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

1,1

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).

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.

Example

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

Mathematica

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

Prog

(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

Crossrefs

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

Sequence in context: A260985 A322448 A374588 * A264164 A045047 A239344

Adjacent sequences: A354178 A354179 A354180 * A354182 A354183 A354184

Keyword

nonn

Author

Amiram Eldar, May 18 2022

Status

approved