A354181 Numbers whose number of divisors is not a 3-smooth number.

16, 48, 64, 80, 81, 112, 144, 162, 176, 192, 208, 240, 272, 304, 320, 324, 336, 368, 400, 405, 432, 448, 464, 496, 512, 528, 560, 567, 576, 592, 624, 625, 648, 656, 688, 704, 720, 729, 752, 784, 810, 816, 832, 848, 880, 891, 912, 944, 960, 976, 1008, 1024, 1040

Offset: 1

Author

Amiram Eldar, May 18 2022

Keywords

• nonn

Comments

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

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).

Examples

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

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• Titus Hilberdink, How often is d(n) a power of a given integer?, Journal of Number Theory, Vol. 236 (2022), pp. 261-279.
• Index entries for sequences computed from exponents in factorization of n.

Crossrefs

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

Programs

Mathematica

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

PARI

is(n) = n>>=valuation(n, 2); n/=3^valuation(n, 3); n>1; \\ A059485
isok(m) = is(numdiv(m)); \\ Michel Marcus, May 19 2022