cp's OEIS Frontend

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

%I A360793 #40 Feb 26 2025 01:55:24
%S A360793 24,40,54,56,88,104,120,135,136,152,168,184,189,232,248,250,264,270,
%T A360793 280,296,297,312,328,344,351,375,376,378,408,424,440,456,459,472,488,
%U A360793 513,520,536,552,568,584,594,616,621,632,664,680,686,696,702,712,728,744,750
%N A360793 Numbers of the form m*p^3, where m > 1 is squarefree and prime p does not divide m.
%C A360793 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.
%C A360793 Subsequence of A126706. - _Michael De Vlieger_, Feb 22 2023
%C A360793 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
%H A360793 Amiram Eldar, <a href="/A360793/b360793.txt">Table of n, a(n) for n = 1..10000</a>
%F A360793 For a(n) = m*p^3, A000005(a(n)) = 2^k, where k = 2 + A001221(m).
%F A360793 Equals A048109 \ A030078.
%e A360793 1608 = 2^3*201 is in this sequence (p = 2; m = 201 is odd and squarefree).
%e A360793 A001221(201) = 2, therefore 1608 has 2^(2+2) = 16 divisors.
%p A360793 filter:= proc(n) local F;
%p A360793   F:= sort(ifactors(n)[2][..,2]);
%p A360793   nops(F) >= 2 and F[-1] = 3 and F[-2] = 1
%p A360793 end proc:
%p A360793 select(filter, [$1..1000]); # _Robert Israel_, Mar 01 2023
%t A360793 Select[Range[1000], (e = Sort[FactorInteger[#][[;; , 2]]])[[-1]] == 3 && Length[e] > 1 && e[[-2]] == 1 &] (* _Amiram Eldar_, Feb 21 2023 *)
%o A360793 (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
%o A360793 (Python)
%o A360793 from itertools import count, islice
%o A360793 from sympy import factorint
%o A360793 def A360793_gen(startvalue=1): # generator of terms >= startvalue
%o A360793     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)))
%o A360793 A360793_list = list(islice(A360793_gen(),20)) # _Chai Wah Wu_, Feb 28 2023
%o A360793 (Python)
%o A360793 from math import isqrt
%o A360793 from sympy import mobius, primepi, integer_nthroot, primerange
%o A360793 def A360793(n):
%o A360793     def bisection(f,kmin=0,kmax=1):
%o A360793         while f(kmax) > kmax: kmax <<= 1
%o A360793         kmin = kmax >> 1
%o A360793         while kmax-kmin > 1:
%o A360793             kmid = kmax+kmin>>1
%o A360793             if f(kmid) <= kmid:
%o A360793                 kmax = kmid
%o A360793             else:
%o A360793                 kmin = kmid
%o A360793         return kmax
%o A360793     def g(x): return sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1))
%o A360793     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)))
%o A360793     return bisection(f,n,n) # _Chai Wah Wu_, Feb 25 2025
%Y A360793 Cf. A000005, A001221, A001222, A013661, A030078, A036537, A048109, A072357, A126706.
%Y A360793 Subsequence of A190641.
%K A360793 nonn
%O A360793 1,1
%A A360793 _David James Sycamore_, Feb 21 2023