A360793 Numbers of the form m*p^3, where m > 1 is squarefree and prime p does not divide m.

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

Amiram Eldar, Table of n, a(n) for n = 1..10000

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

Cf. A000005, A001221, A001222, A013661, A030078, A036537, A048109, A072357, A126706.

Subsequence of A190641.

Sequence in context: A048104 A334801 A362594 * A375076 A297401 A065127

Adjacent sequences: A360790 A360791 A360792 * A360794 A360795 A360796

Keyword

nonn

Author

David James Sycamore, Feb 21 2023

Status

approved