login

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).

A354181
Numbers whose number of divisors is not a 3-smooth number.
3
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).
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
KEYWORD
nonn
AUTHOR
Amiram Eldar, May 18 2022
STATUS
approved