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A371185
Indices of the cubefull numbers in the sequence of powerful numbers.
2
1, 3, 5, 7, 8, 11, 13, 17, 18, 23, 25, 26, 30, 34, 38, 41, 42, 45, 49, 54, 55, 61, 63, 72, 77, 78, 80, 82, 83, 87, 89, 93, 99, 105, 106, 113, 115, 116, 127, 128, 130, 137, 140, 148, 151, 153, 161, 164, 166, 179, 185, 186, 188, 192, 196, 201, 206, 221, 227, 234
OFFSET
1,2
FORMULA
A001694(a(n)) = A036966(n).
a(n) ~ c * n^(3/2), where c = A090699 / A362974^(3/2) = 0.216089803749...
EXAMPLE
The first 5 powerful numbers are 1, 4, 8, 9, and 16. The 1st, 3rd, and 5th, 1, 8, and 16, are also cubefull numbers. Therefore, the first 3 terms of this sequence are 1, 3, and 5.
MATHEMATICA
powQ[n_, emin_] := n == 1 || AllTrue[FactorInteger[n], Last[#] >= emin &]; Position[Select[Range[20000], powQ[#, 2] &], _?(powQ[#1, 3] &), Heads -> False] // Flatten
PROG
(PARI) ispow(n, emin) = n == 1 || vecmin(factor(n)[, 2]) >= emin;
lista(kmax) = {my(f, c = 0); for(k = 1, kmax, if(ispow(k, 2), c++; if(ispow(k, 3), print1(c, ", ")))); }
(Python)
from math import isqrt, gcd
from sympy import mobius, integer_nthroot, factorint
def A371185(n):
def squarefreepi(n): return int(sum(mobius(k)*(n//k**2) for k in range(1, isqrt(n)+1)))
def bisection(f, kmin=0, kmax=1):
while f(kmax) > kmax: kmax <<= 1
while kmax-kmin > 1:
kmid = kmax+kmin>>1
if f(kmid) <= kmid:
kmax = kmid
else:
kmin = kmid
return kmax
def f(x):
c = n+x
for w in range(1, integer_nthroot(x, 5)[0]+1):
if all(d<=1 for d in factorint(w).values()):
for y in range(1, integer_nthroot(z:=x//w**5, 4)[0]+1):
if gcd(w, y)==1 and all(d<=1 for d in factorint(y).values()):
c -= integer_nthroot(z//y**4, 3)[0]
return c
c, l, m = 0, 0, bisection(f, n, n)
j = isqrt(m)
while j>1:
k2 = integer_nthroot(m//j**2, 3)[0]+1
w = squarefreepi(k2-1)
c += j*(w-l)
l, j = w, isqrt(m//k2**3)
c += squarefreepi(integer_nthroot(m, 3)[0])-l
return c # Chai Wah Wu, Sep 12 2024
CROSSREFS
Similar sequences: A361936, A371186.
Sequence in context: A047485 A024969 A296233 * A276112 A228075 A309405
KEYWORD
nonn,easy
AUTHOR
Amiram Eldar, Mar 14 2024
STATUS
approved