login
A360793
Numbers of the form m*p^3, where m > 1 is squarefree and prime p does not divide m.
3
24, 40, 54, 56, 88, 104, 120, 135, 136, 152, 168, 184, 189, 232, 248, 250, 264, 270, 280, 296, 297, 312, 328, 344, 351, 375, 376, 378, 408, 424, 440, 456, 459, 472, 488, 513, 520, 536, 552, 568, 584, 594, 616, 621, 632, 664, 680, 686, 696, 702, 712, 728, 744, 750
OFFSET
1,1
COMMENTS
Can be regarded as the cube version of A072357. Subsequence of A036537 (the number of divisors of any term is a power of 2). Also a subsequence of A048109.
Subsequence of A126706. - Michael De Vlieger, Feb 22 2023
The asymptotic density of this sequence is Sum_{p prime} d(p) = 0.074177741367259601921..., where d(p) = 1/(p^2*(p+1)*zeta(2)) is the density of the subset of terms that are divisible by p^3. - Amiram Eldar, Aug 01 2024
LINKS
FORMULA
For a(n) = m*p^3, A000005(a(n)) = 2^k, where k = 2 + A001221(m).
Equals A048109 \ A030078.
EXAMPLE
1608 = 2^3*201 is in this sequence (p = 2; m = 201 is odd and squarefree).
A001221(201) = 2, therefore 1608 has 2^(2+2) = 16 divisors.
MAPLE
filter:= proc(n) local F;
F:= sort(ifactors(n)[2][.., 2]);
nops(F) >= 2 and F[-1] = 3 and F[-2] = 1
end proc:
select(filter, [$1..1000]); # Robert Israel, Mar 01 2023
MATHEMATICA
Select[Range[1000], (e = Sort[FactorInteger[#][[;; , 2]]])[[-1]] == 3 && Length[e] > 1 && e[[-2]] == 1 &] (* Amiram Eldar, Feb 21 2023 *)
PROG
(PARI) isok(k) = if (k>1, my(f=factor(k), v=f[, 2], m); if (vecmax(v)==3, w=select(x->(x==3), v, 1); if (#w == 1, m = k/f[w[1], 1]^3; (m>1) && issquarefree(m)))); \\ Michel Marcus, Feb 21 2023
(Python)
from itertools import count, islice
from sympy import factorint
def A360793_gen(startvalue=1): # generator of terms >= startvalue
return filter(lambda n:len(f:=sorted(factorint(n).values(), reverse=True))>1 and f[0]==3 and f[1] == 1, count(max(startvalue, 1)))
A360793_list = list(islice(A360793_gen(), 20)) # Chai Wah Wu, Feb 28 2023
CROSSREFS
KEYWORD
nonn
AUTHOR
STATUS
approved