%H Amiram Eldar, <a href="/A354181/b354181.txt">Table of n, a(n) for n = 1..10000</a>

%H Titus Hilberdink, <a href="https://doi.org/10.1016/j.jnt.2021.07.020">How often is d(n) a power of a given integer?</a>, Journal of Number Theory, Vol. 236 (2022), pp. 261-279.

%H <a href="/index/Eu#epf">Index entries for sequences computed from exponents in factorization of n</a>.

%e 16 is a term since A000005(16) = 5 is not a 3-smooth number.

%t smoothQ[n_] := n == 2^IntegerExponent[n, 2] * 3^IntegerExponent[n, 3]; Select[Range[1000], !smoothQ[DivisorSigma[0, #]] &]

%o (PARI) is(n) = n>>=valuation(n, 2); n/=3^valuation(n, 3); n>1; \\ A059485

%o isok(m) = is(numdiv(m)); \\ _Michel Marcus_, May 19 2022

%Y Cf. A000005, A003586, A059485, A036537, A162643.

%K nonn

%O 1,1

%A _Amiram Eldar_, May 18 2022