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

Author

David James Sycamore, Feb 21 2023

Keywords

• nonn

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

Examples

			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.
		

Links

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

Crossrefs

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

Subsequence of A190641.

Programs

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 *)

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

Python

from math import isqrt
from sympy import mobius, primepi, integer_nthroot, primerange
def A360793(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        kmin = kmax >> 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def g(x): return sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1))
    def f(x): return int(n+x+primepi(integer_nthroot(x,3)[0])+sum(sum(-g(x//p**j) if j&1 else g(x//p**j) for j in range(3,x.bit_length())) for p in primerange(isqrt(x)+1)))
    return bisection(f,n,n) # Chai Wah Wu, Feb 25 2025

Formula

For a(n) = m*p^3, A000005(a(n)) = 2^k, where k = 2 + A001221(m).

Equals A048109 \ A030078.